Victoria University of WellingtonSchool of Mathematics and Statistics

Te Kura Mātai Tatauranga

MATH 321/2/3 Applied Mathematics 2018

Introduction to Fluid Mechanics and the Mathematicsof Tsunamis

Contents

I Introduction to the mathematical modelling of water waves 5

1 Introduction 5

1.1 Introduction to waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 What is a fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Systems of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Dimensional Analysis and Scaling 12

2.1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Introduction to Conservation Laws 23

3.1 Classification of 1st order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Traveling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 The characteristic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Nonlinear waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 Determining the breaking time . . . . . . . . . . . . . . . . . . . . . 32

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3.4.2 Solution after the breaking time . . . . . . . . . . . . . . . . . . . . . 34

3.5 Rarefaction waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7 Quasi-linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Inviscid Fluid Flow 43

4.1 Introduction to Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Derivation of the Euler equations . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.3 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.4 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . 51

4.3 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Non-dimensionalization and normalization of the Euler equations . . . . . . 54

4.5 Non-dimensionalization and normalization of the Navier-Stokes equations . . 56

4.6 Potential flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6.1 The streamline function . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.7 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 The Shallow Water Equations: A model for tsunamis 61

5.1 Physical derivation of the shallow water equations . . . . . . . . . . . . . . . 62

5.1.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1.2 Momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Linearization of the shallow water equations . . . . . . . . . . . . . . . . . . 65

5.3 Solution of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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5.4 Systems of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.6 The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.7 The dam-break problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.8 Propagation of bores in shallow water . . . . . . . . . . . . . . . . . . . . . . 83

6 Nonlinear and dispersive waves 89

6.1 Dispersive waves and their characteristics . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Are water waves really dispersive? . . . . . . . . . . . . . . . . . . . 90

6.1.2 Effects of nonlinearity and dispersion . . . . . . . . . . . . . . . . . . 92

6.2 Derivation of nonlinear and dispersive wave equations . . . . . . . . . . . . . 94

6.2.1 Derivation of Peregrine’s system with time dependent bottom . . . . 94

6.3 Derivation of other model equations . . . . . . . . . . . . . . . . . . . . . . . 97

6.3.1 The Serre equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.2 The ‘classical’ Boussinesq system . . . . . . . . . . . . . . . . . . . . 100

6.3.3 Unidirectional model equations: The KdV and the BBM equations . 101

6.3.4 The Shallow Water equations . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Traveling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4.1 Solitary waves of the KdV-BBM equation . . . . . . . . . . . . . . . 103

6.4.2 Solitary waves of the Serre equations . . . . . . . . . . . . . . . . . . 105

6.4.3 Cnoidal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

II Introduction to numerical methods 111

7 Finite-difference methods for hyperbolic conservation laws 111

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7.1 Introduction to finite-difference methods . . . . . . . . . . . . . . . . . . . . 111

7.2 Finite differences approximations to derivatives . . . . . . . . . . . . . . . . 116

7.3 The upwind scheme of O(∆x,∆t) . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3.1 Matlab implementation . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3.2 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.3.3 Convergence and error estimation . . . . . . . . . . . . . . . . . . . . 122

7.3.4 Von Neumann stability analysis . . . . . . . . . . . . . . . . . . . . . 125

7.4 An unstable scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.5 The Lax-Friedrichs scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.6 The Lax–Wendroff scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.7 The Leapfrog scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.8 Norms and the Lax equivalence theorem . . . . . . . . . . . . . . . . . . . . 130

7.9 Finite-difference methods for the two-point boundary-value problem . . . . . 132

7.10 Finite-difference methods for the wave equation . . . . . . . . . . . . . . . . 132

8 Finite-difference methods for nonlinear and dispersive wave equations 132

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Part I

Introduction to the mathematicalmodelling of water waves

1 Introductionsec:introduction

We all have seen water waves propagating on the surface of the water in the sea or in a lake.We have experimented by throwing small rocks in the water and observed the generation ofexpanding waves as they fade away from the point where the rock fell. Some other timeswe observed waves rolling in and break near the beach or waves generated by the motionof a boat or a duck. All these examples of water waves have a different cause, they weregenerated by a different source and their nature and properties are different in each case.In these notes we will try to present the mathematical and numerical models that describewater waves and understand their basic properties.

1.1 Introduction to waves

Let us first discuss some examples of water waves. In general gravitational forces can generatewaves. For example the interaction of the mass of the moon with the oceans is the causefor the generation of tidal waves. Other disturbances such as meteor impacts in the oceans,or underwater earthquakes or landslides can cause the generation of tsunamis and seiches.Another great cause of generation of waves in the sea for example is the wind. The windusually can create from small ripples on the surface of the sea up to large amplitude wavesin the ocean during a storm. The amplitude of a wave is usually referred to the height of thewave from the still water level. Waves are generated when energy is added into the mediumand propagate until their energy is dissipated. What is very interesting it is that wavestransport energy without really transporting matter.

Some water waves are characterised by the depth of the water in relation with their ampli-tude. For example, when waves occur in water of great depth they are called deep-waterwaves. In this case the bathymetry does not affect the speed of the wave drastically butit can affect the direction of the propagation. On the other hand, the behaviour of wavesin shallow waters is different. Not only the friction of the bottom could affect the motionof the wave but also the topography of the bottom is very important. In this last case thewaves are sometimes characterised as shallow-water waves.

As we mentioned before, some important examples of waves are the tides, the tsunamis

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and the wind waves. Tides are usually characterised by huge wavelength and they usuallyappeared as an enormous periodic wave. For example in the open ocean the wavelength (i.e.the distance between two crests or two throughs) can be λ ≈ 15, 000 km .

Another huge and devastating wave is the tsunami wave. The word tsunami is a Japaneseterm derived from the characters “tsu” meaning harbour and “nami” meaning wave. It isnow very common to describe a tsunami wave by a series of traveling waves in water gen-erated by the displacement of the sea floor associated with submarine earthquakes, volcaniceruptions, or landslides. Tsunamis have big wavelength such as λ ≈ 400 km in the deepocean. Tsunamis compared to tides are characterised by their small amplitude in the deepocean. Because tsunamis’ amplitude in the deep ocean is very small we have never beenable to detect them and see them. In order to study them we use specialised buoys thatmeasure the pressure at the bottom of the ocean. On the other hand, as tsunamis approachshallow waters gain amplitude and decrease their wavelength. Their speed of propagation isalso remarkable. In the ocean of an average depth of about 4000 m the speed of a tsunamiis approximately 200 m/sec which is almost 720 km/h (or 450 miles/h), which is similar tothe speed of a jet. When tsunamis approach the shore then they slow down and break.

Wind waves have small wavelength varying in height from 0.1 m up to 100 m. Contrary totsunamis, wind waves have large height in the deep ocean. For example they can be from1 m to 30 m or so tall.

Because all these kinds of waves have some different characteristics we categorise themanalogously. We proceed with a more detailed description of waves. Although we mighthave the impression that water waves transfer matter, this is not usually true. A water waveis a disturbance in the water that transfers energy from one place to another with no or littlemass transport. Usually waves involve a periodic and repetitive motion. To describe waveswe usually use some basic characteristic values. These basic characteristics can be:

• wavelength

• amplitude

• frequency

• period

We start with the wavelength which is usually denoted by λ and is the distance between twosuccessive crests or troughs of the wave. More precisely, wavelength is the fundamental lengthscale over which a wave repeats itself. Therefore, along a wave this might not be constant.Another characteristic of a wave is its amplitude. Amplitude is the maximum displacementof the medium from its rest position and sometimes is the same with the maximum height ofthe wave. Frequency of a wave f is the number of repetitions per second in Hz, while periodis the time for one wavelength to pass a certain point and is usually denoted by T = f−1.

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Other characteristics of waves is the wave number k, which is the number of oscillations in2π units of space at a fixed time. Using this definition it is obvious that the wavelength canbe defined as 2π/k. Finally, we mention the the angular frequency ω, which is the numberof oscillations in 2π units of time observed at a fixed location x.

Figure 1: Some characteristics of a plane wave. fig:planewav

All the wave characteristics described above are sometimes related (especially for simple,linear waves), so the wavelength is λ = 2π/k and the period is T = 2π/ω. The plane wavedepicted in Figure

fig:planewav1 can be described by a simple formula of the form u(x, t) = A cos(kx−ωt)

or in a more descriptive form:

u(x, t) = A cos[k(x− ω

kt)]

. (1.1)

From the last formula we observe that this cosine wave will propagate with phase velocityc = ω/k since it will cover distance equal to ω/k · t units in time t1. When the phase velocityof a wave-train depends on the wave number k the wave is called dispersive since wavesof different wavelength (wavenumber) propagate with different speed, causing the wave todisperse. If ω/k is independent of k then the waves are called non-dispersive or hyperbolic.Since the angular frequency might depend on k sometimes we will write ω = ω(k).

Dispersive waves and dispersion can be both linear and nonlinear and occur very often in thesea or in rivers and lakes. We will discuss the linear and nonlinear aspects of the propagationof water waves later. Sometimes the dispersive effects are crucial in the propagation of somewaves but there are many cases where the dispersion is not important. For this reason wewill study both dispersive and non-dispersive waves. Of course not all waves have the sameform and therefore we cannot represent every wave by a cosine function, but the definitionsof the basic characteristics of the waves will be the same in all cases. Since our focus is on

1f(x− s) is a horizontal translation to the right of the function f(x) by s units

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water waves, and water is a fluid we proceed with a somewhat more formal description offluids.

1.2 What is a fluid

Here we consider the fluid to be a homogeneous medium and we ignore its molecular structureassuming that the matter is continuous and there are no gaps or empty spaces between thefluids particles. In other words we assume that the molecules of a fluid are so close togetherthat we cannot distinguish them. This fundamental assumption is crucial in fluid mechanicssince we can describe the equations of motion using calculus. Because of this approximationwe will restrict ourselves in the part of fluid mechanics known to as continuum mechanicswhere a fluid is treated as a continuum. Of course there are cases such as rarefied gases wherethe air molecules can be found in large concentrations among the gas molecules and then thecontinuum assumption is no longer acceptable. Although we ignore the molecular structureof the fluid we still consider that fluids can be easily deformed (like gases or liquids).

One of the basic differences between fluids and solids is that fluids are continuously de-formable (in other words they flow) under the act of a shearing stress. A shear stress (forceper unit area) is created whenever a tangential force acts on the surface of the fluid. Solidscan be initially deformed by a shearing stress but they will not continuously deform. Al-though some fluids can be continuously deformed by the act of a shearing stress of anymagnitude, there are fluids, where they flow only if the stress exceeds a certain magnitude(often called critical value). An example of such a fluid is the toothpaste.

In fluid mechanics we generalise the fundamental physical laws like Newton’s laws. Suchgeneralisations are the fluids’ mass and momentum conservation. The foundations of clas-sical mechanics has been set by Sir Isaac Newton. Specifically, Newton described the threefundamental laws of motion. The first law states:

“An object will remain at rest or will move at a constant velocity, unless actedupon by a force”

Newton’s second law of motion is widely used in fluid mechanics and according to this law:

“The net force acting on an object is equal to its mass m times its accelerationa”

which isF = m · a .

Finally, the third law asserts that for every action, there is an equal and opposite re-action:

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“When an object exerts a force on second object, the second object exerts a forceequal in magnitude and opposite in direction of the first object”

One of the main characteristics of a fluid is its density. The density determines how muchdense and how easily the fluid can flow. The density of a fluid is usually denoted by the greekletter ρ and is defined as its mass per unit volume. Therefore, the density can characterisethe mass of the fluid. Since density is a characteristic property of a substance, each liquidhas its own characteristic density. It is remarkable that fluids more dense than the fluidsthat they are placed in will sink. For example, if one mixes water and honey you will observethat honey will sink. The density of some common liquids can be found in Table

tab:densities1.

Liquid Temperature T (oC) Density ρ (kg/m3)Alcohol, methyl (methanol) 25 786.5Automobile oils 15 910Benzene 25 873.8Fuel oil 15 890Glycerine 25 1259Methanol 20 791Olive oil 20 860Sea water 25 1025Pure water 4 1000

Table 1: Density of some common liquids. tab:densities

Another characteristic of fluids is the specific weight γ = ρg, where g ≈ 9.81 m/sec2 is theacceleration due to gravity. The specific weight is defined as the weight of the fluid per unitvolume and is used to characterise the weight of the fluid.

According to Newton’s second law, the weight, i.e. the gravitational force of the earth toan object of mass m is W = m · g. A fluid of density ρ that occupy volume V has weightW = ρ · V · g since its mass is ρ · V . Therefore, the weight is W = γ · V , i.e. the specificweight times the volume of the fluid.

Although density and specific weight characterise the “mass” of a fluid it cannot characterisethe fluid completely. Among other physical properties of a fluid we will mention the viscosityof the fluid. Viscosity is the property of a fluid to resist in changing shape. Sometimes it canbe thought of as the inner friction of the fluid. Viscosity is very important to describe thephysics of a fluid and especially the property of the fluid to dissipate the momentum acrossits volume (recall that momentum is a measure to describe how easily an object can changespeed and is defined as mass times velocity). For example, honey is considered as a viscousfluid, while water is much less viscous that in some cases is considered inviscid. As anotherexample we can think a fluid that sticks to the solid boundaries of its container. The forcesgenerated from the solid boundaries generate momentum that diffuses into the fluid volume.

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(Recall that momentum is the product U = m · u, mass time velocity, and it is relevant tothe impulse of an object, i.e. the tendency to move).

Due to viscosity of the fluids usually we observe turbulence in several situations. A disadvan-tage of the turbulent motion is the loss of energy due to turbulence. If a flow is non-turbulentthen the flow is characterised as laminar. When we study waves in the ocean we often neglectviscosity with very accurate results.

Different forces (other than viscous forces) can be developed on the interface between twoimmiscible liquids, or for example at the interface between a liquid and a gas. These forcescause the surface to behave like a membrane. Surface tension is the elastic tendency ofliquids that makes them acquire the least surface area possible. Surface tension is due tothe intermolecular forces of the liquid. Although the intermolecular forces are in balancewith zero total force in the interior of the fluid, the forces on the surface of the fluid (orthe interface of the fluids) are not in balance. The total force is known as surface tension.Surface tension causes insects (e.g. water striders), usually denser than water, to float andstride on the water surface. Surface tension causes capillary effects that can be important inmany cases but again are negligible in cases of long waves such as tsunamis or tidal waves.In order to characterise the surface tension we use the so called capillary length

√τ/ρg,

where τ is the surface tension measure in N ·m−1. For water τwater = 0.07 N ·m−1. Whenthe wavelength λ of the waves is bigger than the capillary length, then the surface tensionis negligible and only gravity waves can be observed. On the other hand, when short wavesare considered, with wavelength λ ≪

√τ/ρh then the surface tension is important and

capillary-gravity waves are generated.

The last property that we will discuss here is the compressibility of the fluids. Compressibilityis a measure to find how easily we can change the volume of a given mass of the fluid bychanging the pressure. In general, gases are considered compressible fluids contrary to liquids.To study water waves we consider the water as an incompressible fluid and thus we assumethat its density (and thus its volume) cannot be changed.

All these properties will be analysed by mathematical equations in the rest of these notesbut in order to be able to quantify characteristic measures we need to introduce first unitsystems.

1.3 Systems of units

All physical properties and measures have dimensions and units. For example the length canbe measured in metres. In some cases we can use a different quantitative measure for lengthknown as foot (1 metre ≈ 3.28 feet). Different measures can be used by different metricsystems adopted in different countries. In 1969 the 11th General Conference on Weights andMeasures adopted the International System of Units (SI) as the international standard. In

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this system the unit of length is the metre (m), the time unit is the second (s), the massunit is the kilogram (kg) and the temperature unit is the Kelvin (K). Although the Celsiusscale is not in itself part of SI, it is a common practice to report temperature in degrees ofCelsius.

Quantity Name Symbol SI base units compact formFrequency Hertz Hz s−1 –Force Newton N kg ·m · s−2 –Pressure Pascal Pa kg ·m−1 · s−2 N/m2

Energy and work Joule J kg ·m2 · s−2 N ·mPower Watt W kg ·m2 · s−3 J/s

Table 2: Some named units derived from SI base units tab:SIunitsder

In order to measure more complicated physical quantities such as speed for example we needderived units. The derived units in the SI are formed by the base units. Therefore, the unitsof the velocity are derived from the bases units of time and length and is metres per second(m/s). In some cases we used named units derived from the SI base units. For example theunit of force, which can be derived by Newton’s second law F = ma, is unit of mass timesunit of acceleration (kg · m/s2) is denoted by N and is called Newton. Another exampleis the unit of pressure (or stress) which is know as Pascal (Pa) and it is kg/m/s2 or evenbetter N/m2 (Force per unit area).

Other system of units includes the imperial and US customary systems both derived fromearlier English unit system. Although imperial and US customary systems have many com-monalities they are different. For example units of mass and weight are different. Units oflength for both of thems are inch, foot, yard, mile, etc.

Next chapter describes in a more mathematical sense the ideas of dimensions of physicalquantities and how we can derive and simplify mathematical models to describe physicalphenomena using dimensional analysis and scaling.

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2 Dimensional Analysis and Scalingsec:dimansca

“Without dimensionlessnumbers, experimental progressin fluid mechanics would havebeen almost nil; It would havebeen swamped by masses ofaccumulated data.”

— R. Olson

Physical laws such as conservation of mass, momentum and energy or other laws such asNewton’s laws usually can be described with the help of mathematical equations. A mathe-matical model commonly consists of a set of physical laws along with other complementarymathematical equations, constitutive relations or other relations based on experimental ev-idence. While studying mathematical models it is important to be able to describe thephysical quantities by relevant independent and dependent variables and parameters. Forexample we usually denote the time by the independent variable t, the location of a particleby the variable x and the speed of the particle by the dependent variable v(x, t). Physicalquantities have dimensions like time, distance, temperature etc. We also measure thesephysical quantities using relevant scales with units such as seconds, meters, and degrees Cel-sius. The knowledge of the relationships between the physical quantities and their relativemagnitudes is very important since it helps us not only to understand better the problem athand but also to make simplifications and finally find a solution.

This section is devoted to the first stages of the mathematical modelling of physical laws,namely to the dimensional analysis and scaling. The methods of dimensional analysis some-times lead to important relations between the physical quantities and help to understand thephysical phenomena even when the governing equations are not available. Using methodsof dimensional analysis we are also able to non-dimensionalise mathematical models and toformulate the model in terms of dimensionless quantities only. Scaling is a technique totransform physical equations into scaled form. This helps in understanding the magnitude(and therefore the importance) of the terms that appear in the physical equations. More-over, reduces the numbers of parameters of the problem. The knowledge of the importanceof a term is crucial in mathematical modelling since the models can be simplified usuallyby discarding terms that are not important or they are very small (negligible) compared toothers terms. An example that shows the relevance of the magnitude is the following: Thespeed of the car is very small compared to the speed of a jet but it is very big comparedto the speed of a turtle. That is why we evaluate usually terms by means of dimensionlessquantities.

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2.1 Dimensional Analysissec:diman

One of the key ingredients of the dimensional analysis are the dimensions of the physicalvariables and parameters. In order to describe physical quantities we use the fundamentaldimensions, which are given in Table

tab:dimensions13 for some commonly used quantities along with their

units in the SI system of units.

Dimension Symbol Unitlength L m (meter)mass M kg (kilogram)time T sec (second)temperature Θ C (degree Celsius)

Table 3: Fundamental dimensions tab:dimensions1

Using these fundamental dimensions we can describe other physical quantities using appro-priate relations. For example, we know that the velocity (or speed) of an object movinghorizontally is defined as the length of the distance covered by the object per time and de-pends on time t. In other words, velocity v is a function of t, v(t), with dimension LT−1

measured in m/sec. Other physical quantities that usually depend on the fundamental di-mensions are combinations of these and the most commonly used in fluid mechanics arepresented in Table

tab:dimensions24.

Quantity (symbol) Dimensions Relation Unitsvelocity (v) LT−1 length per time m/secacceleration (a) LT−2 velocity per time-squared m/sec2

momentum (U) MLT−1 mass times velocity kg ·m/secmass density (ρ) ML−3 mass per volume kg/m3

force (F ) MLT−2 mass times acceleration N (Newtons)energy (E), work (W ) ML2T−2 force times distance J (Joules)power (P ) ML2T−3 energy per time W (Watts)pressure (P ), stress (σ) ML−1T−2 force per area Pa (Pascals)frequency (f) T−1 per time He (Hertz)

Table 4: Dimensions of common quantities in mechanics and thermodynamics tab:dimensions2

It is noted that the dimensions of a quantity q are denoted by [q].

To understand the power of dimensional analysis let’s try to re-derive Newton’s second law,i.e. F = m · a. Assume that there is a physical law of the form of the equation

g(F,m, a) = 0 . (2.1) eq:tay

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This equation relates three dimensioned quantities: the force F , the mass m and the ac-celeration a. Here F has dimensions of mass · length · time−2, and a has dimensions oflength · time−2. We observe that a dimensionless quantity that can be derived easily fromthe three dimensional quantities is the quantity m ·a/F . We make the additional assumptionthat there should be a physical law involving only the dimensionless quantity. This law canbe written as:

f(m · a

F

)= 0 . (2.2) eq:expl

Finally assuming that the physical law is a root of (eq:expl2.2) we get

m · aF

= C ,

and thereforeF = C ·m · a ,

where C is a generic constant. So using dimensional reasoning only we show that the forceF depends linearly on the mass m. In this way we derived Newton’s second law but with aconstant C that can be determined by using experimental data.

The assumption we made in the previous example, i.e. that for a physical law there isan equivalent physical law that relates only dimensionless quantities is the core of the Pitheorem of Buckingham. We mention only the main result while we omit its proof.

Assume that there are m dimensional quantities q1, q2, · · · , qm and assume that they can beexpressed in terms of a minimal set of fundamental dimensions L1, L2, · · · , Ln with n < m.So the dimensions of qi can be written in terms of the fundamental dimensions as:

[qi] = La1i1 La2i

2 · · ·Lanin , i = 1, · · · ,m , (2.3)

for some exponents a1i, a2i, · · · , ani.

It is noted that if [qi] = 1, then the quantity qi is dimensionless. So if π is a quantity of theform

π = qp11 qp22 · · · qpmm ,

we want to find all the exponents p1, · · · , pm such that π is dimensionless. This is

[π] = [q1]p1 [q2]

p2 · · · [qm[pm

= (La111 La21

2 · · ·Lan1n )p1 · · · (La1m

1 La2m2 · · ·Lanm

n )pm

= La11+···+a1m1 La21+···+a2m

1 · · ·Lan1+···+anmn

= 1 .

Since the product La11+···+a1m1 La21+···+a2m

1 · · ·Lan1+···+anmn = 1 this means that the the expo-

nents of Li is equal to zero ai1 + · · ·+ aim = 0 for each i = 1, 2, · · ·n. Since we have n such

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equations with m unknowns (i.e. for i = 1, · · · , n, we obtain a homogeneous system of nlinear equations with m unknowns. This system can be written in matrix form as

Ap = 0 ,

where p = (p1, · · · , pm)T is a column vector and A is a n ×m matrix called the dimensionmatrix. The Pi theorem of Buckingham states that if r = rank (A) then there exists m− rindependent dimensionless quantities π1, π2, · · · , πm−r that can be formed from q1, · · · , qmand the unit-independent physical law

f(q1, qm, · · · , qm) = 0

that relates the dimensional quantities q1, q2, · · · , qm is equivalent to an equation

F (π1, π2, · · · , πm−r) = 0 ,

expressed only in terms of the dimensionless quantities.

The proof of the Pi theorem is omitted but it can be found inLogan2006[Log06].

It is noted that an equation (or a physical law) is unit-independent if it remains the samefor any metric system. This can be explained mathematically as follows: The physical lawf(q1, q2, · · · , qm) = 0 with quantities qi in some metric system is unit-independent if it is thesame for the quantities qi = λiqi in a different metric system, i.e. f(q1, q2, · · · , qm) = 0. Anexample of a unit-independent law is the following Newton’s law:

f(x, t, g) ≡ x− 12gt2 = 0 ,

where in the cgs system of units, x is in centimetres (cm), t in seconds (sec) and g in cm/sec2.Changing the units for the fundamental quantities x and t to inches and minutes, then in thenew system of units x = λ1x and t = λ2t where λ1 = 1/2.54 in/cm and λ2 = 1/60 min/sec.Because [g] = LT−2, we have g = λ1λ

−22 g and so

f(x, t, g) = x− 12gt2 = λ1x− 1

2(λ1λ

−22 g)(λ2t)

2 = λ1(x− 12gt2) = λ1f(x, t, g) = 0 ,

thus f(x, t, g) = 0 and therefore the law is unit-independent. This is a nice property whichenables us to work with the same physical equation in any system of units we prefer.

We continue with some examples:

Example 2.1. Consider the velocity of a car v. We know that the car is related to thedistance l covered by the car and the time t needed to cover this distance. Find a physicallaw that relates all these quantities using dimensional reasoning.

Proof. We have [v] = L/T , [l] = L and [t] = T , which is m = 3 dimensional quantities. Thedimension matrix is

v l t

LT

(1 1 0−1 0 1

),

15

which has rank r = 2. So we have only k = m − r = 1 dimensionless quantity and this isπ1 = va1la2ta3 . So we have

1 = [π1] = [va1la2ta3 ]

= (La1T−a1)(La2)(T a3)

= La1+a2T a3−a1 .

Therefore

a1 + a2 = 0

a3 − a1 = 0 .

Taking a3 = 1 we find a1 = 1 and a2 = −1. So π1 = vlt

and so if the quantities v, l and tare related with a physical law f(v, l, t) = 0 according to the Pi theorem there should be aphysical law which relates the dimensionless variables, here

F (π1) = 0 ,

and so vlt= C which implies that v = Cl/t for some constant C. This is very close to the

well-known formula for the velocity v = l/t.

Example 2.2. The speed v of a wave in deep water is determined by its wavelength λ andthe acceleration g due to gravity. What does the dimensional analysis imply regarding therelationship between v, λ and g?

Proof. Here [v] = LT−1, [λ] = L and g = LT−2. So we have m = 3 dimensional variables.The dimension matrix is:

v λ g

LT

(1 1 1−1 0 −2

),

which has rank r = 2 and so again k = 1. The dimensionless quantity π1 = va1λa2ga3 and so

1 = [π1] = [va1λa2ga3 ]

= (La1T−a1)(La2)(La3T 2a3)

= La1+a2+a3T 2a3−a1 .

This is implies that

a1 + a2 + a3 = 0

a1 + 2a3 = 0 .

Taking a3 = 1 we get a1 = −2 and a2 = 1. Thus,

π1 =gλ

v2.

16

According to Pi theorem there should be a physical law of the form

F (π1) = 0 ,

and if π1 is a root of F then gλv2

= C or equivalently we get:

v = C√gλ .

This means that the speed of deep water waves is proportional to the square root of thewavelength, i.e. longer waves propagate faster than shorter waves.

Remark 2.3. Observe that Pi theorem usually yields a constant multiple of the physicallaw and not the exact physical law. So the choice of the independent parameters in theoverdetermined linear system (in the previous examples a3) it does not matter since thesolution will be equivalent.

Example 2.4. A heater P is heating a room with heat energy e. We assume that the roomhas temperature u = 0 and the heat energy is allowed to diffuse. Let r denotes the radialdistance from the heater, t the time. If c is the heat capacity of the room with dimensions ofenergy per degree per volume and k the thermal diffusivity with dimensions length-squared pertime, find a relation connecting all these physical variables in the form f(t, r, u, e, k, c) = 0.

Proof. The dimensions of the quantities at hand are:

[t] = T ,

[e] = E ,

[r] = L ,

[k] = L2T−1 ,

[u] = Θ ,

[c] = EΘ−1L−3 .

Alternatively, one may use the following dimensions

[t] = T ,

[e] =ML2T−2 ,

[r] = L ,

[k] = L2T−1 ,

[u] = Θ ,

[c] =MΘ−1L−1T−2 ,

but the last set of dimensions is more complicated and instead relates dimensions of M andnot energy which is a characteristic quantity in our problem. Using the first set we can write

17

the dimension matrix A in the form:

t r u e k c

T 1 0 0 0 −1 0L 0 1 0 0 2 −3Θ 0 0 1 0 0 −1E 0 0 0 1 0 1

.

Here m = 6 and n = 4 and the rank of A is r = 4. Therefore, there are m − r = 2dimensionless quantities that can be formed from t, r, u, e, k and c. If π is a dimensionlessquantity that can be deduced from this method then:

1 = [π] = [ta1ra2ua3ea4ka5ca6

= T a1La2Θa3Ea4(L2T−1)a5(EΘ−1L−3)a6

= T a1−a5La2+2a5−3a6Θa3−a6Ea4+a6 .

Equating the exponents we obtain the following set of equations:

a1 − a5 = 0 ,

a2 + 2a5 − 3a6 = 0 ,

a3 − a6 = 0 ,

a4 + a6 = 0 .

Because the system is an underdetermined system we need to use two arbitrarily chosenparameters. For example using a5 and a6 as parameters and taking appropriate values forthem we find two linearly independent solutions:

(a1, a2, a3, a4, a5, a6) = (−1/2, 1, 0, 0,−1/2, 0) ,

and(a1, a2, a3, a4, a5, a6) = (3/2, 0, 1,−1, 3/2, 1) .

It is noted that choosing different values for the parameters a5 and a6 one may find differentlinearly independent solutions. It is noted that we can check if two vectors are linearlyindependent with the help of their dot product.

These two dimensionless quantities are then:

π1 =r√kt, π2 =

uc

e(kt)3/2 .

Therefore, the Pi theorem guranties that the original physical law f(t, r, u, e, k, c) = 0 isequivalent to a physical law of the form:

F (π1, π2) = 0 .

18

Assuming that we can solve for π2 we can write

π2 = g(π1) ,

which leads to the relation:u =

e

c(kt)−3/2g

(r√kt

).

Again, the dimensional analysis didn’t result into the exact physical law but only a relationbetween the variables. Specifically, the previous relation shows that the temperature nearthe source r = 0 falls of like t−3/2.

2.2 Scalingsec:scale

Similitude is another concept applicable to the testing of engineering models. Similitude’smain application is to test fluid flow properties using scaled models. Scaled models are veryuseful in practical studies. It is also a common practice to present scaled experimental datainstead of raw results. Sometimes scaling is very helpful in understanding of the importanceof physical parameters.

In this Section we reformulate mathematical models using new dimensionless variables. Writ-ing the model in dimensionless form is known as non-dimensionalization, or scaling. Thisprocess begins with the choice of the appropriate independent and dependent variables andthe introduction of characteristic scales. These characteristic scales usually are introducedby the physics of the problem at hand. For example a characteristic time scale for the motionof a glacier can be the year (tc = 1 year = 3.15 × 107 sec) while the characteristic timescale for a chemical reaction could be the second (tc = 1 sec). Characteristic scales can beused for any physical quantity such as length and temperature. After determining the char-acteristic scales (for example the characteristic time tc) we introduce a new dimensionlessvariable

t∗ =t

tc.

Then the dimensionless quantity t∗ is usually normalised, i.e. it is neither large nor small,but rather of order one. Using the bigO notation we write that t∗ = O(1). Sometimes, aquantity is said to be of order one but it might be much larger or smaller like 100 or 0.5.

The non-dimensionalization process sometimes leads also to models with fewer parameters,while in some cases the process helps the researchers to understand possible simplificationsof model equations.

Example 2.5. Consider the motion of a projectile thrust vertically upward from the surfaceof the earth. Given that the mass of the projectile is m and the initial velocity V derive thedimensionless model describing its motion.

19

Proof. The motion of the projectile is governed by the differential equation

md2x

dt2= −mg, x(0) = 0,

dx

dt(0) = V , (2.4) eq:projectile1

where x(t) is the distance of the projectile from the ground at time t, and g is the accelerationdue to gravity. Equation (

eq:projectile12.4) is actually Newton’s second law F = ma where F = −mg

and a(t) = x′′(t). The solution of this equation can be found easily after integration twiceto be

x(t) = V t− gt2/2 .

The maximal value for x(t) is attained when dx/dt = 0, i.e. the time tc satisfying V −gtc = 0,which gives tc = V/g. In order to write (

eq:projectile12.4) into dimensionless form we choose characteristic

values for each variable, here x and t using combinations of the given parameters V , g, m.For example we define the characteristic scales:

tc = V/g and xc = V 2/g ,

with the same dimensions as x and t. We then introduce the new non-dimensional variables

t∗ = t/tc = tg/v and x∗ = x/xc = xg/V 2 , (2.5) eq:changev1

and we observe that [t∗] = [x∗] = 1. Using chain rule we have that

d

dt=dt∗

dt

d

dt∗=

g

V

d

dt∗,

anddx

dt=

g

V

d(x∗V 2/g)

dt∗= V

dx∗

dt∗,

andd2x

dt2=

g2

V 2

d2(x∗V 2/g)

d2t∗= g

d2x∗

dt∗2.

Thus (eq:projectile12.4) becomes

d2x∗

dt∗2= −1, x∗(0) = 0,

dx∗

dt∗(0) = 1 . (2.6) eq:projectile2

The last equation is dimensionless and parameter free. This is not always the case.

Example 2.6. If in addition we assume the the air friction is not negligible, then thedifferential equation describing the motion of the projectile is:

md2x

dt2= −kdx

dt−mg, x(0) = 0,

dx

dt(0) = V , (2.7) eq:projectile3

where m, g, V and k are given parameters, and k determines the air resistance. Using thesame characteristic scales write Equation (

eq:projectile32.7) in dimensionless form.

20

Proof. Using the same scaling and performing the change of variables (eq:changev12.5) we write Equation

(eq:projectile32.7) in the form:

d2x∗

dt∗2= −εdx

∗

dt∗− 1, x∗(0) = 0,

dx∗

dt∗= 1 , (2.8) eq:projectile4

whereε =

kV

mg,

is the only dimensionless parameter left in the equation. Morover, if for example ε is verysmall, i.e. 0 < ε ≪ 1, for example when the mass is very large for example (m ≫ 1) thenwe can simplify (

eq:projectile42.8) to (

eq:projectile22.6) by neglecting the small term. Another observation that can

be made is that if the initial velocity is small enough (V ≪ 1) then again the effect of theair friction will be small.

From the previous example we observe that scaling:

• can reduce the number of the parameters in the model,

• can make clear the importance of some physical quantities.

To illustrate further the notions of non-dimensionalisation and scaling we proceed with onemore example.

Example 2.7. Consider a car engine of volume V where a fuel of fixed concentration cigiven in mass per volume, enters at constant rate q. The fuel is burned and consumed inthe engine with rate R, while the remaining mixture exits the engine with the rate q. Theconcentration of the fuel in the engine is denoted by c(t) at any time t. For simplicity assumethat the rate of consumption R is proportional to the concentration c, i.e. that R = kc forsome constant k > 0. After deriving a mathematical model for the concentration of the fuelin the engine, write it in dimensionless variables and study the importance of each term. Itis also given that initially the concentration of the fuel in the engine was c(0) = c0.

Proof. In order to obtain a mathematical model we apply the fundamental law of the con-servation of mass. That is, the time rate of change of the mass of the fuel inside the enginemust be equal to the rate of mass flows in the engine (qci), minus the rate that mass flowsout (qc) plus the mass of the fuel consumed in the engine (V c). This can be written as

d

dt(V c(t)) = qci − qc(t)− V R, c(0) = c0 .

This equation can be simplified using the face that R = V c to

d

dtc(t) =

q

V(ci − c)− kc(t), c(0) = c0 . (2.9) eq:react1

21

The characteristic values of the problem are ci, c0, V , q, k. Observe that there are twoconstants for the concentration, ci and c0, and either one of them is a suitable concentrationscale. Therefore, we define a dimensionless concentration c∗ by

c∗ =c

ci.

To select a timescale we observe that there are two quantities with dimensions of time thatcan be formed from the constants in the problem, namely, V/q and 1/k. The former is basedon the flow rate, and the latter is based on the reaction rate. So the choice of a time scaleis not unique. Taking a general characteristic time T the dimensionless time is denoted by

t∗ =t

T,

where T is either V/q or 1/k. Using chain rule we obtaindc

dt=ciT

dc∗

dt∗,

and thereforedc∗

dt∗=qT

V(1− c∗)− kTc∗ , t∗ > 0 ,

andc∗(0) = c0/ci .

If T = V/q then the dimensionless model is written asdc∗

dt∗= 1− c∗ − εc∗ , t∗ > 0 , (2.10) eq:react2

whereε =

kV

q.

while if T = 1/k then the model reduces to the equationdc∗

dt∗=

1

ε(1− c∗)− c∗ , t∗ > 0 ,

Using those two parameterisations we deduce that if k ≪ q/V then the dimensionless pa-rameter ε ≪ 1 is very small and thus model (

eq:react22.10) is more useful since it can be simplified

to the simpler equationdc∗

dt∗= 1− c∗ , t∗ > 0 .

On the other hand if the reaction is very fast compared to the flow rate, then ε≫ 1 is verylarge and thus the second model is more convenient since it can be reduced to the simplermodel

dc∗

dt∗= −c∗ , t∗ > 0 .

In this example observe again that the non-dimensionalisation leads to fewer parameters andto simpler model equations.

22

The breaking of a wave cannotexplain the whole sea.

— Vladimir Nabokov

3 Introduction to Conservation Lawssec:laws

In physics, a conservation law expresses the principle that a particular quantity of a physicalsystem does not change as the system evolves. In the next chapter we will present somefundamental conservation laws of fluid mechanics. These fundamental laws of nature applyto conservation of energy, mass and momentum. In mathematical physics, a conservationlaw can be expressed as a homogeneous first-order partial differential equation known ashyperbolic conservation law. In this chapter we present the basic mathematical propertiesof first order partial differential equations and methods of solution. We will also study thepropagation of hyperbolic waves and their breaking.

3.1 Classification of 1st order PDEssec:introduction

We will study first-order partial differential equations in one space dimension, i.e. with twoindependent variables, the spatial x and temporal t variables. A general first-order partialdifferential equation has the form:

F (x, y, u, ux, ut) = 0, t ≥ 0, x ∈ D ⊂ R, (3.1) eq:E1

where F is a given function, u = u(x, t) is the unknown function. Also ux = ∂u∂x

and ut = ∂u∂t

are the usual partial derivatives with respect of the independent variables with x the spatialvariable and t the time.

First order equations are divided into linear and non-linear. A linear first-order, partialdifferential equation can be written in the form:

a(x, t) ut + b(x, t) ux + c(x, t) u = d(x, t), (3.2) Eq:lin

for some functions a, b, c and d. When d(x, t) = 0 the equation is called homogeniousotherwise is called inhomogeneous.

Nonlinear equations can also be divided into categories. For example, if equation (eq:E13.1) is of

the forma(x, t) ut + b(x, t) ux = c(x, t, u), (3.3) Eq:slin

23

i.e. it contains nonlinear terms of the function u only, then it is called semi-linear. Onthe other hand, if all the function a, b, c depend on u the equation is called quasi-linear. Aquasi-linear equation is of the form:

a(x, t, u) ut + b(x, t, u) ux = c(x, t, u). (3.4) Eq:qlin

The two most classical examples are the following

ut + c ux = 0, transport equation (3.5)ut + u ux = 0, inviscid Burger’s equation (3.6)

for some real constant c.

3.2 Traveling waves

A wave is a signal or a disturbance in a medium propagating in time while carries energy.Some examples that we are all familiar with are the waves on the surface of the water orsound waves. One of the most basic and fundamental wave is the traveling wave. Travelingwaves propagate with constant speed and without changing their shape for all time. Such awave can be described by a function of the form:

u(x, t) := v(x− ct), (3.7) eq:tw

with c a positive constant. This function represents a right-traveling wave moving at constantspeed c without any change in shape. At t = 0 the wave profile is u = v(x) but at t > 0 thewave has moved to the right by ct units of length, cf. Figure

fig:tw2.

x x− ct x

y

ctt = 0, v(x) t > 0 v(x− ct)

Figure 2: Traveling wave fig:tw

Traveling waves have been observed in rivers, optical fibers and other media and are veryimportant. Denoting ξ = x− ct and taking the derivatives with respect of x and t of such awave we have

ux = v′(ξ), ut = −cv′(ξ). (3.8) eq:der

24

Hence,ut + cux = 0. (3.9) eq:trans

The last equation is a first order linear partial differential equation and it is called thetransport equation (or sometimes the advection equation). If u(x, 0) = v(x) then the generalsolution of (

eq:trans3.9) is the function u(x, t) = v(x− ct) for t > 0.

Not all waves propagate without change in shape and speed. For example observing waveson a beach one will see that as the waves approach the beach they grow in amplitude whiletheir speed is decreased and finally they break, cf. Fig.

fig:bw3. Since the breaking wave cannot

be described by a function we model it by using a discontinuous function which is referredto as a shock wave. This solution usually is not smooth but preserves the mass of the watercolumn of the breaking wave, see Figure

fig:bw3.

x

y

b

b

b

b

b

b

A A

AB B

B

Figure 3: Breaking wave fig:bw

3.3 The characteristic curves

In the previous section we introduced the simplest wave equation, the transport equation,

ut + cux = 0, x ∈ R, t > 0. (3.10) eq:trans1a

If we impose the initial condition

u(x, 0) = v(x), x ∈ R, (3.11) eq:ic1

then the general solution of the initial value problem (eq:trans1a3.10)–(

eq:ic13.11) is the traveling wave

u(x, t) = v(x− ct), (3.12) eq:tw1

propagating at a constant velocity c. The straight lines x − ct = constant, are of centralrole in the analysis bellow. More precisely, the solution propagates along those lines withthe same value. For example, in order to find the value of the solution at the point (x, t) ofFigure

fig:charb5, one should look at the value of the initial condition at the beginning of the line

25

(ξ, 0). Moreover, using the change of variables x(t) the partial differential equation (eq:trans1a3.10)

is reduced to an ordinary differential equation d(u(x(t), t))/dt = 0. Considering the linex(t) = ct+ ξ for some constant ξ. Note also that the slope of these lines is dx/dt = c. Thenthe temporal derivative of the solution u is

d

dtu(x(t), t) = ux(x(t), t)

dx

dt+ ut(x(t), t) (3.13) eq:te1(

since dx(t)dt

= c

)= ux(x(t), t) c+ ut(x(t), t) . (3.14) eq:te2

The right-hand side of the equation (eq:te13.13) coincides with the equation (

eq:trans1a3.10) evaluated for

x = x(t) and therefore we haved

dtu(x(t), t) = 0 , (3.15) eq:E2

for x(t) = ct + ξ for any constant ξ (where actually ξ = x(0) is the beginning of thecharacteristic line). This means that the solution is constant (i.e. it does not change itsshape with time) on the straight lines x(t). The straight lines x − ct = ξ are called thecharacteristic lines (or just the characteristics) of the differential equation.

Keeping the speed of the waves constant is very limiting. For example waves on the surfaceof the ocean propagate with different speeds and interact in a quite complicated way. Forthis reason we continue our study of wave equations with some more general equations. Forexample consider the more general initial value problem

ut + c(x, t)ux = 0, x ∈ R, t > 0, (3.16)u(x, 0) = u0(x), (3.17)

where c(x, t) is a given function. Similarly, we consider the characteristic curves x(t) withslope

dx(t)

dt= c(x, t). (3.18) eq:slope1

Then along a characteristic curve, lets say x(t), and using the chain rule we have

d

dtu(x(t), t) = ux

dx

dt+ ut (3.19)

= ux c(x, t) + ut = 0 . (3.20) eq:E3

Hence u is constant on each curve x(t).

Ex:Ex1 Example 3.1. Find the general solution of the initial value problem:

ut + tux = 0, x ∈ R, t > 0, (3.21)u(x, 0) = exp(−x2), x ∈ R. (3.22)

26

Proof. The characteristic curves are defined by the differential equation

dx

dt= t,

that yields to the family of parabolas

x =1

2t2 + ξ, ξ constant.

Knowing that u is constant on these characteristic curves allows us to find a solution to theinitial value problem. Let (x, t) be an arbitrary point with t > 0. The characteristic curvethrough (x, t) passes through (ξ, 0) and has equation

x =1

2t2 + ξ.

Since u is constant along this curve we have

u(x, t) = exp(−ξ2

)= exp

(−(x− 1

2t2)2)

b

t

x

x = 12t2 + ξ

(ξ, 0)0

Figure 4: The characteristic curves fig:bw2

3.4 Nonlinear wavesnlwaves

In the previous sections the speed of propagation of the waves was independent of the solutionu and it was a simple function of x and t. In this section we will study the case where the

27

speed depends on the magnitude of the solution u. For example, in the water we can observethat from the waves propagating towards the shoreline the larger waves travel faster than theshorter ones as an indication that the speed of propagation depends on the actual solution.

A model equation that describes waves with speed c = c(u) is the so called Burger’s equation(also known to as the inviscid Burger’s equation):

ut + c(u)ux = 0, x ∈ R, t > 0, (3.23) eq:Burgers

with given initial conditionu(x, 0) = ϕ(x), x ∈ R . (3.24) eq:incondb

For simplicity we assume that c′(u) > 0.

Like in the previous section, we define the characteristic curves of (eq:Burgers3.23)–(

eq:incondb3.24) as the solu-

tions of the ordinary differential equationd

dtx(t) = c(u(x(t), t)) , (3.25) eq:charb

with initial condition x(0) = ξ, where ξ is arbitrarily chosen. We observe that using thechain rule we have

d

dtu (x(t), t) = ux (x(t), t)

d

dtx(t) + ut (x(t), t)

= ux (x(t), t) c (u(x(t), t)) + ut (x(t), t)

= 0 ,

since dxdt

= c(u) and thus the solution u(x, t) is constant along the characteristic curvex = x(t).

In addition, taking the second derivative in (eq:charb3.25) we have

d2

dt2x(t) =

d

dt

(d

dtx(t)

)=

d

dtc (u(x(t), t))

= c′(u)d

dtu(x(t), t)

= c′(u) · (c(u(x(t), t)) ux(x(t), t) + ut(x(t), t))

= 0 .

Since the second derivative of a characteristic curve is zero we conclude that the first deriva-tive is constant and thus the characteristic curve is a straight line.

Since the solution is constant along the characteristic lines (cf. Figurefig:charb5), we have that

u(x, t) = u(ξ, 0) , (3.26) eq:solution

28

x

t

(x, t)

(ξ, 0)

Figure 5: Characteristic lines fig:charb

for any x and t. In order to find the equation of the characteristic line passing through thepoint (x, t) we need to solve the ordinary differential equation

d

dtx(t) = c (u(ξ, 0)) (= c(ϕ(ξ))) .

Integrating the last equation with respect of t and since the right-hand side is independentof t we find that

x(t) = c (ϕ(ξ)) t+ C ,

for a generic constant C. Since the line passes through (ξ, 0), then taking t = 0 in the lastequation we conclude that C = ξ and the equation of the characteristic line is given by theequation:

x = c (ϕ(ξ)) t+ ξ , (3.27) eq:lineb

and x(0) = ξ. The slope of the characteristic lines is also called speed of the characteristics.Finally, we conclude that the solution u is such that u(x(t), t) = ϕ(ξ) and x(0) = ξ. Usingξ defined by (

eq:lineb3.27) we can get the general solution u(x, t).

Example 3.2. Solve the initial value problem

ut + u ux = 0, ∈ R, t > 0,

u(x, 0) = ϕ(x) =

2, x < 02− x 0 ≤ x ≤ 11, x > 1

,

with the method of characteristic lines.

Proof. In this example c(u) = u. Taking ξ ∈ R then the characteristic lines starting from(ξ, 0) have inclination (speed) c(ϕ(ξ)) = ϕ(ξ). Therefore, for ξ < 0 the lines have inclination(speed) c = 2, while for ξ > 1 the lines have inclination 1. For 0 ≤ ξ ≤ 1 the lines have

29

−4 0 1 50

1

2

x

t

−4 −3 −2 −1 0 1 2 3 4 5 60

0.5

1

1

1.5

2

t

x

u(x

,t)

Figure 6: Top: Characteristic lines, Bottom: Solutions of Burgers equation fig:sols

inclination 2 − ξ and intersect at the point (2, 1). This means that for 0 ≤ ξ < 1 thesolution cannot be determined on the point (2, 1) since it will have multiple values (equalto the values of the solution on each of the intersecting characteristics. Specifically, thecharacteristics intersecting ξ ∈ (0, 1) can be determined by the equation

eq:lineb3.27 which takes

the form:x(t) = (2− ξ) t+ ξ. (3.28) eq:chars1

The characteristic lines and some instances of the solution are presented in Figurefig:sols6. As

t approaches the value t = 1 the solution (wave) becomes steep. For t > 1 the solutionbecomes multi-valued as it is depicted in Figure

fig:mvshock7. Since the solution is multi-valued for

t > 1 we define a different solution that satisfies the same model equation. This solution willbe discontinuous and it will have the same mass as the original solution. This phenomenonis known as wave breaking and the discontinuous solution is called shock wave.

The breaking of the wave occurs at t = 1 which is the first time where the solution becomesmultiple valued. The region of the x− t plane where the solution is not constant is boundedby the two characteristic lines, namely x = 2t and x = t + 1. Observe that x = 2t is thecharacteristic line passing through (0, 0) while the line x = t+ 1 is the one passing through

30

u

(a) t = 0.0

u

(b) t = 1.0

ξ

u

(c) t = 2.0

Figure 7: Wave breaking of a non-smooth solution and the generation of a shock wave. fig:mvshock

(1, 0) and are designated with a thick blue line in Figurefig:sols6.

To find the solution for 2t < x < t+ 1 we solve (eq:chars13.28) for ξ which is

ξ =x− 2 t

1− t.

The solution is always determined by (eq:solution3.26) which in our case gives

u(x, t) = u(ξ, 0)

= 2− ξ

= 2− x− 2t

1− t

=2− x

1− t.

31

Thus,

u(x, t) =

2, x ≤ 2 t2−x1−t

, 2 t < x < t+ 1

1, x ≥ t+ 1.

We note that the last explicit form of the solution is not defined for the breaking time t = 1.By definition, the breaking time is the time when the solution becomes very steep. On theother hand sometimes the solution can be very steep, or even multivalued and still be ableto get determined after the breaking time.

Example 3.3. For example consider the initial value problem

ut + u ux = 0, ∈ R, t > 0,

u(x, 0) = ϕ(x) = 12π − tan−1(x) .

In this case the characteristic lines are defined as the lines x(t) = ϕ(ξ)t + ξ for all ξ ∈ Rwhile the solution is given analytically as

u(x(t), t) = ϕ(ξ) .

In Figurefig:wavbr8 we observe that the wave becomes steep and finally becomes multivalued as it

breaks.

Although we can find an analytical formula for this solution, it happens that the solutionafter the breaking time is not a single-valued function. For this reason we choose to modelthe breaking wave by constructing a solution which will be discontinuous and will retainthe mass of the initial wave (so as the fundamental law of mass conservation will not beviolated by the solution). To construct such a discontinuous solution, one replaces somepart of the multivalued graph using a vertical line in such a way that the resulting functionis single-valued and has the same area under its graph, cf. Figure

fig:bw3.

3.4.1 Determining the breaking time

Consider the equationut + c(u)ux = 0, x ∈ R, t > 0, (3.29) eq:Burgers2

with the initial conditionu(x, 0) = ϕ(x), x ∈ R. (3.30) eq:incondb2

In order for wave breaking to occur we assume that c′(u) > 0, and for simplicity we assumethat ϕ(x) ≥ 0 and ϕ′(x) < 0. We assume that in general the initial value problem (

eq:Burgers23.29)–

(eq:incondb23.30) has smooth solutions up to a finite time tb where tb is the time where the breaking

32

u

(a) t = 0.0

u

(b) t = 1.0

ξ

u

(c) t = 1.5

Figure 8: Breaking wave of the Burgers equation. fig:wavbr

happens. he breaking of the wave occurs when the solution becomes very steep, or in otherwords, when the gradient of the solution becomes infinite. This phenomenon is also known asthe gradient catastrophe and it can be determined from the singular points of the derivativeof the solution ux.

To determine the time tb we differentiate the solution u(x, t) = u(ξ, 0) with respect to x toobtain

ux(x, t) = ϕ′(ξ)ξx .

We compute ξx by differentiating the equation of the characteristic lines (eq:lineb3.27) i.e. the

equation x = c(ϕ(ξ))t+ ξ with respect to x and we have:

1 = ϕ′(ξ) c′(ϕ(ξ)) ξx t+ ξx .

Solving for ξx we getξx =

1

1 + c′(ϕ(ξ)) ϕ′(ξ)t,

33

and so we haveux =

ϕ′(ξ)

1 + c′(ϕ(ξ)) ϕ′(ξ)t.

The gradient catastrophe will occur at the minimum value of t, which makes the denominatorzero. Hence solving for this t we have

tb = minξ

−1

ϕ′(ξ) c′(ϕ(ξ)), tb > 0.

In the ExampleEx:Ex13.1 where c(u) = u and ϕ(ξ) = 2− ξ we compute easily the minimum tb = 1.

3.4.2 Solution after the breaking time

In the previous sections we solved first order differential equation in the case where thesolutions remain smooth and up to the breaking time. The computation of physically relevantsolutions after the breaking time remains a very important problem that we will try to answerin this section. We first introduce the notion of conservation laws. Any first order partialdifferential equation written in the form

ut + [F (u)]x = 0, x ∈ R, t > 0 (3.31) eq:cl1

is called a conservation law. We also say that the differential equation (eq:cl13.31) is written in

conservative form.

Integrating equation (eq:cl13.31) from x = a to x = b we obtain

d

dt

∫ b

a

u(x, t) dx = −∫ b

a

F (u)xdx. (3.32)

d

dt

∫ b

a

u(x, t) dx = F (u(a, t))− F (u(b, t)). (3.33) eq:eqcl2

The temporal derivative of the integral in (eq:eqcl23.33) represents the rate of change of the total

amount of the quantity u inside the interval [a, b]. The function F (u(x, t)) can be consideredas the flux of the quantity u through a point x, that is, the amount of the quantity per unittime positively flowing across x. Then Equation (

eq:eqcl23.33) states that the rate of change of the

quantity in [a, b] equals the flux in at x = a minus the flux out through x = b.

For example the equationut + u ux = 0, (3.34) eq:eqcl3

can be written as a conservation law of the form

ut +

(1

2u2)

x

= 0,

34

where the flux is given by the function F (u) = 12u2. In general, any equation of the form

ut + c(u) ux = 0, (3.35)

can be written in a conservative form by setting c(u) = F ′(u). If c′(u) > 0 and ϕ′(x) < 0, thenthe characteristics starting from two points (ξ1, 0) and (ξ2, 0) with ξ1 < ξ2 have inclinationsc(f(ξ1)) and c(f(ξ2)), respectively. It follows that c(ϕ(ξ1)) > c(ϕ(ξ2)) and therefore thecharacteristics must intersect at the breaking time tb. After the breaking time the solutionis expected to be discontinuous. In order to find the solution for t > tb we observe that thissolution cannot satisfy the differential equation (

eq:eqcl33.34) because it is not smooth anymore,

but it satisfies the integral equation (eq:eqcl23.33).

In applications the shock solution propagates the discontinuity across a smooth curve, letssay x = s(t). Let us also assume that u is continuous on each side of the curve s(t) and letu0 and u1 denote the right and left limits of u at s(t), respectively, i.e.

ur = limx→s(t)+

u(x, t), ul = limx→s(t)−

u(x, t).

From the conservation law (eq:eqcl23.33) we split the integral

∫ b

ainto the sum of two integrals∫ s(t)

a+∫ b

s(t)which can be written as:

F (u(a, t))− F (u(b, t)) =d

dt

∫ s(t)

a

u(x, t)dx+d

dt

∫ b

s(t)

u(x, t)dx . (3.36)

Recall that in order to differentiate integrals with variable endpoints like the integrals in theprevious relation we use the Leibniz rule:

d

dt

(∫ b(t)

a(t)

f(x, t)dx

)=

∫ b(t)

a(t)

ft(x, t)dx+ f(b(t), t) · b′(t)− f(a(t), t) · a′(t) , (3.37) eq:leibniz1

which in our case gives:

F (u(a, t))− F (u(b, t)) =d

dt

∫ s(t)

a

u(x, t)dx+d

dt

∫ b

s(t)

u(x, t)dx

=

∫ s(t)

a

ut(x, t)dx+ u(s−, t)ds

dt+

∫ b

s(t)

ut(x, t)dx− u(s+, t)ds

dt

=

∫ s(t)

a

ut(x, t)dx+ ulds

dt+

∫ b

s(t)

ut(x, t)dx− urds

dt.

To eliminate the integrals we take the limits a→ s(t)− and b→ s(t)+. Therefore

F (ul)− F (ur) = (ul − ur)ds

dt. (3.38) eq:ds

35

fig:sh

s(t)a bx

t

t

x = s(t)

Figure 9: Breaking time

The last relation is known as the Rankine–Hugoniot jump condition and relates the values ofthe solution u and the flux function F in front of and behind the discontinuity to the speedof propagation ds/dt of the discontinuity. Thus the integral form of the conservation lawprovides a restriction on possible jumps across a simple discontinuity. Equation (

eq:ds3.38) can

also be written in a compact form as

[F (u)] = [u]ds

dt,

where [·] denotes the jump in a quantity across the discontinuity.

Example 3.4. Describe the solution of the initial value problem

ut +

(1

2u2)

x

= 0, x ∈ R, t > 0,

u(x, 0) = ϕ(x) =

2, x < 02− x 0 ≤ x ≤ 11, x > 1

,

with the method of characteristic lines.

fig:char2 Proof. The characteristic diagram is shown in Fig.fig:sols6. The lines starting from the x axis have

inclination c(ϕ(ξ)) = ϕ(ξ). We have already seen that tb = 1 and that a continuous solutionexists for t < 1. For t > 1 we construct a shock wave starting (2, 1). We first observe thatthe solution takes the values u1 = 2 on the left side of the discontinuity and u0 = 1 on theright. Thus, [u] = 2− 1 = 1 and [F ] = 1

2u21− 1

2u20 =

32. We deduce that the discontinuity has

speeds′(t) = [F ]/[u] = 3/2 .

The resulting characteristic diagram of the solution is shown in Figfig:char210.

36

−1 0 1 2 30

1

2

x

t

Figure 10: Characteristic lines

A solution u that satisfies the jump conditionds

dt=F (ul)− F (ur)

ul − ur,

we say that it satisfies the Lax entropy condition if and only if

F ′(ul) >ds

dt> F ′(ur) .

Obviously, when F is a convex function (i.e. F ′′ > 0) then any shock wave with ul > ursatisfies the entropy condition F ′(ul) > ds/dt > F ′(ur). Physically acceptable (discontinu-ous) solutions usually satisfy the entropy condition and therefore we choose to accept onlysolutions with this property and therefore we usually follow the procedure described abovein order to derive these solutions.

In the sea we distinguish three different wave-breaking types: Surging (no breaking), plung-ing, spilling and collapsing. Plunging is the case that all of us have in mind when we arethinking of wave breaking. The wave crest curls over the front and fall with a splashingphenomenon. On the contrary during spilling the top of the wave crest becomes unstablefirst and flows down the wave. Finally collapsing is observe when the wave becomes verysteep and collapses but without observing the rolling effect of the plunging case. In otherwords collapsing characteristics are between those of plunging and surging types.

3.5 Rarefaction waves

Sometimes, in nonlinear conservation laws there are cases where there are regions in thecharacteristic diagram with no characteristics. For these regions, the method of character-istics will be modified to form expanding waves known as rarefaction waves. For exampleconsider the equation:

37

Figure 11: Characteristic lines fig:charare1

ut + uux = 0, x ∈ R, t > 0 , (3.39) eq:raref1

with initial condition given in the form of

u(x, 0) = ϕ(x) =

0, x ≤ 01, x > 0

.

In this case the characteristic speed is c(u) = u and the characteristics x = c(ϕ(x))t+ ξ are:

x = 0 · t+ ξ, if ξ ≤ 0

x = 1 · t+ ξ, if ξ > 0 .

and are given in Figurefig:charare111 The gap in the characteristics diagram can be filled with a “fan

of characteristics” which is known to as the rarefaction fan. Since these lines will start from0 the characteristics will have the form x = ct whose speed c will take values from 0 to 1. Afunction u(x, t) which is constant along each of these inserted characteristics can have theform

u(x, t) = f(x/t) ,

where f is an unknown function. To find the solution f we substitute into the equation andwe get:

− x

t2f ′(x/t) + f(x/t) · 1

t· f ′(x/t) = 0 ,

which is1

tf ′(x/t)

(f(x/t)− x

t

)= 0 .

This means that either f ′ = 0 or f(x/t) = x/t. Since the constant solution has no physicalmeaning in this case we discard the first choice. It is noted that the gap is actually describedas the wedge-shaped region where 0 < x < t. Now using the method of characteristics forx < 0 and x > 1 we get the solution

u(x, t) =

0, x ≤ 0x/t, 0 < x ≤ t1, t < x

.

38

−4 0 1 50

1

2

x

t

−4 −3 −2 −1 0 1 2 3 4 5

0

0.5

1

1.5

20

0.5

1

xt

u(x

,t)

Figure 12: Characteristic lines in the case of a rarefaction wave fig:charare2

Figurefig:charare212 shows several time instances of the generation of the rarefaction wave. In general,

rarefaction waves are solution of the form u(x, t) = g((x−x0)/t) where x0 is the center of thecharacteristics’ fan. Although the solution is constant along those characteristic curves, thecharacteristic curves however, are not constructed using the characteristic equation dx/dt =c(u).

3.6 The Riemann problem

In the general case of the Burger’s equation

ut + uux = 0 , (3.40) eq:geq1

where F (u) = 12u2 is a convex function (i.e. F ′′(u) > 0) the problem with initial data

u(x, 0) =

ul, if x < 0 ,ur, if x > 0 ,

is called the Riemann problem of (eq:geq13.40) and we have two cases:

39

• If ul > ur then the solution is a shock wave of the form of Figurefig:sols6 and is given by the

formulau(x, t) =

ul, if x− at < 0 ,ur, if x− at > 0 ,

with a = ds/dt = [F ]/[u].

• If ul < ur then the solution is a rarefaction wave of the form of Figurefig:charare212 and is given

by the formula

u(x, t) =

ul, if x/t < ul ,x/t, if ul < x/t < ur ,ur, if x/t > ur .

We saw that the profile of a rarefaction solution “rarfies” as time increases. This behaviouris the opposite compared to the generation of a shock wave. Note also that the solutionremains continuous for t > 0, but the derivatives ut and ux do not exist along the linesx = 0 and x = t and so u does not satisfy (like the shock solutions) the differential equationut + uux = 0 at these points. These solutions are considered in some sense weak solutionsand they satisfy an integral equation instead. The integral equation is obtained from themodel equation (

eq:geq13.40) after multiplication by appropriate test functions φ(x, t) such that

• φt and φx are continuous, and

• φ(x, t) = 0 for large values of the variable x.

Specifically, weak solutions of (eq:raref13.39) (such as shock and rarefaction waves) satisfy the equa-

tion: ∫ ∞

0

∫ ∞

−∞

(uφt +

1

2u2φx

)dx dt+

∫ ∞

−∞u(x, 0)φ(x, 0) dx = 0 .

3.7 Quasi-linear equations

In some physical problems the modeling equations are quasi-linear equations of the followingform

a(x, t, u)ut + b(x, t, u)ux = c(x, t, u) t > 0, x ∈ R , (3.41) eq:quasi1

with initial data u(x, 0) = ϕ(x), where a, b, c are appropriate smooth functions. To obtain ageneral solution to this equation we rewrite the equation in a vector form:

(a, b, c) · (ut, ux,−1) = 0 , (3.42) eq:quasi2

indicating that the vector A = (a, b, c) is perpendicular to the vector U = (ut, ux,−1). Thismeans that the vector U = (ut, ux,−1) is the normal vector to the solution surface u =

40

u(x, t). Therefore, the vector A is tangent to the solution surface u = u(x, t). Specifically, ifγ(s) is a path on the solution surface then

du

ds= ut

dt

ds+ ux

dx

ds, (3.43) eq:quasii

which can be written in the form:

(ut, ux,−1) · (dt/ds, dx/ds, du/ds) = 0 . (3.44) eq:quasiib

Although (ts, xs, us) is tangent to the solution surface, it can be different from the vector A.However, we choose the path γ such that it has tangents equal to A. So, combining (

eq:quasi23.42)

and (eq:quasiib3.44) we get

dt/ds

a(x, t, u)=

dx/ds

b(x, t, u)=

du/ds

c(x, t, u), (3.45) eq:quasiic

where we can eliminate the parameter s and obtain the system:

dt

a(x, t, u)=

dx

b(x, t, u)=

du

c(x, t, u), (3.46) eq:quasi4

subject to the initial conditions

x(0) = ξ, t = 0, u(x, 0) = ϕ(ξ), ξ ∈ R . (3.47) eq:quasi5

This system is known as the characteristic system. Sometimes it is very difficult to findan explicit form of the solution u. Instead, we can find an algebraic equation f(x, t, u) =constant which is called the first integral. It is also known that if the equations has twoindependent first integrals ψ1(x, t, u), ψ2(x, t, u), then the general solution is given by theequation H(ψ1, ψ2) = 0. In general, if we the equation has two independent first integrals,ψ1 = c1 and ψ2 = c2, then the parameters c1 and c2 are given by

c1 = ψ1(ξ, 0, ϕ(ξ)), c2 = ψ2(ξ, 0, ϕ(ξ)) .

Eliminating the parameter ξ from these equations, we obtain the general solution in the formH(ψ1, ψ2) = 0.

Example 3.5. Solve the quasi-linear equation

(x+ u)ut + xux = t− x ,

given that the solution satisfies the initial condition u(x, 0) = 1 + x.

Proof. The characteristic system is:

dt

x+ u=dx

x=

du

t− x. (3.48) eq:exq1

41

Adding the first and the last fraction we get the differential equation

d(t+ u)

t+ u=dx

x,

that has solution the first integral

ψ1(x, t, u) =t+ u

x. (3.49) eq:exq2

Similarly, subtracting the first two fractions of (eq:exq13.48) we obtain the ordinary differential

equationd(t− x)

u=

du

t− x,

which has the first integralψ2(x, t, u) = (t− x)2 − u2 . (3.50) eq:exq3

Using the initial condition now we have that

ψ1 =1 + ξ

ξ, ψ2 = −1− 2ξ .

Eliminating the parameter ξ we obtain that

H(ψ1, ψ2) = 0 ,

withH(ψ1, ψ2) = ψ1 −

ψ2 − 1

ψ2 + 1.

Using the formulas (eq:exq23.49), (

eq:exq33.50) we obtain the general solution in the implicit form:

t+ u

x− (t− x)2 − u2 − 1

(t− x)2 − u2 + 1= 0 .

42

Examples... show how difficultit often is for an experimenterto interpret his results withoutthe aid of mathematics

— Lord Rayleigh

4 Inviscid Fluid Flowsec:invicid

An inviscid flow is the flow of an ideal fluid that is assumed to have no viscosity. In fluiddynamics there are problems that are easily solved by using the simplifying assumption of aninviscid flow. The flow of fluids with low values of viscosity agree closely with inviscid floweverywhere except close to the fluid boundary where the boundary layer plays a significantrole. In this Section we describe the equations of the motion for an inviscid fluid.

4.1 Introduction to Kinematicssec:kinematics

Kinematics is the branch of classical mechanics that describes the motion of bodies andsystems of bodies without consideration of the causes of the motion. In this section wepresent the basic tools for studying the motion of a continuous body moving in Rd withd = 1, 2 or 3. For further reading we suggest the books

ChoMar, Hunter2006, Logan2006, Paterson1983, Whitham1999[CM00, Hun06, Log06, Pat83, Whi99]

on which these notes are based on.

Let h be (the label of) a particle of a continuous body that at t = 0 occupies the regionP0. By a fluid motion we mean a mapping ϕ : P0 → Pt, which maps the region P0 intothe region Pt = ϕ(P0) which is occupied by the same fluid at time t. We assume that ϕ isrepresented by the formula

x = X (h, t) , (4.1) eq:eulerian

with X (h, 0) = h. h is the Lagrangian coordinates or the particle’s label at t = 0 while x isthe Eulerian coordinate representing the position of the same particle h at time t. Usually,the mapping X : Rd ×R → Rd, is also referred to as the particle path. We assume that X isa diffeomorphism of Rd, i.e. is smooth and invertible where the derivative DhX =

(∂hj

Xi

)ij

is a nonsingular matrix. Roughly speaking, the last condition means that the motion doesnot crush a nonzero material volume to zero volume.

The invertibility of X guarantees that (eq:eulerian4.1) can be solved in terms of h, i.e.

h = Y (x, t) . (4.2) eq:langrcoo

43

Usually, instead of using the symbols X and Y we write

x = x(h, t) and h = h(x, t) . (4.3) eq:coordsel

The region Pt occupied by the material particles h at time t can be described also as thebounded set with smooth boundary such that

Pt = X (h, t) : h ∈ P0.

For a given fluid motion the velocity is defined by U (h, t) = Xt(h, t) . Then the correspondingspatial velocity u(x, t) is defined by

u(X (h, t)

)= U (h, t).

Conversely, given a smooth spatial velocity u(x, t), we may reconstruct the motion of X (h, t)by solving the system of ODEs

Xt(h, t) = u(X (h, t), t

),

with X (h, 0) = h as initial condition. The fact that we can use either the Eulerian or theLagrangian description of a quantity in an equivalent way is known as the duality principle.

Assume that we take measurements for the material volume at time t. Let a measurementf be a function of the spatial coordinates (x, t), then the corresponding measurement F ofthe material coordinates is a function of (h, t) and they are connected by the formula

F (h, t) = f(X (h, t), t

).

The rate of change of f at a given spatial point is given by the derivative ft (or ∂tf), whilethe rate of change of f following a particle path is given by Ft. The last derivative is calledthe material derivative of f and is usually denoted by Df/Dt, and

Ft(h, t) =Df

Dt(X (h, t), t) . (4.4) eq:mathder1

In order to understand better the importance of the material derivative consider the flowof the water in a river with converging sides. If we measure the fluids velocity at a specificpoint we will observe that it is constant. On the other hand if we follow a particle then wewill observe that the particle’s velocity increases with time. The particle’s acceleration canbe measured by means of the material derivative. Using the chain rule in (

eq:mathder14.4),

Df

Dt(X (h, t), t) = ft(X (h, t), t) + Xt(h, t) · ∇f(X (h, t), t)

= ft(X (h, t), t) + u(Xt(h, t), t) · ∇f(X (h, t), t) .

44

The last relationship implies the compact formD

Dt=

∂

∂t+ u · ∇ ,

where ∇f(x, t) = (fx1 , · · · , fxd) is the vector of the spatial partial derivatives and u · v =∑d

i=1 uivi is the usual inner product of two vectors. In general, material derivative is used todescribe the time rate of change for a given particle. Therefore, the acceleration in Euleriancoordinates can be written in one dimension as as

a =Du

Dt= ut + uux ,

where the first term in the sum is the temporal acceleration and the second term is theadvective acceleration due to the transition of the particle to a new position with differentvelocity. We continue with two examples:Example 4.1. Consider a one-dimensional fluid flow described by the function

x(h, t) = (1 + t3)h , (4.5)

where h is a particle 0 ≤ h ≤ 1. Using Eulerian and Lagrangian coordinates compute thevelocity and the acceleration of the fluid.

Proof. defining the position of the particle h at every time t. Then solving for h we have

h(x, t) = x/(1 + t3) . (4.6) eq:exlangr1

The velocity of a particle a is defined as

U(h, t) = xt(h, t) = 3t2h ,

which is actually the velocity in Langrangian coordinates. To transform this relation intoEulerian coordinates we substitute a from (

eq:exlangr14.6) to obtain

u(x, t) = U(h(x, t), t) = 3t2x

1 + t3.

The Eulerian acceleration can be computed using the material derivative

a(x, t) =Du

Dt= ut(x, t) + u(x, t)ux(x, t)

=−3tx(t3 − 2)

(1 + t3)2+

3t2x

1 + t33t2

1 + t3

=6tx

t3 + 1,

or even simpler by using Lagrangian coordinates A(h, t) = Ut(h, t) = 6th and then substituteh(x, t).

a(x, t) = A(h(x, t), t) =6tx

1 + t3.

45

Example 4.2. Consider the fluid motion described by the function x = X (h, t), t > 0 where

X (h, t) = (t2 + h1, h2et, th1 + h3) ,

and find the velocity of the fluid.

Proof. Here x = (x1, x2, x3) with

x1 = t2 + h1, x2 = h2et, x3 = th1 + h3 .

Inverting we have

h1 = x1 − t2, h2 = x2e−t, h3 = x3 − tx1 + t2 .

ThereforeU =

(∂x1∂t

,∂x2∂t

,∂x3∂t

)= (2t, h2e

t, h1) ,

andu = U (h(x, t), t) = (2t, x2, x1 − t) .

Another kinematic quantity, very useful when we change variables in integrals (from Eulerianto Langrangian and vice versa) is the Jacobian. The Jacobian J is defined as the determinantof the matrix matrix J (h, t) = det[Jij] with Jij =

∂xi

∂hj,

J = det

∂x1

∂h1· · · ∂x1

∂hn... . . . ...∂xn

∂h1· · · ∂xn

∂hn

.

In Lagrangian form the Jacobian satisfies the relation

Jt(h, t) = J (h, t)∇ · u(x, t)∣∣∣h, (4.7)

and in Eulerian formDj

Dt(x, t) = j(x, t)∇ · u(x, t) , (4.8)

where j(x, t) = J (h(x, t), t). For simplicity we give the proof of these relations for a one-dimensional fluid flow. In one dimension the Jacobian is reduced to

J(h, t) = xh(h, t) . (4.9)

Hence

Jt(h, t) =∂

∂hxt(h, t) = Uh(h, t)

= ux(x(h, t), t)xh(h, t) ,

46

or betterJt(h, t) = ux(x(h, t), t)J(h, t) . (4.10)

To obtain the Eulerian form of the previous relation we let h = h(x, t), so

Dj

Dt(x, t) = ux(x, t)j(x, t) , (4.11)

where j(x, t) = J(h(x, t), t). For the proof of the general case we refer toChoMar[CM00].The relation

Jt = J∇ · u is also useful in understanding the incompressibility of a fluid flow. Specifically,a fluid flow is called incompressible if any fluid volume Pt remains constant with time, i.e.

d

dt

∫Pt

dx = 0 .

After changing to Langrangian coordinates the previous integral gives:

0 =d

dt

∫P0

J dh

=

∫P0

Jt dh

=

∫P0

J∇ · u dh

=

∫Pt

∇ · u dx .

Since ∫Pt

∇ · u dx = 0

for all regions Pt, then the fluid is incompressible either when ∇ · u = 0 or when J = 1.

Finally, we define the vorticity as the curl of the velocity field, i.e. ω = ∇× u.

We close this Section with the Reynold’s transport theorem. Reynold’s transport theoremcan be thought of as a generalization of the Leibniz rule

d

dt

(∫ b(t)

a(t)

f(x, t)dx

)=

∫ b(t)

a(t)

ft(x, t)dx+ f(b(t), t) · b′(t)− f(a(t), t) · a′(t) , (4.12) eq:leibniz

for differentiating one dimensional integrals with variable endpoints.

Before presenting Reynold’s theorem we prove the convection theorem.

Theorem 4.3 (The convection theorem). Let f = f(x, t) be a continuously differentiablefunction. Then

d

dt

∫Pt

f dx =

∫Pt

Df

Dt+ f∇ · v dx . (4.13) eq:convextheor

47

Proof. In order to proof (eq:convextheor4.13) we use Langrangian coordinates x = x(h, t). Then, we

transform the integral using the Jacobian dx = J (h, t)dh into:∫Pt

f dx =

∫P0

F (h, t)J (h, t) dh .

Now the integral instead of being over the time dependent region Pt has been transformed toan integral over the time independent region P0. So the temporal differentiation is simpler:

d

dt

∫Pt

f dx =

∫P0

∂

∂t(FJ) dh

=

∫P0

(FtJ + FJt) dh

=

∫P0

(FtJ + F∇ · uJ) dh

=

∫P0

(Ft + F∇ · u)J dh

=

∫Pt

(Df

Dt+ f∇ · u

)dx ,

where in the last step we transform the integral back to Eulerian coordinates and we used(eq:mathder14.4).

Specifically, Reynold’s transport theorem can be stated as:

Theorem 4.4 (The Reynold’s transport theorem). Let f = f(x, t) be a continuously differ-entiable function. Then, we have

d

dt

∫Pt

f dx =

∫Pt

ft +∇ · (fu) dx . (4.14) eq:reynolds1

After using the divergence theorem it can be written as

d

dt

∫Pt

f dx =

∫Pt

ft dx+

∫∂Pt

fu · n dS . (4.15) eq:reynolds2

where n is the outer unit normal to the surface ∂Pt.

Proof. The proof of Reynold’s transport theorem is now straight forward: Equation (eq:reynolds14.14)

occurs from (eq:convextheor4.13) after substituting the integrant on the right-hand side by

Df

Dt+ f∇ · u = ft +∇ · fu .

48

Thus, ∫Pt

(Df

Dt+ f∇ · u

)dx =

∫Pt

ft +∇ · (fu) dx .

Now we have all the necessary tools to derive Euler’s equations.

4.2 Derivation of the Euler equations

The Euler equations consist of a set of physical conservation laws such as the conservation ofmass and momentum, together with the assumption that the density of the fluid is constant,cf.

Whitham1999[Whi99].

4.2.1 Mass conservation

First we derive the conservation of mass. We consider a fluid of density ρ(x, t). The masscontained in a material volume Pt is then given by∫

Pt

ρ dx.

If the mass of the material volume does not change with time, thend

dt

∫Pt

ρ dx = 0.

Hence, using (eq:reynolds14.14), we get that ∫

Pt

ρt +∇ · (ρu) dx = 0.

Since this equality holds for all t ≥ 0 and for an arbitrary smooth region Pt, we concludethat

ρt +∇ · (ρu) = 0. (4.16) eq:mass1

In fluid mechanics, a fluid (or material) with constant density is characterised as incompress-ible. The same with a flow in which the fluid’s density remains constant. When the flow isincompressible it is not always true that the fluid itself is incompressible.

Assuming that the medium is homogeneous and incompressible, i.e. that the density of thefluid is constant throughout, the conservation of mass reduces to the equation

∇ · u = 0. (4.17) eq:mass

49

The last equation is true for both incompressible flow and fluid and it is known to as theincompressibility condition.

Liquids are nearly incompressible fluids that retain approximately a constant volume inde-pendent of pressure. Therefore, water will be assumed incompressible. On the other handair is consider a compressible gas.

4.2.2 Momentum conservation

We proceed with the derivation of the equation of conservation of momentum. The totalmomentum of a material volume Pt is ∫

Pt

ρu dx.

Newton’s second law states that the rate of change of the momentum of a material volumeis equal to the force acting on it. Taking into account that the only forces acting on thematerial volume is the surface pressure force p that acts in the inward normal direction andthe gravity force F = −ρgk, where k is the unit vector perpendicular to the horizontal plane.Thus

d

dt

∫Pt

ρu dx = −∫∂Pt

pk dS +

∫Pt

F dx.

Using (eq:reynolds14.14), and the divergence theorem, we find that∫

Pt

(ρu)t +∇ · (ρu⊗ u) +∇p− F

dx = 0,

where the tensor product u⊗ u = (uiuj)ij and therefore∫Pt

∂

∂t(ρui) +

d∑j=1

∂

∂xj(ρuiuj) +

∂p

∂xi− Fi

dx = 0.

Since this equation holds for any t ≥ 0 and for any arbitrary smooth region Pt, we concludethat

(ρu)t +∇ · (ρu⊗ u) +∇p = F . (4.18) eq:momentum1

Assuming that the density of the fluid is constant, i.e. ρt = |∇ρ|2 = 0, and using (eq:mass4.17), the

conservation of momentum (eq:momentum14.18) reduces to

ut + (u · ∇) u+1

ρ∇p = −gk. (4.19) eq:momentum

50

4.2.3 Vorticity

We close the derivation of the Euler equations by studying the vorticity of the velocity field.The vorticity is defined as ω = ∇ × u. Taking curl on both sides of the conservation ofmomentum (

eq:momentum4.19) we have

ωt + u · ∇ω = ω · ∇u. (4.20) eq:vorticity

If d = 2 then ω = ∂v∂x

− ∂u∂y

, and ω · ∇u = 0, so (eq:vorticity4.20) reduces to the transport equation

ωt + u · ∇ω = 0. (4.21) eq:vorticity1

From equations (eq:vorticity4.20) and (

eq:vorticity14.21) we conclude that if the flow is irrotational initialy (i.e.

∇ × u = 0 for t = 0) then the flow remains irrotational for all time t. The irrotationalitycondition is formulated as

∇× u = 0 . (4.22) eq:irrotationality

One main consequence of the irrotationality condition is that the momentum equation (eq:momentum4.19)

can be written asut +

1

2∇|u|2 + 1

ρ∇p = −gk. (4.23) eq:momentum2

In the three-dimensional case the vorticity is

ω = ∇× u

=

∣∣∣∣∣∣i j k∂x ∂y ∂zu v w

∣∣∣∣∣∣= (wy − vz, uz − wx, vx − uy)

T . (4.24) eq:curl

Now using the vector identity

a× (b× c) = (a · b)c− b(a · c) ,

with a = c = u and b = ∇ one may derive the following very useful identity:12∇|u|2 = (u · ∇)u+ u× (∇× u) . (4.25) eq:vortid

Using (eq:vortid4.25) we can derive (

eq:momentum24.23) for three-dimensional irrotational flows as well.

4.2.4 Boundary and initial conditions

In the context of water waves, we assume that we want to model the ocean. The fluid in theocean is assumed to be bounded above by air, so the surface of the fluid is free, and at thesame time we assume that the fluid is bounded bellow by an impenetrable bottom.

51

So the Euler equations consist of the conservation of mass (eq:mass4.17), conservation of momentum

(eq:momentum4.19) and the irrotationality condition (

eq:irrotationality4.22) defined on a domain bounded above from

the free surface η(x, t) and bellow by the bottom described with the function d(x). (Herex = (x, y)). To close the system of these conservation laws we need to impose two boundaryconditions and two initial conditions. There are two kinds of boundary condition for thefree surface; the kinematic and the dynamic boundary conditions. The kinematic boundarycondition on a free surface states that the surface of the water is impermeable, and thus thefluid velocity u satisfies

u · n = V · n,

on the free surface, where n is the unit normal to the surface and V is the velocity tangentto the surface. Assuming that the free surface has the equation F (x, t) = 0 and that it issmooth we have that the unit normal and the normal velocity are given in terms of F as

n =∇F|∇F |

, V = − Ft

|∇F |n.

Using these expressions, the kinematic boundary condition becomes:

DF

Dt= 0, on F = 0,

meaning that the material particles on the surface remain on the surface. For a surface thatis a graph with equation z = η(x, t) then the kinematic condition can be written as

ηt + u · ∇(−z + η) = 0, on z = η(x, t) , (4.26) eq:kinematic

which is actually the relation:

ηt + uηx + vηy − w = 0, on z = η(x, t) , (4.27) eq:kinematicb

where now x denotes only the horizontal independent variable and z the vertical.

The dynamic boundary condition on the free surface is that the stresses on either side ofthe surface are equal. In the case of an air-water interface, we neglect the motion of theair, because of its smaller density, and assume that the atmospheric pressure is constantp = patm. Because the momentum equation depends only on the ∇p, making the change ofvariables p 7→ p− patm, we can take p = 0 at the free surface.

The boundary condition on the bottom is the impermeability condition:

u · ∇(z + d) = 0, on z = −d(x) ,

which means that there is no normal velocirt through the bottom.

52

4.3 The Navier-Stokes equations

The Navier-Stokes equations extend the Euler equation by incorporating viscous effects.Viscosity measures the dissipation of the momentum across fluid’s volume. In the derivationof the momentum conservation equation of the inviscid Euler equations we assumed thatthe only force acting on the material volume was the surface pressure p acting in the inwardnormal direction and the gravity force. One the other hand there are cases where otherforces (internal forces due to viscosity for example) must be taken into account. In such caseone has to assume that the force on the surface per unit area is something like −pn+ σ · nwhere σ is a matrix called the stress tensor. Then, Newton’s second law, i.e. the balance ofmomentum, is written as:

d

dt

∫Pt

ρu dx = −∫∂Pt

(p− σ)k dS +

∫Pt

F dx . (4.28) eq:visc1

Most fluids obey a linear Newtonian stress-strain relationship, in the sense that the forceacting to the fluid’s particles depends linearly to the velocity gradients. (Contrary to solidswhere the stresses are are proportional to the changes of shape). Therefore, the choice ofthe viscous stress tensor σ for a Newtonian fluid, i.e. a fluid where σ depends linearly onthe rate of deformation D of the fluid by its motion with Dij =

12(∂ui/∂xj + ∂uj/∂xi). The

tensor D can also be written with the help of the Jacobian matrix as D = 12(∇u +∇u T ),

where ∇u denotes the Jacobian matrix of u:

∇u =

(ux uyvx vy

).

Specifically, σ is taken to beσ = λ∇ · uI + 2µD , (4.29) eq:visc2

where λ and µ are the coefficients of viscosity and may depend on temperature or density,but here we assume they are constant. This equation for the stress tensor is also knownto as the Stokes’ stress constitutive equation. Then, substitution into (

eq:visc14.28) leads to the

momentum conservation equation

(ρu)t +∇ · (ρu⊗ u) +∇ (p− (λ+ µ)∇ · u)− µ∆u = F . (4.30) eq:viscmom1

In order to get the term µ∆u one need to compute the divergence of the tensor D, which isdefined as the vector with entries:

[∇ ·D]i =∂Di1

∂x+∂Di2

∂y.

Using the incompressibility condition ∇ · u = 0 we have that ∇ ·D = 12∆u.

53

In the incompressible case where ∇ · u = 0, (eq:viscmom14.30) also reduces to

(ρu)t +∇ · (ρu⊗ u) +∇p− µ∆u = F . (4.31) eq:viscmom2

The momentum conservation equation (eq:viscmom24.31) along with the mass conservation are known

as the Navier-Stokes equations.

4.4 Non-dimensionalization and normalization of the Euler equa-tions

For simplicity, we will restrict the analysis only in the case d = 2. In this case, the Eulerequations for inviscid, incompressible and irrotational flow with a free surface over a hori-zontal bottom at height z = −d0 (the undisturbed level of fluid is at z = 0) may be writtenin dimensional and unscaled variables as follows: Let η(x, t) be the deviation of the freesurface of the fluid above its level of rest and consider the domain Ωt = (x, z) : −∞ < x <∞, −d0 ≤ z ≤ η(x, t). Then for (x, z) ∈ Ωt and t ≥ 0 we have

ut + uux + vuz +1

ρpx = 0, (4.32) E13

vt + uvx + vvz +1

ρpz = −g, (4.33) E14

ux + vz = 0, (4.34) E15

uz = vx, (4.35) E16

where g is the acceleration of gravity, u = u(x, z, t), respectively v = v(x, z, t), denotes thehorizontal, respectively the vertical, velocity component, ρ is the (constant) density of thefluid, and p = p(x, z, t) the pressure. The system (

E134.32)–(

E164.35) is supplemented by the free

surface kinematic and dynamic boundary conditions

ηt + uηx = v, at z = η(x, t), (4.36) E17

p = 0, at z = η(x, t). (4.37) E18

At the bottom z = −d0 we assume that the normal component of the velocity vanishes, i.e.that

v = 0, at z = −d0, (4.38) E19

We also assume that initial conditions for η and u have been specified and let

η(x, 0) = η0(x), x ∈ R, (4.39) E20

u(x, z, 0) = u0(x, z), (x, z) ∈ Ω0, (4.40) E21

54

where η0 and u0 are given real functions. (We assume that u0 satisfies the compatibilitycondition ∂u0

∂z(x, z) = −

∫ z

d0

∂2u0

∂x2 (x, z′) dz′ in Ω0, which follows assuming that (

E154.34), (

E164.35)

and (E194.38) hold at t = 0).

Water waves are usually characterized by their amplitude and their wavelength comparedto the water depth. We assume that a characteristic wavelength is λ, and a characteristicamplitude A. The so called shallow water condition states that the water depth d0 is muchsmaller compared to the wavelength i.e. σ = d0/λ ≪ 1. We define also another importantparameter ε = A/d0. As it will be more clear also later in these notes, ε measures thelocal size of nonlinear effects, while σ measures the local size of dispersive effects. Usually,waves in the deep water are characterized by the small amplitude assumption since ε ≪ 1but as the waves approach the shoreline they gain amplitude, i.e. ε = O(1) and finallythey break σ2 ≪ ε. The smallness of σ is used to derive simple model equations that do notdepend on the vertical coordinate y. It is also common to divide water waves into three maincategories depending on the value of the quotient S = ε/σ2 known as the Stokes number(sometimes also referred to as Ursel number). Stokes number is a measure of the relativestrength of nonlinear and dispersive effects. For example, nonlinear but non-dispersive wavesare characterized by σ2 ≪ ε or S ≫ 1, while for weakly nonlinear and weakly dispersiveσ2 ≪ 1 and σ2 = O(ε), i.e. S = O(1), and finally we speak about strongly nonlinear butweakly dispersive waves when σ2 ≪ 1 but ε = O(1) and S > 1.

To derive simple mathematical models from the Euler equations the first step is to introducenon-dimensional independent variables that are defined as follows:

x∗ =x

λ, z∗ =

z

d0, and t∗ =

σg

c0t. (4.41) E22

We know from the linear theory that the horizontal velocity u(x, y, t) has the following orderof magnitude: u = O(εc0), where c0 =

√gd0 is the linear wave phase velocity. Then the

nondimensional horizontal velocity is:

u∗ =u

ε c0. (4.42) E23

In the case of long waves the motion of the fluid is essentially horizontal, i.e. the verticalcomponent of the velocity is usually very small. That is v = O(σεc0), thus we define

v∗ =v

σεc0, (4.43) E24

also the pressure is scaled by the static pressure, p = O(ρgd0), and thus

p∗ =p

ρgd0. (4.44) E25

Finally, since a typical amplitude of a wave is considered in this analysis to be A, i.e.η = O(A), the nondimensional free surface elevation is given

η∗ =1

εd0η. (4.45) E25b

55

Then, the problem (E134.32)–(

E214.40) is transformed into the equivalent problem that follows, in

which the dependent variables and the initial conditions are of order one, while powers ofε and σ signify the order of magnitude of terms they multiply. We seek u∗ = u∗(x∗, z∗, t∗),v∗ = v∗(x∗, z∗, t∗), p∗ = p∗(x∗, z∗, t∗), η∗ = η∗(x∗, t∗), defined for −∞ < x∗ < ∞, −1 ≤ z∗ ≤εη∗(x∗, t∗), t∗ ≥ 0, such that,

εu∗t∗ + ε2u∗u∗x∗ + ε2v∗u∗z∗ + p∗x∗ = 0, (4.46) E26

εσ2v∗t∗ + ε2σ2u∗v∗x∗ + ε2σ2v∗v∗z∗ + p∗z∗ = −1, (4.47) E27

u∗x∗ + v∗z∗ = 0, (4.48) E28

u∗z∗ − σ2v∗x∗ = 0. (4.49) E29

This system is supplemented with the free surface and bottom boundary conditions, whichnow take the form:

η∗t∗ + εu∗η∗x∗ = v∗, for z∗ = εη∗(x∗, t∗), (4.50) E30

p∗ = 0, for z∗ = εη∗(x∗, t∗), (4.51) E31

v∗ = 0, for z∗ = −1, (4.52) E32

and the initial conditions

η∗(x∗, 0) = η∗0(x∗), u∗(x∗, z∗, 0) = u∗0(x

∗, z∗), (4.53) E33

where, in terms of the functions η0, u0 in (E204.39)–(

E214.40) we have

η∗0(x∗) :=

1

εd0η0

(d0σx∗), u∗0(x

∗, z∗) :=1

εc0u0

(d0σx∗, d0z

∗). (4.54) E34

To derive model equations we will follow the great lines ofBarthelemy2004, BC1998, BCS2002, BS, DM2008, Whitham1999[Bar04, BC98, BCS02, BS76,

DM08, Whi99]. Moreover, in what follows we simplify the notation by dropping the ∗.

We leave the non-dimensionalization and scaling of the three-dimensional Euler equationsas an exercise.

4.5 Non-dimensionalization and normalization of the Navier-Stokesequations

In this section we consider the Navier-Stokes equations for an incompressible fluid of constantdensity under no influence of body forces. These equations can be written in the form:

∇ · u = 0 , (4.55) eq:nvs1

56

andut + (u · ∇)u = −1

ρ∇p+ µ∆u . (4.56) eq:nvs2

Considering a characteristic length L, characteristic speed U and characteristic time to beL/U then using the dimensionless quantities

u ∗ =u

U, x ∗ =

x

L, t∗ =

t

L/U, (4.57)

the incompressible momentum equation takes the form

u ∗t + (u ∗ · ∇)u ∗ = −∇p∗ + µ

LU∆u ∗ , (4.58) eq:nvs3

where p∗ = p/ρU2 and while the incompressibility condition remains the same ∇ · u ∗ = 0.The dimensionless number LU/µ multiplying the viscous term is very important since itdetermines the viscosity of the fluid and is known as the Reynolds number denoted by

Re =LU

µ.

Using this notation we rewrite the Navier-Stokes equations in dimensionless form as:

u ∗t + (u ∗ · ∇)u ∗ = −∇p∗ + 1

Re∆u ∗ . (4.59) eq:nvs4

In this setting the term ∆u is called the diffusion term while the term (u · ∇)u is called theconvective term.

From Equation (eq:nvs44.60) we observe that for small values of the Reynolds number the viscous

effects are important and the Navier-Stokes equations can be simplified to the so calledStokes’ equations where

u ∗t = −∇p∗ + 1

Re∆u ∗ . (4.60) eq:nvs4

contrary to the case of large Reynolds numbers where the effects of the dissipative termare important even if the term is small and therefore the Navier-Stokes equations cannotbe simplified to the Euler equations except for the case where there is no viscosity at all.For example turbulence can be observed for large Reynolds numbers. On the other hand, anon-turbulent flow is called Laminar flow. It is noted that Equation (

eq:nvs44.60) is parabolic and

can be solved quite easily.

57

4.6 Potential flow

We return back to the Euler equations:

ηt + u · ∇η = w, on z = η(x, y, t) , free surface condition (4.61) eq:eu2dai

ut + u · ∇u+ 1

ρ∇p+ gk = 0 , momentum equation (4.62) eq:eu2dbi

∇ · u = 0 , mass equation (incompressibility) (4.63) eq:eu2dciu · ∇(z + d(x, y)) = 0, on z = −d(x, y) . bottom boundary condition (4.64) eq:eu2ddi

Sometimes it is more convenient if instead of using the vector function of the velocity u weuse a scalar function ϕ(x, y, z, t) such that

u = ∇ϕ . (4.65) eq:vpot

The function ϕ is called the velocity potential and it was first introduced by Joseph-LouisLagrange in 1788.

Substituting the (eq:vpot4.65) into the incompressibility condition (

eq:eu2dci4.63) we get that ∇ · ∇ϕ = 0

which is the Laplace equation∆ϕ = 0 . (4.66) eq:laplace

The Laplace’s equation is linear and can be solved in general easily numerically and theo-retically.

Similarly, the momentum conservation equation after substitution of the velocity takes theform:

ϕt +1

2|∇ϕ|2 + gz, for z = η(x, y, t) . (4.67) eq:potmom

Finally, the boundary condition on the free surface and at the bottom become

ηt + ϕxηx + ϕyηy − ϕz = 0, for z = η(x, y, t) , (4.68) eq:codf1

andϕxdx + ϕydy + ϕz = 0, for z = −d(x, y) , (4.69) eq:codf2

or written in compact form:

ηt +∇ϕ · ∇(−z + η) = 0, for z = η(x, y, t) , (4.70) eq:codf1b

and∇ϕ · ∇(z + d) = 0, for z = −d(x, y) , (4.71) eq:codf2b

respectively.

58

It is noted that the velocity potential is not unique. If for example ϕ is the velocity potentialsuch that u = ∇ϕ then the function ψ = ϕ + C is a velocity potential too since it satisfiesthe same equation u = ∇ψ.

Moreover, if the flow is potential, i.e. there exists a velocity potential function ϕ such thatu = ∇ϕ then the flow is irrotational. To see that take

∇× u = ∇×∇ϕ = 0 ,

where the last equality holds for example by (eq:curl4.24). We therefore conclude that if the flow

is potential then it is irrotational too.

4.6.1 The streamline function

In addition to the velocity potential, in two-dimensional flows the velocity can be expressedin terms of a stream function too. We observed that the velocity potential ϕ was chosento satisfy the irrotationality condition. The stream function ψ is selected to satisfy thecontinuity equation. Specifically, choosing u = ∇× ψ, or equivalently

u = ψy, v = −ψx . (4.72) eq:streq1

We observe that again the velocity satisfies the continuity equation:∇ · u = ux + vy = ψyx − ψxy = 0 .

A contour on which the stream function is constant is call a streamline. On the other hand,a contour on which the velocity potential function is constant is called iso-potential line. Itturns out that these two contours are orthogonal at any point in the fluid. To see this wetake the inner product ∇ϕ · ∇ψ which is

∇ϕ · ∇ψ = ϕxψx + ϕyψy = u(−v) + vu = 0 .

From the definition of the streamline and potential velocity functions we have thatu = ϕx = ψy , (4.73)u = ϕy = −ψx . (4.74)

The last two equations are known in Complex analysis as the Cauchy-Riemann equations.Solution of the Cauchy-Riemann equations are called analytic and in general are very smoothfunctions. Taking into account this property we define the complex potential function

F (z) = ϕ(x, y) + ψ(x, y)i , (4.75) eq:cpot

where now z = x+ yi is a complex variable and i =√−1 is the imaginary unit. Taking the

derivative of the complex potential function we define the complex velocity w(z):w(z) = F ′(z) = ϕx + ψxi

= u− vi . (4.76) eq:cvel

59

4.7 Energy conservation

In classical mechanics we have learned that there are two fundamental forms of energy, thepotential and kinetic energy. The potential energy P is due to the position of the mass whilethe kinetic energy K is due to the motion of the mass. Although there might be other formsof energy, such as thermal or internal energy, we will assume that the total energy E of anideal fluid is the sum of the kinetic and potential energy, i.e. E = K + P. In the context ofwater waves, we consider an incompressible and inviscid flow where ∇ · u = 0.

The kinetic and potential energy of the water bounded in a domain D are defined by thefollowing formulas:

K =

∫DK, P =

∫DP , (4.77) eq:energies

where the quantities K = 12ρ|u|2 , P = ρgz denote the kinetic and potential energy densities

(measured in J/m3 in SI).

Multiplying the momentum conservation equation (eq:momentum4.19) by u yields

Kt = u · ∇(K + p) + gw = 0 . (4.78) eq:kine1

From the equation of potential energy we obtain

Pt = u · ∇P − gw = 0 , (4.79) eq:poten1

Adding the previous equations we obtain the conservation law for the total energy

Et + u · ∇(E + p) = 0 , (4.80) eq:total1

and using the continuity equation ∇·u = 0, we obtain the conservation law in its conservativeform:

Et +∇ · (u(E + p)) = 0 . (4.81) eq:total1

Integrating the last equation we obtain the conservation of energy.

For a more detailed study of energy equations for water waves we refer toFKD2014[FKD14].

60

Water is the driving force of allnature.

— Leonardo da Vinci

5 The Shallow Water Equations: A model for tsunamissec:swe

Apart from scalar conservation laws there are also systems of conservation laws. The Eulerequations and the shallow water equations are examples of systems of conservation laws.In real-life applications such as the generation and the propagation of tsunamis the shal-low water equations with variable bathymetry has been widely used. Figure

fig:botex13 shows the

topography of the ocean bottom south of Java. The 17 July 2006, a tsunami struck thesouthern coast of Java causing hundreds of casualties. The triggering earthquake happened225 km off the coast of Panandaran (9.2S, 107.3E) in a region where the indian and thepacific plates meet and form an underwater canyon presented in Figure

fig:botex13. Using the Euler

Figure 13: An example of a variable ocean bottom topography: Underwater trench south ofJava. (Horizontal scale in geographical coordinates; vertical scale in meters). fig:botex

61

equation it appears to be a very difficult task due to the moving domain bounded bellow bythe bottom and above by the free surface of the sea. In order to study such problems weneed to introduce two-dimensional models that approximate the Euler equations.

5.1 Physical derivation of the shallow water equations

To derive the Shallow Water Equations we rewrite the Euler equations of Sectionsec:invicid4.

ηt + u · ∇η = w, on z = η(x, y, t) , free surface condition (5.1) eq:eu2da

ut + u · ∇u+ 1

ρ∇p+ gk = 0 , momentum equation (5.2) eq:eu2db

∇ · u = 0 , mass equation (incompressibility) (5.3) eq:eu2dcu · ∇(z + d(x, y)) = 0, on z = −d(x, y) , bottom boundary condition (5.4) eq:eu2dd

Here, p denotes the pressure, η the free surface elevation around an undisturbed level z = 0,u = (u, v, w)T the velocity of the fluid, ρ the density, g the acceleration due to gravity, andd(x, y) the bottom topography.

Rewriting (eq:eu2db5.2) component-wise we get the momentum equations in the form:

Du

Dt≡ ut + uux + vuy + wuz = −1

ρpx , (5.5) eq:eu2di

Dv

Dt≡ vt + uvx + vvy + wvz = −1

ρpy , (5.6) eq:eu2dii

Dw

Dt≡ wt + uwx + vwy + wwz = −1

ρpz − g . (5.7) eq:eu2diii

In order to simplify the Euler equations we make the long-wave assumption (also known asthe shallow water assumption), i.e. that the length of the waves is much larger compared tothe depth of the fluid. However, we do not assume that the waves are of small amplitudelike in the case of the Boussinesq equations. A rigorous derivation might include the non-dimentionalisation and scaling of the Euler equations but it is left as an exercise. It hasbeen observed that the vertical acceleration of long waves is negligible and thus because ofthe long-wave assumption, we can neglect the vertical acceleration term in the momentumequation (

eq:eu2diii5.7). Specifically, we can take Dw/dt = 0 in (

eq:eu2diii5.7), which is equivalent to the

assumption that the horizontal velocity is constant along z. To see this take Dw/dt = 0 in(eq:eu2diii5.7) and then observe that pz = −ρg, which after integration along the vertical column of

the fluid gives p = ρg(η − z). This is called the hydrostatic pressure relation (see also the

62

Sectionsec:momsw5.1.2). Moreover, differentiation with respect to x and y of the hydrostatic pressure

equation gives thatpx = −ρgηx, py = −ρgηy .

Since η does not depend on z we deduce that the pressure p is independent of z. Therefore,the accelerations given in (

eq:eu2di5.5)–(

eq:eu2dii5.6) by Du/Dt = −1

ρpx and Dv/Dt = −1

ρpy are also

independent of z and thus, the horizontal velocities are constant along z. This is the crucialobservation at a physical level that we make in order to simplify the Euler equations at thisstage.

5.1.1 Mass conservationsec:massw

In order to derive the conservation of mass, we integrate the continuity equation (eq:eu2dc5.3) along

z:

0 =

∫ η

−d

∇ · u dz =∫ η

−d

[ux + vy + wz] dz = [using Leibniz rule (eq:leibniz13.37)]

=∂

∂x

∫ η

−d

u dz − u∣∣z=η

ηx + u∣∣z=−d

(−dx) +∂

∂y

∫ η

−d

v dz − v∣∣z=η

ηy + v∣∣z=−d

(−dy)+

+ w∣∣z=η

− w∣∣z=−d

and using the bottom boundary condition (eq:eu2dd5.4) we get

∂

∂x

∫ η

−d

u dz − u∣∣z=η

ηx +∂

∂y

∫ η

−d

v dz − v∣∣z=η

ηy + w∣∣z=η

= 0 . (5.8) eq:mass0

Applying the free surface boundary condition (eq:eu2da5.1) we have

ηt +∂

∂x

∫ η

−d

u dz +∂

∂y

∫ η

−d

v dz = 0 . (5.9) mass

Since the velocity u is independent of the vertical coordinate z, the conservation of masstakes the final form:

ηt +∇ · [(η + d)u] = 0 , (5.10) eq:cmass

but here u = (u, v)T .

5.1.2 Momentum conservationsec:momsw

Neglecting the term Dw/Dt in (eq:eu2db5.2), we deduce the hydrostatic pressure equation pz = −ρg.

Integrating with respect to z we obtain:∫ η

z

pz dz = −∫ η

z

ρg dz

p(x, y, η, t)− p(x, y, z, t) = −ρg(η − z) .

63

Using the condition of the pressure on the free surface (p(x, y, η, t) = 0 for all t ≥ 0) we getthe hydrostatic pressure equation:

p(x, y, z, t) = ρg(η − z) . (5.11) eq:pres0

Further, assuming that there are no vertical variations in (u, v) the horizontal momentumequations of the shallow water system can be written:

ut + gηx + uux + vuy = 0 , (5.12)vt + gηy + uvx + vvy = 0 . (5.13)

Introducing the notation of the Jacobian matrix

∇u .=

(ux uyvx vy

), (5.14)

the momentum conservation equation can be written in the form:

ut + g∇η + [∇u] u = 0 . (5.15)

Summarising the shallow water equations are the following:

ηt + [(η + d)u]x + [(η + d)v]y = 0 , (5.16) eq:swe1ut + gηx + uux + vuy = 0 , (5.17) eq:swe2vt + gηy + uvx + vvy = 0 . (5.18) eq:swe3

If we restrict the analysis in one horizontal dimension then the equations take the simpleform:

ηt + [(η + d)u]x = 0 , (5.19) eq:swea1ut + gηx + uux = 0 . (5.20) eq:swea2

Both systems are systems of first order partial differential equations. It is noted also thatequations (

eq:swea15.19)–(

eq:swea25.20) are nonlinear equations and therefore we expect to have similar

properties with the scalar nonlinear equations of Chaptersec:laws3. If the waves are very small in

amplitude (and very smooth) then one can further simplify the shallow water equations.

Remark 5.1. Denoting the total depth η+ d by h, also known to as the hydraulic depth, theshallow water equations (

eq:swea15.19)–(

eq:swea25.20) are written in the form

ht + [hu]x = 0 , (5.21) eq:sweab1ut + ghx + uux = gdx . (5.22) eq:sweab2

64

Although system (eq:sweab15.21)–(

eq:sweab25.22) can be written in a conservative form and the first equation

(eq:sweab15.21) expresses the conservation of mass correctly, the second equation expresses the conser-

vation of particles speed u, which does not make sense physically speaking. For this reason weusually prefer to work with other conservative variables such as the mass h and the impulsehu. Specifically, we can re-write system (

eq:sweab15.21)–(

eq:sweab25.22) in the physically correct conservative

form:

ht + (hu)x = 0 , (5.23) eq:shba1(hu)t + (1

2gh2 + hu2)x = 0 . (5.24) eq:shba2

It is worth noting that equations (eq:sweab15.21)–(

eq:sweab25.22) have different shock wave speed compared

to the equations (eq:shba15.23)–(

eq:shba25.24) which gives the physically correct shock wave speed. So in

physical problems we use the conserved quantities that make physical sense and we try towrite our system in terms of the analogous variables.

5.2 Linearization of the shallow water equationssec:linearised

To linearise the shallow water equations, we assume that the solutions are of very smallamplitude such as small perturbations of the undisturbed surface of the water. That is,

η = 0 + ϵη, u = 0 + ϵu, v = 0 + ϵv , (5.25) eq:tiny

where ϵ≪ 1 are very small.

Substituting (eq:tiny5.25) into the shallow water equations (

eq:swe15.16)–(

eq:swe35.18) and neglecting the second-

order terms (i.e. terms of O(ϵ2), we obtain the linearised shallow water equations:

ηt + (ud)x + (ud)y = 0 , (5.26) eq:lsh1ut + gηx = 0 , (5.27) eq:lsh2vt + gηy = 0 . (5.28) eq:lsh3

Multiplying (eq:lsh15.26) by √

g, and both (eq:lsh25.27) and (

eq:lsh35.28) by

√d, we obtain:

(√gη)t + (u

√d ·√gd)x + (v

√d ·√gd)y = 0 , (5.29) eq:lshb1

(u√d)t +

√gd(

√gη)x = 0 , (5.30) eq:lshb2

(v√d)t +

√gd(

√gη)y = 0 . (5.31) eq:lshb3

Eliminating the terms u√d and v

√d from the above equations we obtain the linear wave

equation:ηtt = ∇ · [gd∇η] . (5.32) eq:linwe

65

Remark 5.2. Since the typical depth of the ocean is about 4 km while the wave length of atsunami is about 100 km it is reasonable to use the linearised shallow water equations to studytsunamis propagating across the open sea and away from the shore. The linearised shallowwater equations also predict that the local speed of propagation of a tsunami, in any directionis c(x, y) =

√g d(x, y) with no dispersion, i.e. waves with different amplitude propagate

with the same speed over the same sea-floor.

As a tsunami approaches the shore, the waves grow in amplitude while the wavelength becomessmaller. For this reason the linear approximation is no more valid and the nonlinear equationsmust be used.

The relative importance of linear and nonlinear effects can be measured by the Ursell (orStokes) number S = aλ2/d3, where a is a typical wave amplitude, λ is a typical wavelengthand d is the mean water depth. For S ≫ 1, the nonlinear effects control wave propagationand only nonlinear models are applicable. Ursell

Ursell1953[Urs53] proved that near the wave front S

behaves like S ∼ t1/3. Hence, regardless of how small are the nonlinear effects initially theywill become important. For long waves (λ ≫ d) with small Ursell number, S ≪ 100, linearwave theory is usually valid.

5.3 Solution of the wave equation

We have seen that the linearised shallow water equations can be written in the form of asingle linear equation, the wave equation

ηtt = ∇ · [gd∇η] . (5.33)

Considering the case of flat bottom topography in one space dimension the wave equationis then written in the simple form

ηtt = c2ηxx , (5.34)with c =

√gd. In order to find a unique solution we need two initial conditions, since the

wave equation is a second order equation in time. For this reason we assume that it is giventhat η(x, 0) = f(x) and ηt(x, 0) = g(x). We saw that the solution of this equation for x ∈ Rconsists of a sum of two functions ϕ and ψ that represent waves propagating in differentdirections. Therefore, we search for a solution of the form η(x, t) = ϕ(x − ct) + ψ(x + ct).The solution then satisfy η(x, 0) = ϕ(x)+ψ(x) and ηt(x, 0) = −cϕ′(x)+cψ′(x), and therefore

ϕ(x) + ψ(x) = f(x) , (5.35) eq:weq1−cϕ′(x) + cϕ′(x) = g(x) . (5.36)

Dividing the last equation by c and integrating from 0 to x we obtain

− ϕ(x) + ψ(x) = −ϕ(0) + ψ(0) +1

c

∫ x

0

g(s)ds . (5.37) eq:weq2

66

Solving the system (eq:weq15.35)–(

eq:weq25.37) for ϕ and ψ we obtain

ϕ(x) = 12f(x)− 1

2(−ϕ(0) + ψ(0))− 1

2c

∫ x

0

g(s)ds , (5.38)

ψ(x) = 12f(x) + 1

2(−ϕ(0) + ψ(0)) +

1

2c

∫ x

0

g(s)ds . (5.39)

Finally, the general solution is given

η(x, t) = ϕ(x− ct) + ψ(x+ ct)

= 12f(x− ct)− 1

2(−ϕ(0) + ψ(0))− 1

2c

∫ x−ct

0

g(s)ds

+ 12f(x− ct) + 1

2(−ϕ(0) + ψ(0)) +

1

2c

∫ x+ct

0

g(s)ds

= 12(f(x− ct) + f(x+ ct)) +

1

2c

∫ x+ct

x−ct

g(s)ds .

The resulting for of the solution

η(x, t) = 12(f(x− ct) + f(x+ ct)) +

1

2c

∫ x+ct

x−ct

g(s)ds , (5.40)

is called the d’Alambert solution to the wave equation.

Example 5.3. Verify that the initial-value problem

utt = c2uxx, −∞ < x <∞, t > 0 ,

u(x, 0) = e−x2

,

ut(x, 0) = 0 ,

is given by the formulau(x, t) = 1

2

(e−(x−ct)2 + e−(x+ct)2

). (5.41)

The initial condition evolves into two pulses propagating in different directions. Figurefig:waveqsol114

shows the solution at t = 0, 2 and 4.

Remark 5.4. Since the wave equation is linear if we take two waves that are far away apartone of each other the solution is the sum of those two waves. For example taking

u(x, t) = 12

(e−(x+5)2 + e−(x−5))2

), (5.42)

as initial condition to the wave equation, the two counter-propagating waves will interactlinearly and then separate and propagate in different directions. Figure

fig:waveqsol215 shows the solution

at different time.

67

−10 10−0.2

0

1.5

η

(a) t = 0

−10 10−0.2

1.5

η

(b) t = 2

−10 10−0.2

0

1.5

x

η

(c) t = 4

Figure 14: Solution to the wave equation at different time t. fig:waveqsol1

Although the wave equation is a scalar equation it comes from the linearisation of theshallow water equation and therefore it can be written as a system of two coupled, linearequations. Since both the nonlinear and the linear shallow water equations are first orderpartial differential equations we proceed with an introduction to systems of conservationlaws.

5.4 Systems of conservation laws

In Chaptersec:laws3 we studied equations of the form

ut + [F (u)]x = 0 , (5.43) eq:claw

where u(x, t) was a scalar function and F (u) a function of a single variable. Here we considersystems of equations of the form (

eq:claw5.43). Specifically we consider the system of partial

differential equations written in the form

ut + F (u)x = 0 , (5.44) eq:syslaw1

68

0

(a) t = 0

0

(b) t = 2

0

η(c) t = 5

0

(d) t = 7

−10 10

0

1.5

x

(e) t = 12

Figure 15: Solution to the wave equation at different time t. fig:waveqsol2

whereu = (u1, u2, · · · , um)T , F (u) = (f1, f2, · · · , fm)T . (5.45)

Sometimes it is very useful to rewrite system (eq:syslaw15.49) using the Jacobian matrix:

def:jacm Definition 5.5. The matrix

J(u).=∂F (u)

∂u=

∂f1/∂u1 ∂f1/∂u2 · · · ∂f1/∂um∂f2/∂u1 ∂f2/∂u2 · · · ∂f2/∂um

... ... ...∂fm/∂u1 ∂fm/∂u2 · · · ∂fm/∂um

, (5.46) eq:jacm

is called the Jacobian matrix of the vector flux function F . If we denote the entries of theJacobian matrix with aij then aij = ∂fi/∂uj.

Using the chain rule we observe that∂F (u)

∂x=∂F (u)

∂u· ∂u∂x

(5.47) eq:jachelp

69

or simplerF (u)x = J(u) · ux . (5.48) eq:jachelp2

Substituting (eq:jachelp25.48) into (

eq:syslaw15.49) then the system of conservation laws becomes

ut + J(u) · ux = 0 . (5.49) eq:syslaw1

The eigenvalues and eigenvectors of the Jacobian matrix play an important role in the studyof systems of conservation laws. Recall that:

def:eigs Definition 5.6. If A is an m × m matrix and I is the m × m identity matrix, then theequation

|A− λI| = det(A− λI) = 0 , (5.50) eq:charact

is called the characteristic equation of A. The solutions of the characteristic equation of thematrix A are called eigenvalues of A. The characteristic equation might have several solutionλi, i = 1, 2, · · · ℓ where k might be less or equation to m. A solution of the singular linearsystem (A− λiI)ki = 0 is called the i-th eigenvector ki of A.

In the case of scalar equations the eigenvalue is the solution of the equation f ′(u)− λ = 0.So, for example, if we consider the scalar conservation law ut + f(u)x = 0 with f(u) = 1

2u2

then f ′(u) = u and thus λ = u is the eigenvalue of the Jacobian. In the case of systems thesituation a little bit more difficult.

ex:exmplwave1 Example 5.7. Consider the linearized shallow water system of equation (eq:lsh15.26)–(

eq:lsh25.27) re-

stricted in one spatial dimension (ignore the velocity component v), with constant depthd = const written in the form:

ηt + dux = 0 , (5.51) eq:lshb1ut + gηx = 0 , (5.52) eq:lshb2

where we have dropped the tildes. Because this is a system of linear advection equations weneed two initial conditions, for example it would be adequate to know η(x, 0) = η0(x) andu(x, 0) = u0(x). Other initial conditions could be η(x, 0) and ηx(x, 0) since from the secondwe can solve for u(x, 0). Equations (

eq:lshb15.51)–(

eq:lshb15.51) can take the matrix form:

ut + F (u)x = 0 , (5.53) eq:vecformsw

with u = [η, u]T and F (u) = [du, gη]T . The Jacobian of F is

J(u) =

(0 dg 0

). (5.54)

The characteristic equation of J is

det(J − λI) =

∣∣∣∣−λ dg −λ

∣∣∣∣= λ2 − gd = 0 . (5.55) eq:charlw1

70

The solutions of the characteristic equation (eq:charlw15.55) are the real eigenvalues

λ1 =√gd, λ2 = −

√gd . (5.56) eq:reigslw

In order to find the eigenvector k1 = (x1, x2)T of J we write the system

(J − λ1I)k1 = 0 , (5.57)

which is equivalent with the equation

−√gdx1 + dx2 = 0 ,

gx1 −√gdx2 = 0 .

Multiplying the first equation with −√gd we get the second equation. Thus, solving the second

equation for x2 we get x2 =√g/dx1 for any x1 ∈ R. Choosing x1 = 1, the eigenvector k1 is

k1 = (1,√g/d)T . (5.58)

Similarly, we obtain the second eigenvector

k2 = (1,−√g/d)T . (5.59)

Since

Jk1 = λ1k1

Jk2 = λ2k2

defining the matrix

K = (k1, k2) =

(1 1√g/d −

√g/d

),

where its columns are the eigenvectors of the Jacobian matrix J , we observe after performingthe matrix-matrix multiplication that

JK = ΛK , (5.60)

where Λ is the diagonal matrix with the eigenvalues of J on the main diagonal, i.e.

Λ =

(λ1 00 λ2

). (5.61)

Since the eigenvectors of a matrix are linearly independent, i.e. det(K) = 0, we concludethat

K−1JK = Λ . (5.62)

71

This means that the matrix J is diagonilizable and equivalent to the diagonal matrix λ. Thus,making the change of variable u = Kw and substituting into (

eq:vecformsw5.53) we have

Kw = J ·Kww = K−1JKw

w = Λw .

The last equation is written

∂

∂t

(w1

w2

)=

(λ1 00 λ2

)∂

∂x

(w1

w2

), (5.63) eq:redusw

which is comprised of two un-coupled equations, namely:

w1t = λ1w1x ,

w2t = λ2w2x ,

or equivalently

w1t =√gdw1x ,

w2t = −√gdw2x .

The last equations are two separate advection equations with solution w1(x, t) = ϕ(x−√gd·t)

and w2(x, t) = ψ(x +√gd · t). Obviously, ϕ is a wave traveling to the right and ψ to the

left. Both waves have a phase speed√gd. To specify the function ϕ and ψ we need initial

and boundary conditions for the wave equation. Since η(x, 0) = η0(x) and u(x, 0) = u0(x)we have that w(x, 0) = K−1u(x, 0) with

K−1 = 12

(1

√d/g

1 −√d/g

), (5.64)

and thus ϕ(x) = w1(x, 0) = η0(x) +√d/gu0(x) and ψ(x) = w2(x, 0) = η0(x)−

√d/gu0(x).

Now, going back to the original variables we have that u = Kw we find the general solutionof the linearised shallow water equations to be:

η(x, t) = ϕ(x−√gd · t) + ψ(x+

√gd · t) , (5.65)

u(x, t) =√g/d

[ϕ(x−

√gd · t)− ψ(x+

√gd · t)

]. (5.66)

Later we will compute a more general solution of the wave equation in a bounded domain. Ingeneral, because real world problems appear in finite regions we restrict the equations boundeddomains and impose appropriate boundary conditions.

72

In the previous example we saw that the system can be transformed into a new system ofdecoupled equations. The transformation was successful because the 2× 2 Jacobian matrixhad two real eigenvalues and two linearly independent eigenvectors. And this is because thechange of variables was based on the matrix formed by the two independent eigenvectors.In general we have the following definition:Definition 5.8 (Diagonalisable matrix). A matrix A is said to be diagonalisable if it isequivalent to a diagonal matrix Λ i.e. if there is an invertible matrix K such that

A = KΛK−1 . (5.67)

If a m×m matrix A has m eigenvalues λi, i = 1, ·,m and m linearly independent eigenvectorski, i = 1, · · · ,m then it is diagonalisable with

Λ = diag(λ1, λ2, · · · , λm), and K = [k1, k2, · · · , km] .

A system of the formut + Aux = 0 , (5.68) eq:sysor

with diagonalisable matrix A is called diagonalisable system and it can be transformed intoa new system of completely uncoupled equations that can usually be solved easier.Definition 5.9 (Hyperbolic system of equations). A system of equations ut + F (u)x = 0is called hyperbolic system if the Jacobian matrix J(u) has m distinct, real eigenvaluesλ1 < λ2 < · · · < λm, and m linearly independent eigenvectors ki, i = 1, 2, · · · ,m.

A hyperbolic system is thus diagonalisable. To diagonalise a system we consider the changeof variables

w = K−1u , (5.69)with w = (w1, w2, · · · , wm)

T . The new variable w is called the characteristic variable. If Ahas constant entries then K do so and thus

ut = Kwt, ux = Kwx ,

and after substitution to system (eq:sysor5.68) we get

Kwt + AKwx = 0 .

Multiplying the last equation with K−1 from the left we obtain the uncoupled system wt +K−1AKw = 0, or equivalently

wt + Λwx = 0 . (5.70) eq:uncoupled

System (eq:uncoupled5.70) is known to as the canonical or characteristic form of system (

eq:sysor5.68). Writing

system (eq:uncoupled5.70) analytically we have

∂t

w1

w2...wm

=

λ1 0 · · · 00 λ2 · · · 0... ... . . . ...0 0 · · · λm

· ∂x

w1

w2...wm

,

73

where the i-th equation is the equation

wit + λiwix = 0 ,

for i = 1, 2, · · · ,m. This equations has only one unknown function, the dependent variablewi(x, t) and thus the system is uncoupled. For each of these equations the characteristicspeeds are the values λi and therefore there are m characteristic curves satisfying the ordinarydifferential equation

dx

dt= λi, i = 1, 2, · · · ,m . (5.71)

The characteristic curve related to the i-th eigenvalue λi is called the i-caracteristic. Theexistence of multiple characteristic lines starting from the same point is one of the basicdifferences between scalar conservation laws and systems. On each of the these characteristiclines a different wave will propagate unchanged. For example instead of the generation ofone shock there might be more than one or there might be also a rarefaction wave generatedin addition to the shocks. We will study such an example later when we will try to solve theRiemann problem for the shallow water equations.

Returning back to the original variables we observe that

u(x, t) = Kw(x, t)

= w1(x, t) · k1 + w2(x, t) · k2 + · · ·+ wm(x, t) · km

=m∑i=1

wi(x, t) · ki

=m∑i=1

w0i (x− λit) · ki , (5.72) eq:seriesol

where w0i (x) = wi(x, 0) and therefore wi(x, t) = w0

i (x − λit). Therefore, the solution u is asuperposition of m traveling waves, which are known to as simple waves. More specifically,any solution wi of the system that remains constant along the characteristics is called i-simplewave.Remark 5.10. Since a m × m system has m characteristic lines then contrary to scalarconservation laws from any point P (x, t) of the t-x plane there are m characteristics linespassing from it. On each of these characteristic lines a contact discontinuity is propagating.

We proceed with a specific example, the Riemann problem.

5.5 The Riemann problem

Consider a linear hyperbolic system with constant coefficients of the form ut + Aux = 0 forx ∈ R and t > 0. We further assume that A is a real m × m real matrix with constant

74

coefficients. Furthermore, the Riemann problem is based on the initial condition of the form

u(x, 0) = ϕ(x) =

uL, x < 0 ,uR, x ≥ 0 ,

(5.73) eq:riemic

Since we assumed that the system is hyperbolic, there are m real and distinct eigenvaluesλ1 < λ2 < · · · < λm. Each eigenvalue determines a characteristic line as shown in Figure

fig:sysrp116.

The characteristic lines start from the origin since the initial condition changes type aroundx = 0. The solution left of the characteristic line with speed λ1 will be constant to uL andsimilarly the solution right of the characteristic line with speed λm will be uR. We only needto determine the solution between these two characteristic lines. If ki, i = 1, · · · ,m are them linearly independent eigenvectors of A they form a basis of Rm. Thus we can the vectorsuL and uR as linear combinations of the vectors ki:

uL =m∑i=1

αiki, uR =m∑i=1

βiki . (5.74) eq:solsinitrmn

Recall that the solution is given as linear combination of the functions w0i in (

eq:seriesol5.72). Then

from (eq:solsinitrmn5.74) and (

eq:seriesol5.72) we conclude that

w0i (x) =

αi, x < 0βi, x > 0

. (5.75)

Therefore, the solution of the characteristic equations

wit + λiwix = 0 , i = 1, · · · ,m ,

is givenwi(x, t) = w0

i (x− λit) =

αi, x− λit < 0βi, x− λit > 0

. (5.76)

If now we take a point (x, t) then there are two characteristic lines, lets say with speeds λjand λj+1, that include the point (x,t). Then,

λj <x

t< λj+1 , (5.77)

that is x − λjt < 0 and x − λj+1t > 0. Thus the solution changes around this point and itcan be written in the form

u(x, t) =

j∑k=1

αiki +m∑

k=j+1

βj ki . (5.78)

Remark 5.11. The solutions wi that remain constant along the characteristic lines are alsoknown to as the Riemann invariants of the system.

75

λ1

λ2λi

λm

λm−1

λI+1

λI

x

t

0

Right data uRLeft data uL

b

Figure 16: Characteristic lines of a system fig:sysrp1

Example 5.12. In the case of a 2 × 2 system with two eigenvalues λ1 < λ2 the solutioninitially will be uL = α1k1 + α2k2 and uR = β1k1 + β2k2. Then the solution for any point(x, t) such that λ1 < x/t < λ2 takes the form

u(x, t) = β1k1 + α2k2 , (5.79)

since x− λ1t > 0 while x− λ2t < 0.

P (x, t)

0 x

t

x1 x2

λ1λ2

b

Figure 17: Domain of dependent of a point P (x, t) fig:sysrp2

The jump discontinuity in the solution u is [u] = uR− uL = (β1−α1)k1+(β2−α2)k2, whichis the eigenvector expansion with coefficients βi − αi known to as wave strengths.

It is noted that the interval [x1, x2] in Figurefig:sysrp217 is called the domain of dependence of the

point P while the area bounded by the two characteristics and the domain of dependence ofP is called the domain of determinacy of the point P .

Remark 5.13. In case of systems with constant coefficients, the i-characteristic lines arealways parallel but when the system is nonlinear the there might be i-characteristic lines with

76

different speeds depending on the initial condition. We will study the shallow water equations,which is a 2× 2 nonlinear system of conservation laws, in the next section.

As we have discussed before the discontinuous solutions of scalar conservation laws mustsatisfy the Rankine-Hugoniot condition. Similarly, for systems of conservation laws of theform ut + f(u)x = 0 these conditions take the following form:

[F ] = Si[u] , (5.80) eq:rankhogsys

where Si = dsi/dt is the speed of the characteristic lines on which the discontinuities prop-agate, [u] = uR − uL, [F ] = F (uR)− F (uL), with uL and uR are the respective states imme-diately to the left and right of the discontinuity. It is noted that unlike scalar conservationlaws, it is very difficult to compute the speeds Si for nonlinear systems of conservation laws.For linear systems the situation is simpler since the speeds Si coincide with the eigenvaluesof the coefficient matrix λi.

In general the Riemann problem for a general nonlinear hyperbolic system with initial data(eq:riemic5.73) has a similarity solution u(x/t) consisting of m + 1 constant states separated by m

waves. These waves may be discontinuities such as shock waves, contact discontinuities orsmooth transition waves such as rarefactions. The difference between a shock wave anda contact wave is the speed of propagation of the two states uL and uR. In the case ofshock waves the two states propagate with different speeds satisfying the entropy conditionλi(uL) > Si > λi(uR) , while in contact discontinuities the two states have the same speedλi(uL) = λi(ur) = Si. For more information we refer to

ChoMar[CM00].

5.6 The method of characteristics

Although the shallow water system is not linear with constant coefficients, we can transformit into a new system of uncoupled equations using eigenvalues and eigenvectors that nowdepend on η and u. The shallow water equations can be written in the matrix form:

ut + Aux +Buy = −s (5.81) eq:matform

where u = (η, u, v)T , s = (u(η + d)x + v(η + d)y, 0, 0)T , and

A =

u η + d 0g u 00 0 u

, B =

u 0 η + d0 v 0g 0 v

. (5.82)

The eigenvalues of A areu, u±

√g(η + d) , (5.83)

and the eigenvalues of the matrix B are

v, v ±√g(η + d) . (5.84)

77

When these eigenvalues are real and distinct, the shallow water equations are hyperbolicpartial differential equations. The equations admit discontinuous (weak) solutions. Such adiscontinuity is called a bore and approximates a breaking wave. In order to describe sucha discontinuous solution we can apply the method of characteristics introduced in Sectionsec:laws3. In order to simplify the computations we will consider the one-dimensional form of theshallow water system by assuming that v = vx = uy = 0:

(η + d)t + u(η + d)x + (η + d)ux = 0 , (5.85) eq:sh1ut + g(η + d)x + uux = gdx , (5.86) eq:sh2

and written in more compact form

ht + [uh]x = 0 , (5.87) eq:sh1aut + ghx + uux = gdx , (5.88) eq:sh2b

where h = η + d is the total depth of the water.

Denoting c =√gh and multiplying (

eq:sh1a5.87) by g we have

(c2)t + u(c2)x + c2ux = 0 , (5.89)

orc [(2c)t + u(2c)x + cux] = 0 , (5.90)

so(2c)t + u(2c)x + cux = 0 . (5.91) eq:c1

Similarly (eq:sh25.86) becomes

ut + uux + c(2c)x = gdx . (5.92) eq:c2

Adding (eq:c15.91) and (

eq:c25.92) and then subtracting from (

eq:c15.91) equation (

eq:c25.92) we have

(u+ 2c)t + u(u+ 2c)x + c(u+ 2c)x = gdx , (5.93)(u− 2c)t + u(u− 2c)x − c(u− 2c)x = gdx , (5.94)

or better:

(u+ 2c)t + (u+ c)(u+ 2c)x = gdx , (5.95) eq:cons1(u− 2c)t + (u− c)(u− 2c)x = gdx , (5.96) eq:cons2

which is a system of uncoupled equations. In the case of a flat bottom, the last equationsimply that the quantity u+ 2c is constant along the curves in the (x, t) plane defined by

dx

dt= u+ c , (5.97) eq:char1

and the quantity u− 2c is also constant along the curvesdx

dt= u− c . (5.98) eq:char2

Therefore, if h is constant, the quantities R+ = u+ 2c and R− = u− 2c are called Riemanninvariants (i.e. functions that remain constant along the curves).

78

Remark 5.14. The method of characteristics in some cases can be generalised in morespatial dimensions but it is more difficult. For hyperbolic problems in more dimensions werefer to the book

ZT[ZT86].

The same eigenvalues and eigenvectors can be found for the shallow-water system written inthe conservative form:

ht + (hu)x = 0 , (5.99) eq:csw1a(hu)t + (1

2gh2 + hu2)x = ghdx . (5.100) eq:csw1b

Specifically, (eq:csw1a5.99)–(

eq:csw1b5.100) can be written in the form:

u

The nonlinear nature of the shallow water equations make the computation of analyticalsolution and more precisely the solution of the Riemann problem very difficult task. Althoughit is very difficult there are some cases (initial conditions) that can be solved. We proceedwith a specific example known to as the one-dimensional dam-break problem.

It is noted that the eigenvalues of the matrices A and B play an important role in thetheory of water waves and in general in continuum mechanics. Their sign depend on themagnitude of u and c. It is remarkable that the quantities u and c can determine differentflow regimes. For this reason a special attention has been given to the ratio u/c. Specifically,a dimensionless number has been introduced known to as the Froude number:

Fr =u√gh

,

with h = η + d. Usually, u is a characteristic velocity of the flow (know to as the flowvelocity) or the average flow velocity, averaged over a cross-section for example. In somespecial cases, such as the propagation of traveling waves with constant speed v, the Froudenumber is defined as the ration cs/c.

Based on the values of the Froude number we can distinguish three regimes where the flowis characterised as:

• critical, when Fr = 1,

• supercritical, when Fr > 1, and

• subcritical, when Fr < 1.

The denominator represents the speed of a small wave on the water surface relative to thespeed of the water, called also wave celerity. Supercritical flow is controlled upstream and

79

disturbances are transmitted downstream. Usually critical flows are unstable while the term“supercritical” and “subcritical” is due to high or low energy states of the flow. At criticalflow celerity equals flow velocity. Any disturbance to the surface will remain stationary. Insubcritical flow the flow is controlled from a downstream point and information is transmittedupstream. This condition leads to backwater effects.

Wave propagation can be used to illustrate these flow states: An object placed in the watercreates a V pattern of waves downstream. If the flow is subcritical waves will appear infront of the object. If the flow is supercritical no upstream waves will appear and the waveangle will be less than 45 while if the flow is critical then the waves will have a 45 angleapproximately.

It is noted that when the flow is subcritical, then the eigenvalues of the matrices A and Bhave opposite signs. The Froude number determines also the number of boundary conditionsrequired in order to solve the shallow water equations in a bounded domain lets say [a, b]. Iffor example the flow is subcritical, then Fr < 1 and therefore λ1 = u+c > 0 and λ2 = u−c <0 (where c =

√gh). We have seen that the eigenvalues λ1 and λ2 determine the direction of

the characteristic lines along which the Riemann invariants are being transmitted. Therefore,when Fr < 1 there are information coming from both boundaries and therefore we need twoboundary conditions, one to determine the information coming from the boundary x = aand also the information coming from the boundary x = b.

λ1 = dx/dt λ2 = dx/dt

x

t

a b

reflected wave R− transmitted wave R+

Figure 18: Riemann invariants propagating along the characteristics with slopes λ1 and λ2at the boundary x = b with Fr < 1 fig:bcsr

For example, assume that we want to impose totally reflective boundary conditions fora subcritical flow on the boundary x = b. The condition is set only for the invariantcorresponding to the outgoing characteristic, because the other invariant value is carried tothe boundary along the incoming characteristic line. This means that we need to set theincoming wave R− to have the same magnitude but opposite direction with R+, i.e.

R− = −R+ , (5.101) eq:rm1

80

on the boundary x = b, see Figurefig:bcsr18. Substitution of h and u into (

eq:rm15.101) gives u(b, t) = 0,

which means that there is no horizontal velocity on the wall x = b in agreement with thephysics of an impenetrable wall. Similarly, if we want to impose radiation (or absorbing)boundary conditions, the requirement of no information transmitted inside the domain fromthe outside region of the domain [a, b] at the boundary x = b, implies that u = 0 and h = 0,on R−, i.e. R− = −2

√gd, where d = d(b) the value of the bottom at the right boundary

x = b.

Although, supercritical flows rarely occur, we mention that when Fr > 1 then the shallowwater equations need two boundary conditions on the left boundary and no boundary con-ditions on the right boundary since all of the characteristic lines have the direction from theleft to the right.

5.7 The dam-break problem

In this section we consider the shallow water equations with flat bottom d = const. Thetotal depth of the water usually is denoted by h = η + d. Because this problem is of greatphysical importance, we use the shallow water equations written in the conservative formwhich makes physical sense:

ht + (hu)x = 0 , (5.102) eq:shb1(hu)t + (1

2gh2 + hu2)x = 0 , (5.103) eq:shb2

which using vector notation isut + F (u)x = 0 , (5.104)

where u = (h, hu)T , F = (f1, f2)T with f1(h, u) = hu and f2(h, u) =

12gh2 + hu2.

The eigenvalues are again λ± = u ±√gh, the eigenvectors are k1 = (1, u +

√gh)T , k2 =

(1, u−√gh)T , and the Riemann invariants are R± = u± 2

√gh.

0x

x xxx

wall

h0 Water

Figure 19: Sketch of the dam-break problem fig:damsk

81

We consider the idealised problem where a vertical dam located at x = 0 prevent the flowof a mass of water with total depth h = h0. This means that initially

h(x, 0) =

h0, x < 00, x ≥ 0

(5.105) eq:dam1

while u(x, 0) = 0 everywhere. Suddenly, we assume that the dam is removed and the wavestart flowing under gravity. At t = 0 the characteristic lines C± can be defined for x < 0 andhave slope λ± = u±

√gh = ±

√gh0 while the Riemann invariants at t = 0 are R± = ±2

√gh0.

Denoting the linear wave speed by c0 =√gh0, we have that λ± = ±c0 and R± = ±2c0. The

most extreme right characteristic line is the line x = −c0t. The characteristic diagrampresenting the characteristics C± is in Figure

fig:damchar120. We observe that if C+ and C− intersect

then u + 2√gh = 2c0, u − 2

√gh = −2c0 and hence u = 0 and h = h0. This means that

the water remain undisturbed for x ≤ −c0t. We also observe that the characteristics will bestraight lines due to the intersection.

C−C−

C−

C−

C−

C−

C−

C−

C+

C+

C+

x=−c0 t

x=2c

0t

t

x

0

Figure 20: The characteristic diagram for the dam-break problem fig:damchar1

For values x > −c0t we expect that the water will flow. Therefore, the characteristics C+ willcross the characteristic x = −c0t and enter the domain x > −c0t. Since the characteristicslines C+ are continuations of the characteristic straight lines for x < −c0t we deduce thatu + 2

√gh = 2c0. Also u and h must be constant on each characteristic line C−. Since the

characteristic lines C− are determined by the slope dx/dt = u −√gh for x > −c0t where

the Riemann invariants R− = u− 2√gh are constant we deduce that the characteristic C−

82

are again straight lines (i.e. dx/dt = const). Since the fluid occupies initially the regionx < 0 the characteristics C− must start from at the origin and since x = (u −

√gh)t then

the slope must be u−√gh = x/t for x > −c0t. We ended up with two equations for h and

u for x > −c0t,

u+ 2√gh = 2c0 , (5.106)

u−√gh = x/t . (5.107)

Solving for h and u we get

h =h09

(2− x

c0t

)2

, u =2

3

(c0 +

x

t

). (5.108)

Note that on the dry front of the water mass where h = 0 is x = 2c0t. This also suggeststhat no characteristic C+ reaches the region x > 2c0t so that u = h = 0 there.

Summarising the solution of the shallow water equations for the dam-break problem is givenfor the depth h:

h(x, t) =

h0, x < −c0t ,h0

9

(2− x

c0t

)2, −c0t ≤ x ≤ 2c0t ,

0, x > 2c0t ,

(5.109) eq:damsolh

and for the velocity u:

u(x, t) =

0, x < −c0t ,23

(c0 +

xt

), −c0t ≤ x ≤ 2c0t ,

0, x > 2c0t ,(5.110) eq:damsolu

It is noted that the dam-break problem and the generation bores in shallow waters have beentwo of the main problems studied analytically using the shallow water equations,

stoker1992water[Sto92].

These problems are also considered standard benchmarks for most of the numerical codesfor the shallow water equations.

5.8 Propagation of bores in shallow water

Bores occur in rivers and in the oceans and are usually tidal waves entering a narrowingestuary. Because of their shape, bore is used also to describe shock waves in water. Wecontinue with a more general dam-break problem. Consider the idealised problem wherefluids of different depth are initially at rest on either side of a dam, i.e. on the downstreamas well as in the upstream side of the dam. The dam is described by a jump discontinuity,cf. Figure

fig:initdb21. This is equivalent with a Riemann problem where the initial conditions for

the free surface is a step-like function while the velocity is u = 0. Thus, we consider the

83

dam break problem as a type of Riemann problem posed for the shallow water equations(eq:sha1??)–(

eq:sha2??). In this problem the initial condition is assumed to be the water bounded by a

vertical dam at rest, i.e.h(x, 0) =

hL, x < 0hR, x > 0,

(5.111)

where we assume hL > hR ≥ 0, and the initial velocity

u(x, 0) = 0 . (5.112)

x

hR

hL

Figure 21: Initial states at rest of the total depth h of the water at t = 0. fig:initdb

Removing the dam, by a somehow magical way, some water on the left side of the disconti-nuity is accelerated instantaneously causing the water to flow over the slower moving fluidon the right side of the discontinuity. This will cause a shock formed at the right of the ini-tial disturbance. The solution behind this shock will remain constant. However this cannotremain constant for all time since the velocity of this region is non-zero and a section of theupstream domain will still be undisturbed and thus have no velocity. This means that theremust be some form of rarefaction wave connecting these two constant states.

At t = 0 the dam wall is removed instantaniously and the spatial domain is divided into fourregions. A shock wave forms to the right of the initial disturbance as fast moving water,accelerated by removing the dam, flows over the stationary water downstream. A constantregion denoted by 3 is behind the shock wave where the solution remains constant. Thisregion is connected with the undisturbed fluid of region 1 by a rarefaction wave withparabolic shape in region 2 . The point at which these two regions meet is determined bythe characteristic line

x = −√ghLt . (5.113)

The characteristic line separating the rarefaction fan of region 2 and the constant wave ofregion 3 is

x = (um −√ghm)t . (5.114)

Finally the characteristic line along of which the shock wave is travelling with speed S =ds/dt is given by

x = St . (5.115)

Let ci =√ghi for i = L,R, r and m, where the subscripts L and R are related to the left

and right constant states around the initial discontinuity and r and m are associated withthe rarefaction wave and the shock wave of regions 2 and 3 .

84

t

x

hL

hR

hm

−t√ghL t(um −

√ghm) tS

0

1 2 3 4

1

2 3

4

shockrarefaction

hr

x = s(t)

Figure 22: Sketch of the solution of the dam break problem in different regions of the domain. fig:dbchara

Solving the Riemann problem requires determining how the left and right states of theconserved variables are connected through the intermediate states hi, ui, for i = r and m.In this analysis in order to determine the shock wave we will use the Rankine-Hugoniotcondition, which is

S(ui − um) = F (ui)− F (um) . (5.116) eq:rhcond

where ui is either uL or uR. Using the same notation the Rankine-Hugoniot conditions takethe form

S(hi − hm) = hiui − hmum , (5.117) eq:rankhogdb1S(hiui − hmum) =

12g(h2i − h2m) + hiu

2i − hmu

2m . (5.118) eq:rankhogdb2

These equations can be also obtained by applying the conservation of mass and momentumto the column of water around the shock wave.

Combining the first equation (eq:rankhogdb15.117) we gethR(S − uR) = hm(S − um) . (5.119)

Since the initial velocity uL = 0 we havec2Rc2m

=hRhm

=S − umS

. (5.120) eq:veldb

85

Similarly from the second equation (eq:rankhogdb25.118) of the system we obtain

12(c2m − c2R) = hm(S − um)(um − uR) . (5.121)

Again, since uR = 0 and substituting um from (eq:veldb5.120) yields

12(c2m − c2R) = hm(S − um)S

hm − hRhm

, (5.122)

which can also be written in a more convenient form:

c2m + c2R = 2S(S − um) . (5.123) eq:veldb3

Eliminating cm from (eq:veldb35.123) using (

eq:veldb5.120) we obtain the quadratic equation

2S(S − um)− c2R(S − um)− c2RS2 = 0 , (5.124)

which admits the solutionum = S − cR

4S

(1 +

8S2

c2R

). (5.125) eq:soldbp

It is noted that um and s are always positive. The positive sign was used to ensure thatum − S and −S have the same sign.

Eliminating um from (eq:soldbp5.125) using (

eq:veldb5.120) yields

c2mc2R

=1

2

√1 +

8S2

c2R− 1

2. (5.126) eq:dbeq

Equations (eq:soldbp5.125) and (

eq:dbeq5.126) provide expressions of the velocity um and the wave speed cm

in the undisturbed region 3 behind the shock as functions of the shock speed s and thewave speed cR.

Consider the rarefaction fan in region 4 . Along each of the characteristic lines in this regionthe Riemann invariant u+ 2c is constant, so on the left side of the rarefaction fan it holds

u+ 2c = 2cL , (5.127)

while on the right sideu+ 2c = um + 2cm . (5.128)

Equating the right hand sides of these equations we obtain

2cL = um + 2cm . (5.129)

86

Substituting the values of um and cm obtained from (eq:soldbp5.125) and (

eq:dbeq5.126) leads to a nonlinear

expression for the shock speed s

cL =S

2− c2R

8S

(1 +

√1 +

8S2

cR

)+

√√√√c2R2

√1 +

8S2

cR− c2R

2, (5.130)

or equivalently

S = 2cL +c2R4S

(1 +

√1 +

8S2

cR

)−

√√√√2c2R

√1 +

8S2

cR− 2c2R . (5.131)

Now that a relation for the shock speed has been determined we can evaluate um and cmand hence determine the nature of the solution in region 2 . In this region the characteristiclines are determined by

dx

dt=x

t= ur − cr . (5.132)

Along the characteristic lines in this region the Riemann invariant u+ 2c is constant, so onthe left side of the rarefaction fan holds

u+ 2c = 2cL . (5.133)

On the rarefaction fan we haveu+ 2c = ur + 2cr , (5.134)

so2cL = ur − 2cr . (5.135)

Finally, solving these equations for ur and cr we get

ur =2

3

(cL +

x

t

), (5.136)

andcr =

2

3cL − x

3t. (5.137)

Concluding, at any time t > 0 for the idealised dam-break problem with zero initial velocitywe have the solution

h(x, t) =

hL, x ≤ −t

√ghL

hr =49g

(√ghL − x

2t

)2, −t

√ghL < x ≤ t(um −

√ghm)

hm = hR

2

√1 + 8

(2hm

hm−hR

√hL−

√hm√

hR

)2− hr

2, t(um −

√ghm) < x < tS

hR, x ≥ tS(5.138)

87

and

u(x, t) =

0, x ≤ −t

√ghL

ur =23

(√ghR + x

t

), −t

√ghL < x ≤ t(um −

√ghm)

um = s− ghr

4s

(1 +

√1 + 8s2

ghR

), t(um −

√ghm) < x < tS

0, x ≥ tS

(5.139)

where the shock speed is given by

S = 2√gh+

ghR4S

(1 +

√1 +

8S2

ghR

)−

√√√√2ghR

√1 +

8S2

ghR− 2ghR . (5.140)

Although we manage to find analytical expressions for the solution of the dam-break problem,it is rather difficult to use them since the computation of the speed s and the rarefaction wavehr require the solution of nonlinear systems. The problem can be simplified by assumingthat there is no water on the right of the dam wall, i.e. hR = 0. Then the solution to thisproblem is given by the formula

h(x, t) =

hL, x ≤ −t

√ghL

hr =49g

(√ghL + x

2t

)2, −t

√ghL < x ≤ 2t

√ghL

0, x ≥ 2t√ghL

(5.141)

and

u(x, t) =

0, x ≤ −t

√ghL

ur =23

(√ghL + x

t

), −t

√ghL < x ≤ 2t

√ghL

0, x ≥ 2t√ghL

(5.142)

which are in agreement with the solution (eq:damsolh5.109) and (

eq:damsolu5.110).

Analytical solutions are very useful not only in understanding the physics of a problembut also as a reference solution to test and develop methods for the numerical solution ofthe differential equations. In the second part of these notes we present methods for thenumerical solution of the shallow water system and we will test the numerical solutionsagainst analytical solutions.

88

All the effects of Nature areonly the mathematicalconsequences of a small numberof immutable laws.

— Pierre-Simon Laplace

6 Nonlinear and dispersive wavessec:modeq

Although waves appear in most of the cases to be nonlinear, like shock waves, there is an-other very important property characterising especially water waves where waves of differentwavenumber propagate with different speeds. This property is known to as dispersion. Inthis section we will introduce the notion of the dispersion and we will derive some of themost important models for water waves that are nonlinear and dispersive.

6.1 Dispersive waves and their characteristicssec:disperseqsc

For simplicity we consider the simple transport equation introduced in Chaptersec:laws3:

ut + cux = 0 x ∈ R , (6.1) eq:trnseq

with initial condition u(x, 0) = u0(x). Instead of using the method of characteristics weassume that this equation admits a general traveling wave solution of the form

u(x, t) = ei(kx−ωt) , (6.2) eq:trvwavsol

where k is the wavenumber and ω is the frequency of the wave. Recall that the period ofthe wave is defined as T = 2π/ω, while the wavelength is λ = 2π/k.

Substitution this solution into the transport equation we observe that −ω + ck = 0 or inother words the frequency depends on the wavenumber.

ω = ck . (6.3)

This is in general true and we write ω = ω(k) indicating the dependence of ω on k. Writingthe traveling-wave solution (

eq:trvwavsol6.2) in the form

u(x, t) = eik(x−ωkt) ,

we observe that the phase velocity of the wave (i.e. the speed of propagation of the wave) is

c(k) =ω(k)

k, (6.4) eq:phasevel

89

where in the case of the transport equation this is c(k) = c. Although in this examplethe phase speed is independent of the wavenumber it is not always the case. For exampleconsider the equation

ut + uxxx = 0 . (6.5) eq:linkdv1

Again, considering the general traveling wave solution u(x, t) = ei[kx−ω(k)t] after substitutionto (

eq:linkdv16.5) yields that

ω(k) = −k3 , (6.6) eq:disperq

and thus the phase speed depends on the wavenumber and is c(k) = −k2. This is charac-teristic of dispersive waves, where waves of different wavenumber propagate with differentspeed.

Usually, dispersive waves appeared in groups of waves that propagate with different speedswhile as a group has the group velocity defined as

cg(k) = ω′(k) . (6.7)

Mathematically a wave is called dispersive wave if ω(k) is real-valued and the group velocitydepends on k, i.e. ω′′(k) = 0. The relation for ω(k) is called dispersion relation and becauseusually we can only find this relation for linear equations sometimes it is called the lineardispersion relation.

6.1.1 Are water waves really dispersive?

The answer to this question is yes. Consider the Euler equations in potential flow (eq:laplace4.66)–

(eq:codf24.69) over a flat bottom d = d0 and linearised around z = 0. For simplicity we write these

equations in two dimensions in the form:

ϕxx + ϕzz = 0, −d0 < z < 0 , (6.8) eq:linela1ηt = ϕz, z = 0 , (6.9) eq:linela2ϕt + gη = 0, z = 0 , (6.10) eq:linela3ϕz = 0, z = −d0 , (6.11) eq:linela4

where the first equation is the Laplace equation and the rest are the boundary conditions onthe free surface and the bottom. It is noted that the boundary condition on the free surfacecan be combined in the single equation

ϕtt + gϕy = 0, z = 0 . (6.12) eq:linela23

In order to study the dispersion relation of the linearised Euler equations it is required tostudy the general periodic waves of the form:

ϕ(x, z, t) = Φ(z)ei(kx−ωt) . (6.13)

90

Substitution in the Laplace equation gives the second order ordinary differential equation

Φzz − k2Φ = 0, −d0 < z < 0 , (6.14) eq:linsoleq1

along with the boundary conditions:

Φz −ω2

gΦ = 0, z = 0 , (6.15) eq:linsoleq2

Φz = 0, z = −d0 . (6.16) eq:linsoleq3

From the theory of ordinary differential equations we know that Equation (eq:linsoleq16.14) admits a

solution of the formΦ(z) = cosh(k(d0 + z)) . (6.17)

Applying the boundary conditions (eq:linsoleq26.15) at the free surface to the solution Φ(z) yields the

relation:ω2(k) = gk tanh(kd0) , (6.18) eq:ldrww

which is the linear dispersion relation for water waves. It is worth to mention that in (eq:ldrww6.18)

the frequency ω depends on k and the phase velocity is c2 = g/k tanh(kd0) again depends onk. This means that a general disturbance on the surface of the water will disperse. Moreover,we observe again (like in the case of the linear wave equation) the speed of the wave dependsalso on the depth of the water column d0.

Back to the solution of the linearised Euler equations we observe that substituting thesolution ϕ(x, z, t) = cosh(k(d0 + z))ei(kx−ωt) into (

eq:linela36.10) we have

η(x, t) = −1

gϕt(x, 0, t)

=iω

gcosh(kd0)e

i(kx−ωt) .

Hence the solution of the linearised Euler equations is of the form

η(x, t) = Aei(kx−ωt) ,

where A = iωgcosh(kd0). For more information about the linearised Euler equation we refer

toWhitham1979[Whi79]

Remark 6.1. In shallow water wave theory (or else for long waves) we assume that thewavelength λ = 2π/k is large compared with the depth d0, i.e. d0 ≪ λ, from which yieldsthat kd0 ≪ 1. As kd0 → 0, we can approximate tanh(kd0) ≈ kd0 and then we have theapproximate dispersion relation ω ≈ ±

√gd0k or the known phase speed from the linearised

shallow water equations (wave equation)

c ≈ ±√gd0 . (6.19)

On the other extreme case of deep water waves, we have kd0 ≫ 1 then the dispersion relationis approximated by ω ≈ ±

√gk and the phase speed is c ≈ ±

√g/k.

91

From this remark we conclude that the wave equation and the shallow water equationsare not dispersive models, or in other words their solutions are not dispersive waves. Thisdrawback is important in some cases and therefore we need to derive other (dispersive) waveequations that will include this important physical quality of waves.

6.1.2 Effects of nonlinearity and dispersion

In 1834, John Scott Russel, a British civil engineer, while was working along a canal betweenEdinburgh and Glasgow, he discovered a wave that he described as the wave of translation.The description of the generation of this wave is remarkable,

russell1844[Rus44]. His original description

is quoted here:

“I was observing the motion of a boat which was rapidly drawn along a narrowchannel by a pair of horses, when the boat suddenly stopped – not so the massof water in the channel which it had put in motion; it accumulated round theprow of the vessel in a state of violent agitation, then suddenly leaving it behind,rolled forward with great velocity, assuming the form of a large solitary elevation,a rounded, smooth and well-defined heap of water, which continued its coursealong the channel apparently without change of form or diminution of speed. Ifollowed it on horseback, and overtook it still rolling on at a rate of some eightor nine miles an hour [14 km/h], preserving its original figure some thirty feet[9 m] long and a foot to a foot and a half [300-450 mm] in height. Its heightgradually diminished, and after a chase of one or two miles [2-3 km] I lost itin the windings of the channel. Such, in the month of August 1834, was myfirst chance interview with that singular and beautiful phenomenon which I havecalled the Wave of Translation.”

These waves of translation are known to as Solitons or Solitary waves and have applicationsin oceanography, medicine, even in telecommunications. Their shape also is presented inFigure

fig:solitawave123.

Scott Russell tried to re-generate these waves by doing experiments in a wave tank and bytheoretical investigations. Although, his observations were very accurate, the theory at thatcouldn’t explain or describe these kind of waves. Specifically, Russell’s findings contradictto what Airy’s, Newton’s, Bernoulli’s or other theories could predict. In 1876 Lord Rayleighpublished a paper in support of John Scott Russell. The first theoretical evidence of theexistence of solitons came out in 1871 in the work of Joseph Valentin Boussinesq whileKordeweg and de Vries derived a model equation which was nonlinear and dispersive andwas able to describe solitons with great accuracy. This equation is know to as the KdVequation and written in nondimensional but unscaled variables has the form:

ut + ux + uux + uxxx = 0 . (6.20) eq:kdv1

92

Figure 23: Shape of a solitary wave. fig:solitawave1

This equation extends the nonlinear Burger’s equation with the addition of the high-orderterm uxxx. This extra term makes the equation dispersive. It’s linearisation ut+ux+uxxx = 0has dispersion relation ω(k) = k − k3. The KdV equation has several favourable properties.For example it is integrable, which means that can be solved analytically using the inversescattering transform. Moreover, its solitons have analytical formulas that we will studylater. Although the KdV equations seems to be a nice model for dispersive waves, there areseveral difficulties, in the theory and in the numerical analysis of this equation that lead tothe derivation of even simpler models such as the BBM equation. In 1974 Benjamin, Bona,and Mahony, derived and study thoroughly a similar model known to as the BBM equation.Although the BBM equation is asymptotically equivalent to the KdV equation, it appearsto have significant differences. The BBM equation in non-dimensional and unscaled variableis written in the form:

ut + ux + uux − uxxt = 0 . (6.21) eq:BBM1

Both the KdV and BBM equations are generalisations of the Burger’s equation ut+ux+uux =0 incorporating the correction terms uxxx and uxxt. A combination of these two equations willresult to the KdV-BBM equation ut + ux + uux + uxxx − uxxt = 0. All these models admitsolitary waves as solutions. We will compute analytical formulas for their solitary waveslater. A solitary wave usually has the shape as it is depicted in Figure

fig:solitawave123. It is noted that

there exist in the nature and as solutions to model equations other traveling waves differentthan the solitary waves described by Russell. These are known as generalised solitary wavesand although the have a main pulse, they decrease and connected to oscillatory solutions. Insome cases the oscillations are periodic and in come other cases they are damped oscillationsdecreasing to zero. If the traveling wave is periodic without oscillations then the travelingwave is called cnoidal wave because it can be described by the Jacobi elliptic functions suchas the cn function. We will discuss traveling wave solutions of several model equations laterin this chapter.

93

6.2 Derivation of nonlinear and dispersive wave equations

In the previous sections we studied the shallow water equations that do not include thephysical quality of dispersion. The only dispersive model we considered so far was thefull Euler equations. The Euler equations are the original equations derived using classicalmechanics and they are very complicated and very difficult to be solved. For this reason wewill try to make simplifications and derive other simplified mathematical models from theEuler equations that are also dispersive. These models aim in modelling specific phenomenaor waves, and unfortunately, are not universal models that can be used for any purpose.For example the tsunamis are long waves of small-amplitude. So in order to study tsunamisone need ideally a mathematical model for small-amplitude long waves that might not beso accurate to study large-amplitude waves. In this section we will derive several modelsapproximating the Euler equations. We begin with the so called Peregrine’s system.

6.2.1 Derivation of Peregrine’s system with time dependent bottom

Tsunamis are small-amplitude long waves and usually are generated by the deformation ofthe ocean’s floor. The motion of the bottom of the ocean generate analogous motion onthe free surface of the ocean and thus waves. In order to model the waves generated bya moving bottom we derive in this section model equations with time dependent bottom.Specifically, we derive an approximation of the Euler equations in three dimensions andwith time dependent bottom. Peregrine’s system is a Boussinesq type system, actually verysimilar to the original system derived by Boussinesq,

B1872[Bou72], but improved in many ways.

To simplify the notation later, we will denote by (x, y, z) a cartesian coordinate system. Let x,y be the horizontal coordinates and z measured upwards from the still water level. Consider athree-dimensional wave field with water-surface deviation propagating from its rest position,η(x, y, t), at time t, over a variable bottom d bellow the undisturbed surface of the waterd(x, y, t) = D(x, y) + ζ(x, y, t). The fluid is assumed to be inviscid and incompressible, andthe flow is assumed to be irrotational. The fluid velocity is denoted by u = (u, v, w)T in thex, y and z directions, respectively. The Euler equations which describe three dimensionalwave propagation on the free surface are written in the form

ut + (u · ∇)u+1

ρ∇p = −gk, (6.22)

∇ · u = 0, for − d < z < η(x, y, t) (6.23)∇× u = 0, (6.24)

where p is the pressure field, ρ is the density and g is the acceleration due to gravity and k =(0, 0, 1)T . The first equation expresses the conservation of momentum, where the other twoequations express the conservation of mass and the irrotationality of the flow, respectively.

94

The kinematic boundary condition at the free surface and seabed can be expressed as

ηt + u · ∇η = 0, for z = η(x, y, t), (6.25) E2.4

anddt + u · ∇d = 0, for z = −d(x, y, t), (6.26) E2.5

respectively. The fluid is assumed to satisfy the dynamic boundary condition p(x, y, t) =p0(x, y) at the free surface z = η(x, y, t).

Consider a characteristic water depth d0, a typical wavelength λ0 and a typical wave heighta0. A natural scaling of the independent and dependent variables for the purposes of longwave modelling is chosen, cf.

P1967[Per67], so

x∗ =x

λ0, y∗ =

y

λ0, z∗ =

z

d0, t∗ =

c0λ0t,

and

u∗ =d0a0c0

u, v∗ =d0a0c0

v, w∗ =λ0a0c0

w, η∗ =η

a0, d∗ =

d

d0, D∗ =

D

d0, ζ∗ =

ζ

a0,

where c0 =√gd0.

Then, the governing equations for the fluid motion in the non-dimensional and scaled form,and after dropping the asterisk from the variables, take the following form:

εut + ε2((u · ∇)u+ wuz) +1

ρc20∇p = 0, (6.27) E2.6

εσ2wt + ε2σ2(u · ∇w) + wwz) +1

ρc20pz = −1, (6.28) E2.7

for −d < z < εη. The parameters ε = α0/d0 and σ = d0/λ0 are assumed to be small. Theconservation of mass is formulated as

∇ · u+ wz = 0 for − d < z < εη, (6.29) E2.8

the irrotationality condition is given by the formulas

uy − vx = 0, (6.30)uz − σ2∇w = 0, (6.31)

for −d < z < εη, while the boundary conditions take the form

ηt + ε(u · ∇η)− w = 0 on z = εη, (6.32) E2.11

ζt + u · ∇h+ w = 0 on z = −d, (6.33) E2.12

95

where u = (u, v)T and ∇ = (∂x, ∂y). We note that d = D + εζ and thus dt = O(ε).Integrating equation (

E2.86.29) with respect to z from −h to z, and using (

E2.126.33) we have

w = −u · ∇h−∫ z

−h

∇ · u− ζt. (6.34) E2.13

After integration of equation (E2.106.31) and using (

E2.136.34), we observe that

u = ub +O(σ2), (6.35) E2.14

where ub is the horizontal velocity of the fluid at the bottom z = −h. Substitution of (E2.136.34)

into the irrotationality condition (E2.106.31), and using (

E2.146.35), yields

uz = −σ2∇(∇ · (hub))− σ2z∇(∇ · ub)− σ2∇ζt +O(σ4, εσ2). (6.36) E2.15

Integration of (E2.156.36) with respect to z from −h to z gives

u = ub − σ2(z + h)∇(∇ · (hub))− σ2 z2 − h2

2∇(∇ · ub)− σ2(z + h)∇ζt +O(σ4, εσ2). (6.37) E2.16

We note that using (E2.146.35), equation (

E2.136.34) takes the form

w = −∇ · (hub)− z∇ · ub − ζt +O(σ2), (6.38) E2.17

and thus,wt = −∇ · (hub)t − z∇ · ubt − ζtt +O(σ2). (6.39) E2.18

Assuming that p = 0 at z = εη, integrating (E2.76.28) with respect to z from z to εη, and using

(E2.186.39) we have

p

ρc20= εσ2(z∇ · (hub)t +

z2

2∇ · ubt) + εσ2zζtt + εη − z +O(εσ4, ε2σ2). (6.40) E2.19

Using (E2.166.37), (

E2.196.40), for z = −h, we have

ubt +∇η+ ε(ub · ∇)ub − σ2h∇(∇ · (hubt)) + σ2h2

2∇(∇ · ubt)− σ2h∇ζtt = O(σ4, εσ2), (6.41) E2.20

while we made use of the fact that dt = O(ε).

Integration of the mass equation (E2.86.29) with respect of z from −d to εη yields

w(εη)− w(−d) = −∫ εη

−d

∇ · u dz, (6.42) E2.21

and thus, using the boundary conditions (E2.116.32) and (

E2.126.33) we have

ηt +∇ ·∫ εη

−d

u dz + ζt = 0. (6.43) E2.22

96

Denote the depth-average horizontal velocity of the fluid by

u =1

d+ εη

∫ εη

−d

u dz. (6.44) E2.23

then (E2.226.43) becomes

ηt +∇ · [(d+ εη)u] + ζt = 0. (6.45) E2.24

It is noted that Equation (E2.246.45) is another form of the mass conservation equation and is

exact, i.e. no approximation has been done.

Moreover, using (E2.236.44), the equation (

E2.166.37) gives

ub = u+ σ2d

2∇(∇ · (du))− σ2d

2

3∇(∇ · u) + σ2d

2∇ζt +O(σ4), (6.46) E2.25

and thus, the equation (E2.206.41) gives

ut +∇η + ε(u · ∇)u− σ2d

2∇(∇ · (hut)) + σ2d

2

6∇(∇ · ut)− σ2d

2∇ζtt = O(εσ2, σ4). (6.47) E2.26

Note that the system of equations (E2.246.45), (

E2.266.47) is the Boussinesq system of equations derived

by Peregrine,P1967[Per67].

Remark 6.2. Discarding the terms of O(σ2) in equations (E2.246.45), (

E2.266.47) we obtain the shallow

water equations for time dependent bottom.

6.3 Derivation of other model equations

From the previous derivation it can be observed that the horizontal velocity of the fluid isalmost uniform across the fluid depth. For example equation (

E2.146.35) shows that the horizontal

velocity at any depth approximates the horizontal velocity at the bottom. Similarly, fromthe irrotationality condition (

E294.49) we deduce that the vertical acceleration is very small.

More precisely it holds that uz = O(σ2) and thus u is almost constant along z. Takinginto account this observation it is convenient to study either the depth averaged velocity orthe velocity of the fluid at a certain height above the bottom. It is expected that in bothcases the values of the velocities will be close. Here we will restrict the derivation to twodimensions only but the results can be generalised into three dimensions as well. We willobtain approximate models to the Euler equations by using the mean velocity with respectto depth:

u =1

1 + εη

∫ εη

−1

u(x, z, t) dz. (6.48) eq:dav

97

Integrating the equation of the conservation of mass (continuity equation) (E284.48) we have∫ εη

−1

ux dz + v(x, εη, t)− v(x,−1, t) = 0. (6.49) E36

Using Leibniz rule2 we have∫ εη

−1

ux dz =∂

∂x

∫ εη

−1

u dz − u(x, εη, t) · εηx = [hu]x − u(x, εη, t) · εηx. (6.50) E36b

Applying the boundary conditions (E304.50) and (

E324.52) we obtain from (

E36b6.50) the (exact) equa-

tionηt + [hu]x = 0, (6.51) Eeta

where h denotes the total depth 1 + εη.

Similarelly, integrating the momentum equation (E264.46) and using Leibniz rule, the continuity

equation (E284.48) and the boundary conditions (

E304.50)–(

E324.52) we have

εhut + ε2huux + ε2∂

∂x

∫ εη

−1

(u2 − u2) dz = −∫ εη

−1

px dz. (6.52) E38

6.3.1 The Serre equations

In this subsection we will derive model equations for strongly nonlinear long waves known toas the Serre equations (sometimes also referred to as also as the Su and Gardner equations),cf.

S1953I,S1953II,SG1969,Green1974[Ser53a, Ser53b, SG69, GLN74]. For the derivation of model equations, crucial role plays

the assumptions on the pressure field. Using Leibniz rule and boundary condition (E314.51)

gives ∫ εη

−1

px dz = [hp]x − hxp(h) = [hp]x. (6.53) E39

To compute p we write the z momentum equation (E274.47) as

− py = 1 + εσ2Γ(x, z, t), (6.54) E40

whereΓ(x, z, t) = vt + εuvx + εvvz. (6.55) E41

Integrating (E406.54) from z to εη we get

− p(x, z, t) = (y + 1− h)− εσ2

∫ εη

z

Γ(x, ζ, t) dζ, (6.56) E42

2 ddt

(∫ b(t)

a(t)f(x, t)dx

)=∫ b(t)

a(t)ft(x, t)dx+ f(b(t), t) · b′(t)− f(a(t), t) · a′(t).

98

and taking the mean value

− hp = −1

2h2 − εσ2

∫ εη

−1

∫ εη

z

Γ(x, ζ, t) dζ dz, (6.57) E43

and therefore, equation (E386.52) is written

ut + ηx + εuux +σ2

h

∂

∂x

∫ εη

−1

∫ εη

z

Γ(x, ζ, t) dζ dz = − ε

h

∂

∂x

∫ εη

−1

(u2 − u2) dz (6.58) E44

To compare u and u we compute the Taylor3 polynomial of u around the bottom −1. De-noting by ub the horizontal velocity at the bottom and using the boundary condition (

E324.52)

and the irrotationality condition (E294.49), the Taylor polynomial for u is

u(x, z, t) = ub(x, t)−1

2σ2(z + 1)2

∂2ub∂x2

+O(σ4) (6.59) E45

and for vv(x, z, t) = −(z + 1)

∂ub∂x

+1

3!σ2(z + 1)3

∂3ub∂x3

+O(σ4). (6.60) E46

Integrating (E456.59) it follows that

ub = u+1

6σ2h2

∂2u

∂x2+O(σ4, εσ4), (6.61) E48

and consequently,

u(x, z, t) = u+1

6σ2h2

∂2u

∂x2− 1

2σ2(z + 1)2

∂2u

∂x2+O(σ4, εσ4). (6.62) E49

Taking squares in the previous equation we have:

u2(x, z, t) = u2 +1

3uσ2h2

∂2u

∂x2− uσ2(z + 1)2

∂2u

∂x2+O(σ4, εσ4). (6.63) E49b

Integrating (E49b6.63) from −1 to εη and after simplifications follows that:∫ εη

−1

(u2 − u2) dz = O(σ4, εσ4). (6.64) E47

Moreover, the vertical velocity

v(x, z, t) = −(z + 1)∂u

∂x+O(σ2). (6.65) E50

Substituting the last relation into (E416.55) gives

Γ(x, z, t) = −(z + 1)[uxt + εuuxx − ε(ux)2] +O(σ2, εσ2). (6.66) E51

3f(x) = f(a) + (x− a)fx(a) +12! (x− a)2fxx(a) +O(x− a)3.

99

Combining (E516.66), (

E476.64), (

E446.58) and taking into account that the quantity uxt+εuuxx−ε(ux)2

is independent of y we have

ut + ηx + εuux −σ2

3h

∂

∂x[h3(uxt + εuuxx − ε(ux)

2)] = O(σ4, εσ4). (6.67) E52

In this setting we assume that σ ≪ 1 and ε = O(1). Summarising, we derived the Serresystem of equations given by

ηt + [(1 + εη)u]x = 0, (6.68) E53

ut + ηx + εuux −σ2

3h

∂

∂x[h3(uxt + εuuxx − ε(ux)

2)] = O(σ4, εσ4). (6.69) E54

Dropping the high-order terms and going back to dimensional variables is easy to see thatsystem (

E536.68)–(

E546.69) writes

ht + [hu]x = 0, (6.70) E55

ut + ghx + uux −1

3h

∂

∂x[h3(uxt + uuxx − (ux)

2)] = 0, (6.71) E56

with h = d0 + η.

Remark 6.3. The vertical acceleration of the fluid can be estimated by Equation (E416.55).

Specifically, the vertical acceleration vt is

vt = Γ(x, z, t) +O(ε). (6.72) Eqva

Using (E516.66) the vertical acceleration becomes:

vt(x, z, t) = −(z + 1)[uxt + εuuxx − ε(ux)2] +O(ε) , (6.73) Eqva2

and so the analogous part of the quantity inside the square brackets in the Equation (E546.69),

i.e. γ = h(uxt + uuxx − (ux)2) has the physical sense of the vertical acceleration of fluid

particles computed at the free surface z = η.

6.3.2 The ‘classical’ Boussinesq system

Considering long waves of small amplitude, i.e. when σ ≪ 1 and ε ≪ 1 Serre system couldbe simplified more. For example, keeping the terms of O(ε, σ2) in (

E546.69) we obtain the

‘classical’ Boussinesq system,B1872,P1967,BCS2002[Bou72, Per67, BCS02]:

ηt + [(1 + εη)u]x = 0, (6.74) E57

ut + ηx + εuux −σ2

3uxxt = O(σ4, εσ2), (6.75) E58

100

and in dimensional variables, setting the right-hand side equal to zero, we have

ht + [hu]x = 0, (6.76) E59

ut + ghx + uux −1

3uxxt = 0, (6.77) E60

with h = d0 + η.

6.3.3 Unidirectional model equations: The KdV and the BBM equations

The previous systems describe the two-way propagation of water waves. In this section wewill derive equations that describe water waves traveling mainly in one direction, such asthe KdV and the BBM equations.

From the equations (E576.74) and (

E586.75) of the ‘classical’ Boussinesq system we observe that

ηt + ux = O(ε) and ut + ηx = O(ε, σ2) respectively. From these equations the wave equationfollows ηtt + ηxx = O(ε, σ2), from which we choose the component traveling to the right, i.e.the solutions such that ηt + ηx = O(ε, σ2). If we choose such η and set u = η then we obtainan O(1) solution, traveling mainly towards one direction. Moreover, since ηt+ ηx = O(ε, σ2)we deduce the general low order approximation ∂t + ∂x = O(ε, σ2).

To improve the accuracy of the solution we assume that

u = η + εA+ σ2B +O(ε2, σ4), (6.78) Eonew

where A, B are functions of x and t. Substitution into the system (E576.74)–(

E586.75) and collecting

the terms of the same order we have

ηt + ηx + ε(Ax + 2ηηx) + σ2Bx = O(ε2, σ4), (6.79) EA1

ηt + ηx + ε(At + ηηx) + σ2(Bt −1

3ηxxt) = O(ε2, σ4). (6.80) EA2

Using the low order approximation ηt = −ηx+O(ε, σ2), the equations (EA16.79)–(

EA26.80) are then

consistent if A = −14η2 and B = 1

6ηxx. Therefore, (

Eonew6.78) reduces to

u = η − ε

4η2 +

σ2

6ηxx +O(ε, σ2)

and so (E576.74) reduces to the KdV equation:

ηt + ηx +3

2εηηx +

σ2

6ηxxx = O(ε2, σ4) . (6.81) EKdV

Ignoring the high-order terms and turning back to the dimensional and unscaled variables,KdV takes the form

ηt +√gd0ηx +

3

2

√g

d0ηηx +

d306

√g

d0ηxxx = 0 . (6.82) EKdV2

101

Using the usual notation for the linear speed of propagation c0 =√gd0 we rewrite the KdV

equation in the formηt + c0ηx +

3

2

c0d0ηηx +

c0d20

6ηxxx = 0 . (6.83) EKdV3

To derive the BBM equation we use the fact that ηt + ηx = O(ε, σ2) and thus ηxxx =−ηxxt +O(ε, σ2). Substituting the last identity into (

EKdV6.81) we get the BBM equation

ηt + ηx +3

2εηηx −

σ2

6ηxxt = O(ε2, σ4) , (6.84) EBBM

which in dimensional and unscaled variables could be written in the form:

ηt + c0ηx +3

2

c0d0ηηx −

d206ηxxt = 0 . (6.85) EBBM2

The BBM equation was proposed as an alternative of the KdV with some theoretical andnumerical advantages but in practice both models have advantages and disadvantages atthe same time and although the are asymptotically equivalent models they have some bigdifferences in their solutions. We will explore their solutions in more detail later in a laterchapter.

6.3.4 The Shallow Water equations

Assuming that the pressure is hydrostatic, i.e. the vertical accelerations are negligible, andpy = −1, (or in dimensional variables py = −ρg), equation (

E436.57) reduces to hp = 1

2h2.

Therefore, equation (E446.58) with the help of (

E476.64) leads to

ut + ηx + εuux = O(σ4, εσ4), (6.86) Esw1

and the Shallow Water Wave system:

ht + ε[hu]x = 0, (6.87) Esw2ut + ηx + εuux = O(σ4, εσ4). (6.88) Esw3

In dimensional variables, dropping the high order terms the Shallow Water equations arewritten as

ht + [hu]x = 0, (6.89) Esw4ut + ghx + uux = 0, (6.90) Esw5

which is a nonlinear system of conservation laws without dispersive terms. This system canalso be derived using the shallow water assumption σ2 ≪ 1 and keeping only the terms ofO(ε) in (

E606.77).

102

6.4 Traveling waves

John Scott Russell was first who discovered traveling water waves. In his reportrussell1844[Rus44] not

only describe very accurately the shape and the properties of the solitary waves but alsothe relation between the amplitude and the (linear) speed of propagation c =

√gh for water

waves. In this section we will study traveling wave solution of some nonlinear and dispersivewave equations.

6.4.1 Solitary waves of the KdV-BBM equation

We begin with the solitary waves for the general KdV-BBM equation in the form:

ηt + αηx + βηηx − γηxxt + δηxxx = 0 , (6.91) eq:kdvbbm

with α, β, γ, δ ≥ 0.

Solitary waves are traveling wave solutions and therefore we have to search for solution ofthe form

η(x, t) = ϕ(x− ct) , (6.92) eq:swa1

with a given phase speed c > 0. Denoting s = x − ct, using the symbol ϕ′ = ddsϕ(s) and

substituting (eq:swa16.115) into (

eq:kdvbbm6.91) we get the ordinary differential equation

−cϕ′ + αϕ′ + βϕϕ′ + (cγ + δ)ϕ′′′ = 0 .

Integrating and multiplying with ϕ′ the last equation becomes:

−cϕ′ϕ+ αϕ′ϕ+β

2ϕ′ϕ2 + (cγ + δ)ϕ′ϕ′′ = Aϕ′ ,

and integrating once more we have

− c

2ϕ2 +

α

2ϕ2 +

β

6ϕ3 +

cγ + δ

2(ϕ′)2 = Aϕ+B , (6.93) eq:kdvbbm2

where A, B are integration constants. This equation can be re-written in the form(d

dsϕ

)2

=1

3(cγ + δ)p(ϕ), with p(ϕ) = 3(c− α)ϕ2 − βϕ3 + 6Aϕ+ 6B . (6.94) eq:swa2

Depending on the properties of the polynomial p(ϕ) we can compute different kind of solu-tions to (

eq:swa26.94). According to Russell’s description the solitary wave should be symmetric and

decay to 0 for large values x. It is therefore natural to assume that lim|s|→∞|ϕ′(s)|+|ϕ(s)| =0, and the second derivative of ϕ is bounded. This implies that A = B = 0 and (

eq:swa26.94) is

equivalent with the equation(d

dsϕ

)2

=β

3(cγ + δ)ϕ2

(3(c− α)

β− ϕ

). (6.95) eq:swa3

103

Since we are looking for real and bounded solutions, it mush hold:

ϕ ≤ 3(c− a)

β, (6.96) eq:swa4

for s ∈ R. From (eq:swa36.95) we observe that ϕ should be an even function of s that increases from

0 to the maximum value 3(c−α)β

as s increases from −∞ to 0 and then the solution returnsto 0 as s→ ∞. Moreover, for s ≤ 0

d

dsϕ =

√β

3(cγ + δ)ϕ

(3(c− a)

β− ϕ

)1/2

. (6.97) eq:solwavcc

Settingg2 =

3(c− α)

β− ϕ ,

the Equation (eq:solwavcc6.97) becomes:

dg3(c−α)

β− g2

=1

2

√β

3(cγ + δ).

Using the method of partial fractions and integration the last equation gives:

ln

√

3(c−α)β

+ g√3(c−α)

β− g

=c− α

cγ + δs+ d .

Solving for g yields:

g(s) =

√3(c− α)

β

e−√

(c−α)/(cγ+δ)s+d − 1

e−√

(c−α)/(cγ+δ)s+d + 1= −

√3(c− α)

βtanh

[1

2

√c− α

cγ + δs− d

],

which back to the original variables and taking d = 0 is written as

ϕ(s) =3(c− α)

βsech 2

(√c− α

4(cγ + δ)s

),

and thus the solution u(x, t) of the KdV-BBM equations is:

η(x, t) =3(c− α)

βsech 2

(√c− α

4(cγ + δ)(x− ct)

), (6.98) eq:solkdvbbm

Following the previous derivation we actually proved the following lemma:

104

lemma64 Lemma 6.4. The nonlinear third-order ordinary differential equation

αϕ′(s)− βϕ′′′(s) = ϕ(s)ϕ′(s), s ∈ R , (6.99) eq:genode

with α, β > 0 admits solutions of the form

ϕ(s) = 3α sech 2

(1

2

√α

βs

). (6.100) eq:gensolw

It is noted that one can take any translation of the solutions ϕ(s + s0) and observe that itsatisfies the model equation again. We can use this result and find solitary waves for othermodel equations.

Returning back to the KdV and BBM equation in dimensional variable we have α = c0,β = 3

2c0d0

, γ =d206

and δ = d206c0. Therefore, we observe that both KdV and BBM have solitary

waves with the same speed-amplitude relation:

c = c0

√1 +

A

2d0(6.101) eq:spdvampl

but with different shape, cf. Figurefig:swskdvbbm24.

6.4.2 Solitary waves of the Serre equations

We continue with the computation of the solitary-wave solutions of the more challengingSerre equations. For simplicity we consider the Serre equations in a dimensionless butunscaled form:

ht + [hu]x = 0, (6.102) Es1

ut + hx + uux −1

3h

∂

∂x[h3(uxt + uuxx − (ux)

2)] = 0, (6.103) Es2

where η denotes the free-surface elevation, u the depth-averaged horizontal velocity andh = 1 + η. Like before we consider the unknown solutions to be of the form of travelingwaves, i.e. h(x, t) = h(ξ) and u(x, t) = u(ξ) with ξ = x − ct. Substituting into the first(mass) equation (

Es16.102) and integrating with respect to ξ we obtain the relationship between

u and η:u = c

h− 1

h. (6.104) etau1

Substitution into the second (momentum) equation we obtain:

− cu′ + uu′ + h′ =1

3h[h3(−cu′′ + uu′′ − (u′)2)]′ . (6.105) etau2

105

That equation equation can be simplified using the relations between η and u (etau16.104) and

their derivativesu′ = c

h′

h2, u′′ = c

h′′h− 2(h′)2

h3. (6.106) udev1

Hence we rewrite (etau26.105) using the formulas (

etau16.104) and (

udev16.106) in the form:

− ch′

h2+ c2

(h− 1)h′

h2+ h′ = −c2 1

3h

(hh′′ − (h′)2

h

)′

. (6.107) etau3

After simplification of same terms this gives:

− c2h′

h2+ hh′ = −c2 1

3h

(hh′′ − (h′)2

h

)′

, (6.108) etau4

which after integration and division by h becomes

c21

h2+h

2= −c

2

3

(h′

h

)′

+

(c2 +

1

2

)1

h. (6.109) etau5

In order to reduce further the order of the derivatives appeared in (etau56.109) we multiply this

equation with h′/h integrate once again. This leads to the first order ordinary differentialequation:

− c2

h2+ h = −c

2

3

(h′

h

)2

− 2c2 + 1

h+ (c2 + 2) . (6.110) etau6

Rearranging (etau66.110) for the derivative h′ we obtain

c2

3(h′)2 = c2 − (2c2 + 1)h+ (c2 + 2)h2 − h3 . (6.111) etau7

Using now the formula h = 1 + η in (etau76.111) we obtain the simple first order equation

c2

3(η′)2 = (c2 − 1)η2 − η3 , (6.112)

which after differentiation and some simplifications takes the form

c2 − 1

3η′ − c2

9η′′′ = ηη′ , (6.113)

which is of form (eq:genode6.99). From Lemma

lemma646.4 now we obtain the solution:

η(x− ct) = (c2 − 1)sech 2

(1

2

√3(c2 − 1)

c2(x− ct)

). (6.114) swserre

106

Denoting the amplitude of the solitary by A = c2 − 1 and the wavelength parameter λ =√3A/4(1 + A) we write the solitary wave solutions of the Serre system in the form:

η(x, t) = Asech 2[λ(x− ct)], u(x, t) = c0

(1− 1

1 + η(x, t)

), (6.115) eq:swa1

λ =

√3A

4(1 + A)and c =

√1 + A , (6.116) eq:swb1

or in dimensional form:

η(x, t) = Asech 2[λ(x− ct)], u(x, t) = c0

(1− d0

d0 + η(x, t)

), (6.117) eq:swa

λ =

√3A

4d20(d0 + A)and c = c0

√1 +

A

d0. (6.118) eq:swb

It is noted that the solitary wave of the Serre system have the speed-amplitude relationpredicted by Scott Russell in 1844 and which is different than the analogous relation of theKdV and BBM equations. Unfortunately, still there is no exact analytical formula relatingthe speed and the amplitude of the solitary waves of the full Euler equations but onlyasymptotic solutions, such as the one obtained in

Fenton1972,SM1980[Fen72, SM80]

c = c0

(1 +

1

2

A

d0− 3

20

(A

d0

)2

+3

56

(A

d0

)3

+ · · ·

). (6.119) eq:fentonca

A comparison between the three approximations is presented in Figurefig:amplspd25 where we observe

that the solitary waves of the Serre equations are closer to the solitary waves of the Eulerequations.

-5 5x

-0.1

1.1

η

KdV

BBM

Serre

Figure 24: Solitary wave solution of amplitude A = 1 for the KdV, BBM and Serre equations. fig:swskdvbbm

107

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Amplitude

1

1.1

1.2

1.3

1.4

1.5

1.6

Speed

KdV-BBM

Serre

Euler

Figure 25: Comparison between the speed-amplitude relations for the KdV-BBM, Serre andEuler equations. fig:amplspd

For the classical Boussinesq system there are no analytical formulas known, and thus itis a common practice to construct solitary waves numerically for practical use. Numericalgeneration of solitary waves will be discussed later.

Remark 6.5. Scientists were able to construct experimentally stable solitary waves withmaximum amplitude A/d0 ≈ 0.7 and maximum velocity c/c0 ≈ 1.29. Although there is nocomplete theory on the solitary waves of the Euler equations, theoretical approximations andnumerical estimations on the solitary waves of the Euler equations lead to computation ofsolitary waves with maximum amplitude A/d0 ≈ 0.82. Then, instabilities appear in thepropagation of the waves. The solitary waves becomes narrow and steeper as they grow inamplitude,

Fenton1972[Fen72].

6.4.3 Cnoidal waves

Cnoidal waves are periodic traveling waves. Sometimes, are used to describe the oscillationsthat follow an undular bore or just to describe periodic waves in shallow waters. When theirperiod becomes very large then they tend to solitary waves. In this section we will constructsolitary wave solutions for the KdV-BBM equation.

We consider the KdV-BBM equation (eq:kdvbbm6.91). Since cnoidal waves are periodic traveling waves

we assume again that the solution propagates with constant speed c and can be written interms of the unknown function ϕ(s) as η(x, t) = ϕ(x− ct). Substitution into (

eq:kdvbbm6.91) leads to

the ordinary differential equation (eq:swa26.94). Because the cnoidal waves do not necessarily tend

to 0 at infinity we must keep the generic constants A and B in the equation (eq:swa26.94) where we

rewrite here for the shake of convenience:

3(cγ + δ)

(d

dsϕ

)2

= −βϕ3 + 3(c− α)ϕ2 + 6Aϕ+ 6B . (6.120) eq:swb1

108

Without loss of generality we are searching for bounded solutions which have a minimumvalue zero and a maximum value η0 and oscillate between the two values. For boundedsolutions the polynomial on the right hand side of (

eq:swb16.120) must have three real roots. We

assume that B = 0 and that the resulting equation has real solution ϕ1 = 0, ϕ2 = η0 andϕ3 = η0 − κ with κ > η0 > 0 (since the root ϕ3 . Then, equation (

eq:swb16.120) becomes:

3(cγ + δ)

(d

dsϕ

)2

= −βϕ(ϕ− η0)(ϕ− η0 + κ) . (6.121) eq:swb2

Equating the coefficients of the polynomials in (eq:swb16.120) and (

eq:swb26.121) we have 2η0 − κ =

3(c − α)/β and (κ − η0)η0 = 6A/β. Solving the other two equations for positive values weget the solution:

η0 =12

(3(c− α) +

√24Aβ + 9(c− α)2

)/β κ =

√24Aβ + 9(c− α)2/β . (6.122)

Solving the first equation in terms of c we obtain the speed-amplitude relationship c =[(3α+ βη0)η0 − 6A]/3η0. In order to solve (

eq:swb26.121) we change in the variable ϕ = η0 − p2 and

we rewrite (eq:swb26.121) as

dp√(η0 − p2)(κ− p2)

=

√β

12(cγ + δ)ds . (6.123) eq:swb3

Changing the variable again by setting p = √η0q we have

dp√(1− q2)(1−m2q2)

=

√βκ

12(cγ + δ)ds, with m =

√η0κ. (6.124) eq:swb4

The last equation is known to admit solutions of the form

q(s;m) = sn[(

βκ

12(cγ + δ)

)1/2

s;m

],

which can be written as

η(x− ct) = η0 cn 2

[(βκ

12(cγ + δ)

)1/2

(x− ct);m

]. (6.125) eq:kdvcnoidal

It is known that these solutions have period Π = 2(

12(cγ+δ)βκ

)1/2K(m), where K(m) is the

complete elliptic integral of the first kind defined by:

K(m) =

∫ π/2

0

(1−m2 sin2 θ)−1/2 dθ .

Figurefig:cnoidal26 shows a cnoidal wave for the KdV-BBM equation with small value of A. It is noted

that there are two limiting cases with great interest: (i)m→ 0 and (ii)m→ 1. Asm→ 0 the

109

-21.5863 21.5863x

-0.1

0.8

η

Cnoidal wave

Figure 26: A cnoidal wave of the KdV-BBM equation. fig:cnoidal

solution tends to correspond to small-amplitude linear waves since cn (z) → cos(z) while asm→ 1 then the period of the cnoidal wave tends to ∞ and the wave tends to a solitary wavesince cn (z) → sech (z). In analogy with the KdV-BBM equation, the Serre and Boussinesqsystems also admit periodic traveling wave solutions. Specifically, the cnoidal wave solutionsof the Serre equations are given by the formulas:

h(x, t) = a0 + a1sech 2(K (x− ct)) , (6.126) eq:solwavea

u(x, t) = c

(1− a0

h(x− ct)

), (6.127) eq:solwaveb

where

K =

√3a14a20c

2, c =

√a0 + a1,

a0 > 0, and a1 > 0. By taking a0 = 1 and a1 = A the formulas for the classical solitarywaves are obtained.

We will now stop discussing the properties of the dispersive waves and we will proceed withsome numerical methods for water wave equations.

110

Part II

Introduction to numerical methods

7 Finite-difference methods for hyperbolic conserva-tion laws

In the first part of this book we studied mathematical models that describe wave phenomena.We derived scalar conservation laws in the form of first order partial differential equationsand also high order models such as the KdV, BBM equations and other Boussinesq typemodels. In this chapter we will study numerical methods for first order partial differentialequation starting with finite difference methods.

7.1 Introduction to finite-difference methods

We start the presentation of the numerical methods with the simple transport equationwritten in the form

ut + c ux = 0 , (7.1) eq:fdtransport

with c > 0 and initial condition u(x, 0) = u0(x). Because we aim to study practical waterwave problems by numerical means we need to consider the mathematical models in a finiteinterval. For this reason we consider the equation (

eq:fdtransport7.1) in the finite interval I = [a, b].

Moreover, we assume that we want to study the problem up to a finite time t = T . We knowthat in general the solution to this problem can be found analytically and it is u(x, t) =u0(x − ct). Here, the characteristic lines are ξ = x − ct and since c > 0 they start fromthe lines t = 0 and x = a with direction to the right as it is shown in Figure

fig:fdcharact127. Thus,

the values of the solution u(x, t) at a given point (x, t) with x ∈ [a, b] and t ∈ [0, T ] canbe determined using the data at t = 0 and at x = a. Therefore, in order to solve theequation (

eq:fdtransport7.1) in a finite interval we need to impose in addition to the initial condition, an

appropriate boundary condition at only one boundary from which characteristic lines start.For this reason we assume that the solution is given at x = a and it is u(a, t) = f(t). Youmay think this boundary condition as a wave maker that generates waves propagating tothe right.

Remark 7.1. If c was negative, then we should have provided the solution on the otherboundary x = b since the characteristic lines would have had different direction. On theother hand, if we just want to study localised structures such as simple waves of the formu(x, t) = exp(−(x − ct)2) then we could just take the interval I sufficiently large so as thesolution fit in it and take u(a, t) = f(t) = 0 for 0 ≤ t ≤ T . Although the specific initial

111

Figure 27: Characteristic lines of transport equation in a bounded domain. fig:fdcharact1

condition is not zero at all, it decays exponentially to zero and can reach a specific thresholdclose to the machine epsilon very fast (for example 10−10. Recall, that in a computer systemthe number 0 practically does not exist and we approximate it by the number ε known as themachine epsilon, which in double precision arithmetic is approximately ε ≈ 2× 10−16.

In order to construct a numerical method we need first to discretise the domain [a, b]× [0, T ].Since available computer memory is finite we will only be able to approximate the solution ofthe differential equations at a finite number of points in the intervals I and only for certaintimesteps. The set of points in the interval I, will be denoted by Ih = xi, i = 0, · · · , N,which is called the mesh or grid while the points xi are called grid points or nodes. Forsimplicity, we assume that the nodes xi are uniformly distributed in I. More precisely thenodes are such that a = x0 < x1 < · · · < xN = b with

∆x := xi+1 − xi = constant.

112

The distance between the nodes ∆x is called meshlength and in our case we have

xi+1 = xi +∆x

= x0 + i∆x, for i = 0, · · · , N .

Similarly, we consider a temporal grid 0 = t0 < t1 < · · · < tM , where

tj+1 = tj +∆t

= t0 + j∆t, for j = 0, · · · ,M ,

where now ∆t = tj+1− tj is the (uniform) timestep. Usually, we take ∆t,∆x < 1 for reasonsthat will become obvious later.

Since our intention is to approximate the solution u(x, t) at the nodes (xi, tj) we will denotethe approximation of u(xi, tj) by U j

i , i.e.

U ji ≈ u(xi, t

j) , for i = 0, · · · , N, j = 0, · · · ,M .

Figurefig:fdgrids128 shows the computational grid. The blue nodes represent nodes with given initial

and boundary data while the white nodes represent the nodes on which we need to computethe solution U j

i . In this figure we can observe that in order to compute the solution, forexample at the node x1 for t = t1 we must use the data for t = t0 and for x = x0.

Another key ingredient in the discretisation of differential equations using finite differencesmethods is the approximation of the derivatives. The simplest approximation is based onthe definition of the derivative known from calculus. The derivative of a function u(x) is thelimit

ux(x) = limh→0

u(x)− u(x− h)

h= lim

h→0

u(x+ h)− u(x)

h= lim

h→0

u(x+ h)− u(x− h)

2h.

(Note that in the previous relation we do not show the dependence of the function u on t soas to keep the notation simple.) If we want to compute the x-derivative at the point xi (forany t), then according to the definition we can write:

ux(x, t) = limh→0

u(xi, t)− u(xi − h, t)

h,

and if we fix h = ∆x > 0 very small then we have that

ux(xi, t) ≈u(xi, t)− u(xi −∆x, t)

∆x=u(xi, t)− u(xi−1, t)

∆x.

Similarly, we consider the temporal derivative of the function u at t = tj for all x,

ut(x, tj) = lim

h→0

u(x, tj + h)− u(x, tj)

h,

113

Figure 28: A computational grid for the transport equation. fig:fdgrids1

where now if we fix k = ∆t > 0 very small we can approximate the limit by the differencequotient:

ut(x, tj) ≈ u(x, tj +∆t)− u(x, tj)

∆t=u(x, tj+1)− u(x, tj)

∆t.

In this way we can approximate the derivatives either by using the boundary values (forx = x0) or other internal nodes if it is for the node x5. The difference quotient in theapproximation of the derivatives are called finite difference approximations, and for examplethe approximation of the spatial derivative at (xi, t

j)

ux(xi, tj) ≈ u(xi, t

j)− u(xi−1, tj)

∆x, (7.2) eq:backdif

it is know to as the backward finite difference formula for the first derivative. In a similarmanner, the approximation

ux(xi, tj) ≈ u(xi, t

j+1)− u(xi, tj)

∆t, (7.3) eq:backdif

114

is the forward finite difference approximation of the first derivative.

As we mentioned before, at each timestep we will approximate the solution u(xi, tj) by U j

i ,i.e. U j

i ≈ u(xi, tj). Therefore, the derivatives ut(xi, tj) and ux(xi, t

j) can be approximatedas:

ut(xi, tj) ≈ U j+1

i − U ji

∆t; ux(xi, t

j) ≈U ji − U j

i−1

∆x. (7.4) eq:fdapprox1

It is noted that taking the limits ∆x → 0 and ∆t → 0 these approximations of the firstderivatives become exact and coincide with the definition of the derivative. Using equations(eq:fdapprox17.4) in the differential equation (

eq:fdtransport7.1) we reduce the problem from a differential equation to

an algebraic equation:

U j+1i − U j

i

∆t+ c

U ji − U j

i−1

∆x= 0 , for i = 1, · · · , N, j = 0, · · · ,M . (7.5) eq:fdapprox2

If we look carefully the discrete equation (eq:fdapprox27.5) we observe that if the solution (or its ap-

proximation) is given at the time t = tj and for all the nodes xi, for i = 0, · · · , N , then thesolution for the next timestep t = tj+1 at each node xi is given as

U j+1i = U j

i −c ∆x

∆t(U j

i − U ji−1) , for i = 1, · · · , N, j = 0, · · · ,M , (7.6) eq:fdapprox3

and given the initial condition U0i for i = 0, · · · , N , at t0 = 0 and the boundary data U j

0 atx0. This numerical scheme is known to as the Upwind scheme since it uses the data fromleft to right.

For example, given the solution U0i for i = 0, · · · , N we compute the solution U1

i for i =0, · · · , N from the formula

U1i = U0

i − c ∆x

∆t(U0

i − U0i−1) , for i = 0, · · · , N .

With this very simple scheme we can advance the solution to the next time step (from t = tj

to tj+1) using only

• the solution at the previous time step

• data from the left boundary

In the same way we compute the solution U2i at t = t2 using the solution U1

i computed fort = t1, and so on.

Remark 7.2. In the numerical scheme (eq:fdapprox37.6) the approximation of the solution at a new

time step t = tj+1 can be computed in an explicit way from the solution at the previous timet = tj, U j

i and it is the simplest explicit numerical scheme for the solution of the transportequation. This numerical method is known to as the Euler method for time integration.

115

Remark 7.3. If we have used the forward differences approximation for the spatial and thetemporal derivatives then the resulting scheme would have been

U j+1i = U j

i −c ∆x

∆t(U j

i+1 − U ji ) , for i = 0, · · · , N, j = 0, · · · ,M , (7.7) eq:fdapprox4

and so the computation of the value U25 = U1

5 − c ∆x∆t

(U16 −U1

5 ) for example requires the valueU16 which is not included in our grid, cf. Figure

fig:fdgrids128. This scheme works only in the case

of the transport equation with c < 0 where the characteristics have the opposite inclinationand we would have the boundary condition on the right boundary of the domain I and it isknown to as the downwind scheme.

7.2 Finite differences approximations to derivatives

The approximation of the derivatives in the previous section lead to a very simple numericalscheme, but we still know almost nothing about this numerical method. For example wedo not know how good are the approximations, how stable is this numerical scheme tosmall perturbations, (for example the initial data in the computer system will differ fromthe actual data U0

i ≈ u(xi, 0) due to finite precision arithmetic), what appropriate valuesof the discretisation parameters ∆t and ∆x would be appropriate so as to approximatethe solution with some certain accuracy. To answer all these questions we will re-derivethe Upwind method for the advection equation in a more formal way and using Taylor’spolynomial.

The Taylor polynomial for a smooth function u ∈ Cn+1[a, b] at around a point c ∈ [a, b],then

u(x) = u(c) + (x− c)ux(c) +(x− c)2

2!uxx(c) + · · ·+ (x− c)n

n!u(n)(c) +Rn(x) , (7.8) eq:taylorp

where u(k) denotes the k-th derivative of u, Rn is the residual

Rn(x) =(x− c)n+1

(n+ 1)!u(n+1)(ξ)

for some ξ ∈ [a, b]. In other words |Rn(x)| ≤ C|x− c|n+1 for some constant C > 0 or simplerRn(x) = O((x− c)n+1). In general we will use the big-O notation to denote a quantity of acertain order (i.e. f = O(g) means that |f | ≤ C|g|).

Therefore, if we take two consecutive nodes x = xi−1 and c = xi then, according to (eq:taylorp7.8)

u(xi−1) = u(xi) + (xi−1 − xi) ux(xi) +O((xi−1 − xi)2) ,

and since xi−1 − xi = xi −∆x− xi = −∆x we have that

u(xi−1) = u(xi)−∆x ux(xi) +O(∆x2) .

116

Solving for the first derivative we obtain the backward finite difference formula of the deriva-tive:

ux(xi) =u(xi)− u(xi−1)

∆x+O(∆x) . (7.9) eq:fdbackwardfd1

Now it is clear that the approximation of the first derivative by the backward finite differenceformula

ux(xi) ≈u(xi)− u(xi−1)

∆x, (7.10) eq:fdbackwardfd2

will introduce a discretisation error of the order O(∆x) since∣∣∣∣ux(xi)− u(xi)− u(xi−1)

∆x

∣∣∣∣ ≤ C∆x , (7.11)

where C = 12maxa≤s≤b|uxx(s)|.

Similarly if take x = xi+1 and c = xi we have:

u(xi+1) = u(xi) + (xi+1 − xi) ux(xi) +O((xi − xi+1)2) ,

and solving for the first derivative we obtain the forward finite difference formula of thederivative:

ux(xi) =u(xi+1)− u(xi)

∆x+O(∆x) , (7.12) eq:fdbackwardfd3

which has the same order of accuracy with the backward formula.

We can use the same formulas for the approximation of the temporal derivatives by changingxi to tj, xi+1 to tj+1 and ∆x into ∆t and get

ut(x, tj) =

u(tj+1)− u(tj)

∆t+O(∆t) . (7.13) eq:fdforwardfd2

We can obtain higher-order approximation of the first derivative by considering the Taylorpolynmials:

u(xi +∆x) = u(xi) + ∆xux(xi) +∆x2

2uxxu(xi) +O(∆x3) , (7.14) eq:taylorh1

u(xi −∆x) = u(xi)−∆xux(xi) +∆x2

2uxxu(xi) +O(∆x3) . (7.15) eq:taylorh2

Subtracting the two Taylor expansions the second derivatives are canceled and we obtain:

u(xi+1)− u(xi−1) = 2∆xux(xi) +O(∆x3) ,

which gives the second order finite-difference formula for the first derivative:

ux(xi) =u(xi+1)− u(xi−1)

2∆x+O(∆x2) . (7.16) eq:fdcenterfd1

117

High-order derivatives can also be approximated by finite-difference formulas. For example,a finite-difference approximation for the second derivative uxx(xi) can be obtained by addingthe fourth order Taylor polynomials:

u(xi +∆x) = u(xi) + ∆xux(xi) +∆x2

2uxxu(xi) +

∆x3

3!uxxx(xi) +O(∆x4) , (7.17) eq:taylorh3

u(xi −∆x) = u(xi)−∆xux(xi) +∆x2

2uxxu(xi)−

∆x3

3!uxxx(xi) +O(∆x4) . (7.18) eq:taylorh4

and after simplifying the odd derivatives on the right-hand side we obtain:

u(xi −∆x) + u(xi +∆x) = 2u(xi) + ∆x2uxx(xi) +O(∆x4) .

The finite-difference approximation of the second derivative follows after solving the lastequation for the second derivative:

uxx(xi) =u(xi−1)− 2u(xi) + u(xi+1)

∆x2+O(∆x2) . (7.19) eq:fdsecder1

In the rest of the chapter we shall use the finite-differences approximations to discretise(write in a discrete form) partial differential equations. We initiate this study by derivingfinite-differences methods for the transport equation:

ut + c ux = 0 , (7.20) eq:transport2

where c > 0 and given to the initial and boundary conditions u(x, 0) = u0(x) and u(a, t) =f(t) for x ∈ [a, b] and t ≥ 0. Given a spatial and a temporal grid with uniform meshlength∆x and timepstep ∆t the grid nodes are defined as xi+1 = x0 + i ∆x for i = 0, · · ·N andtj+1 = t0 + j ∆t for j = 0, · · · ,M . We will denote the approximations of the solution at thenodes u(xi, tj) by U j

i . Given the approximations of the solution at t = tj (for some givenj) and for all xi, i = 0, · · · , N , the spatial derivative ux(xi, tj) can be approximated by thefinite-differences formulas:

ux(xi, tj) ≈

U ji+1 − U j

i

∆x, with error of O(∆x) , (7.21) eq:fdformula1

ux(xi, tj) ≈

U ji − U j

i−1

∆x, with error of O(∆x) , (7.22) eq:fdformula2

ux(xi, tj) ≈ Ui+1 − Ui−1

2∆x, with error of O(∆x2) . (7.23) eq:fdformula3

Similarly, we will consider the approximation of the temporal derivative by the finite-difference formula:

ut(xi, tj) ≈ U j+1

i − U ji

∆t, with error of O(∆t) , (7.24) eq:fdformula4

We shall use these approximations to derive finite-difference schemes for the numerical so-lution of the transport equation (

eq:transport27.20) with the relevant initial and boundary value data.

118

7.3 The upwind scheme of O(∆x,∆t)

As we have already seen in the previous section, given the initial data U0i for i = 0, · · · , N

and the boundary data U j0 then the upwind scheme for the approximation of the solution

u(xi, tj+1) by the value U j+1

i is

U j+1i = U j

i −c ∆t

∆x(U j

i − U ji−1) , (7.25) eq:upwind

for i = 1, · · · , N and j = 0, · · · ,M . The question is how good is this method. We first tryto explore the properties of this method by numerical means.

7.3.1 Matlab implementation

As an example we consider the equation (eq:fdtransport7.1) with speed c = 1 in the interval I = [0, 2]. The

initial condition we choose is u(x, 0) = e−100(x−0.5)2 . This initial condition is very smooth andcentred at x = 0.5. On the boundary a = 0 the solution has value which is of order 10−11

and it is considered to be 0. Actually, we impose the boundary condition u(a, t) = 0. Forthe discretisation of the differential equation we use a uniform grid with N = 200 intervalsor else N + 1 = 201 nodes with a uniform mesh length ∆x = 2/200 = 0.01. Initially, andfor reasons that we will explain later we choose p = c∆t/∆x to be 1 i.e. we tale ∆x = ∆t(since c = 1).

There are several ways to implement a numerical method depending on the needs and theadopted programming style of the author. In this book we present an implantation whichis easy to understand and quite efficient. In this implementation we first introduce theparameters of the problem such as the discretisation parameters ∆x, ∆t, a, b, c we definethe computational grid (line 15). In line 17 we define the solution matrix U(i,j) to store thesolution U j

i . The difference is that now the indices take the values i = 1, 2, · · · N+ 1 insteadof i = 0, 1, 2, · · ·N , and so in order to impose the boundary condition we write U(1, :) = 0.0,which means that the solution at the first node is zero for each timestep.

Finally, we iterate over the timesteps j = 1, M and we store the solution of the new timestepin the next column of the matrix U(i, j+ 1). We have chosen to store the solution at eachtime step in the matrix U but we could have avoided the use of the matrix by using vectorsand storing the solution in a datafile but we leave this for the reader.

DescriptiveLabel1 %advection1.m2 % Solution of the transport equation with the upwind method3 clear all; close all;4

5 % initialisation6

7 a = 0.0; % Left boundary8 b = 2.0; % Right boundary

119

9 c = 1.0; % advection parametre10 Tfinal = 1.0; %final time11 N = 200; % Number of nodes12 M = 100; % Number of timesteps13 Dx = (b-a)/N; % Uniform mesh length14 Dt = Tfinal/M; % Timestep15 x = a:Dx:b; % Uniform grid16

17 U=zeros(N+1,M+1);18

19 U(:,1) = exp(-100*(x-0.5).^2); % The initial condition20 U(1,:) = 0.0; % The boundary condition21

22 % plot the initial condition23 plot(x,U(:,1),'ko-','MarkerFaceColor', 'k', 'MarkerEdgeColor','k', ...24 'LineWidth', 1.5, 'MarkerSize', 2); hold on;25

26 p = c * Dt / Dx;27

28 for j = 1: M29 for i=2:N+130 U(i,j+1) = U(i,j) - p * (U(i,j)-U(i-1,j));31 end32 end33

34 % plot the solution at the final time T=135 plot(x,U(:,M+1),'ko-','MarkerFaceColor', 'k', 'MarkerEdgeColor','k', ...36 'LineWidth', 1.5, 'MarkerSize', 2);

Listing 1: Matlab implementation of the upwind method

Executing the code we obtain the numerical results depicted in Figurefig:figure1fd29 (a). We observe

that the numerical solution is very close to the exact solution. They are actually so closethat it is impossible to observe differences with eye accuracy. For this reason we will try tointroduce systematic ways of studying the accuracy of a numerical method later. Trying toimprove the accuracy of the method by choosing a smaller value for ∆t for example by taking∆t = ∆x/2 (i.e. p = 0.5) we observe that the numerical solution at t = 1 is very differentfrom the exact solution and the numerical solution for ∆t = ∆x. More precisely, the newnumerical solution has been dissipated, cf. Figure

fig:figure1fd29 (b). On the other hand, taking larger

values for ∆t (i.e. for p > 1 the solution becomes unstable and is increasing in amplitudevery fast. For example when we took p = 1.25 the results are presented in Figure

fig:figure1fd29 (c).

Taking even larger values for p then the solution lead to a blow-up phenomenon, or in otherwords it is destroyed very fast.

This instability phenomenon that depends on the choice of the discretisation parameters ∆x,∆t and c is common in the numerical studies of differential equations and for this reasonfurther analysis of the numerical method is required.

120

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1.2

Figure 29: Propagation of a smooth solution of the transport equation (c = 1) with theupwind method for ∆x = ∆t = 0.01 fig:figure1fd

7.3.2 Consistency

The upwind method can be derived by approximating the derivatives of the equation

ut(xi, tj) + cux(xi, t

j) = 0 , (7.26) eq:origeq

by the finite differences (eq:fdforwardfd27.13) and (

eq:fdbackwardfd27.10). Specifically, it gives

u(xi, tj+1)− u(xi, t

j)

∆t+ c

u(xi, tj)− u(xi−1, t

j)

∆x= O(∆t,∆x) . (7.27) eq:consist

Taking the difference between the original equation (eq:origeq7.26) and the approximation (

eq:consist7.27) we

get that

lim∆t,∆x→0

(ut(xi, t

j) + cux(xi, tj)− u(xi, t

j+1)− u(xi, tj)

∆t− c

u(xi, tj)− u(xi−1, t

j)

∆x

)= 0 ,

(7.28)

121

since tj+1 = tj+∆t and xi−1 = xi−∆x. This means that the numerical scheme is consistent.The definition of the consistency is the following:Definition 7.4. A finite difference scheme L(∆t,∆x)U = f derived to approximate thesolutions of a differential equation Lu = f is called consistent if for any smooth functionv(x, t)

lim∆t,∆x→0

Lv − L(∆t,∆x)v = 0 . (7.29)

Although the numerical scheme is consistent we still don’t know how accurate it is. Specifi-cally, we don’t know what is the error between the actual and the numerical solution.

7.3.3 Convergence and error estimation

The first question that arrises observing the dependence of the numerical solution ∆x and∆t is apart from the discretisation error, which already has been addressed and it is expectedto be of O(∆t,∆x), is the nature of this error. In order to understand what are the effects ofthe discretisation error we define the local truncation error of the numerical method, whichis the error during the computation of the solution at a new timestep by using the exactsolution in the previous timestep.

Rji :=

u(xi, tj+1)− u(xi, t

j)

∆t+ c

u(xi, tj)− u(xi−1, t

j)

∆x. (7.30)

To estimate the local truncation error we consider the Taylor expansions:u(xi, t

j+1) = u(xi, tj) + ∆tut(xi, t

j) + 12∆t2utt(xi, t

j) + C1∆t3

u(xi−1, tj) = u(xi, tj)−∆xux(xi, t

j) + 12∆x2uxx(xi, t

j) + C2∆x3 ,

where C1 = uttt(xi, τ)/6 and C2 = uxxx(ξ, tj) for some ξ ∈ (xi−1, xi) and τ ∈ (tj, tj+1). Then,

the effective equation becomesRj

i = [ut(xi, tj) + 1

2∆tutt(xi, t

j)] + c[ux(xi, tj)− 1

2∆xuxx(xi, t

j)] + (C1∆t2 + C2∆x

2)

= [ut(xi, tj) + c ux(xi, t

j)] + 12[∆t utt(xi, t

j)− c∆x uxx(xi, tj)] + (C1∆t

2 + C2∆x2) .

since u satisfies (eq:fdtransport7.1) then ut = −c ux and so

Rji = [ut(xi, t

j) + c ux(xi, tj)] + 1

2[∆t utt(xi, t

j)− c∆x uxx(xi, tj)] + (C1∆t

2 + C2∆x2)

= [ut(xi, tj) + c ux(xi, t

j)] + 12[∆t c2uxx(xi, t

j)− c∆x uxx(xi, tj)] + (C1∆t

2 + C2∆x2)

= [ut(xi, tj) + c ux(xi, t

j)] + ∆x1

2

(c ∆t

∆x− 1

)cuxx(xi, t

j) + (C1∆t2 + C2∆x

2) .

Because the high-order terms can be again considered negligible we have that if p = 1, i.e.c ∆t/∆x = 1 the local truncation error Rj

i is of order O(∆x2,∆t2) since ut + cux = 0. Onthe other hand if c ∆t/∆x = 1 then we actually solve the following effective equation:

ut + cux +1

2(p− 1) cuxx = 0 . (7.31) eq:diffus1

122

Remark 7.5. The behaviour of the numerical solutions can be better understood by analysingthe differential equation that the numerical scheme satisfies up to the order of the method.This equation is usually called the effective equation and it is not always possible to .

From (eq:diffus17.31) we can see that if p − 1 < 0 the energy of the system dissipates because it is

like solving the diffusion equation and if p − 1 > 0 the energy is increased. With the termenergy we mean a function (which we also call L2-norm) of a solution of (

eq:diffus17.31), defined as:

E(u) =

∫ ∞

−∞u2(x) dx . (7.32)

To see this multiply (eq:diffus17.31) by u and integrate. This leads to:∫ ∞

−∞uut dx = −c

∫ ∞

−∞uux dx−

1

2(p− 1)c

∫ ∞

−∞uuxx dx

1

2

d

dt

∫ ∞

−∞u2 dx = −c1

2

∫ ∞

−∞(u2)x dx+

1

2(p− 1)c

∫ ∞

−∞(ux)

2 dx

d

dtE(u) = (p− 1)cE(ux) .

The last relation shows that

• ddtE(u) > 0 if p− 1 > 0 and thus the energy is increasing with time,

• ddtE(u) < 0 if p− 1 < 0 and thus the energy is decreasing with time, and

• ddtE(u) = 0 if p− 1 = 0 and thus the energy is conserved.

It is noted that the square root of this energy functional coincides with the standard L2-norm i.e. ∥u∥ =

√E(u). A norm in general determines the distances between two objects

(in this case the distance of between the function u and the trivial function 0). Becausethese functions are in general important in mathematical analysis we will consider themagain later.

In other words if p = 1 the local truncation error is of O(∆x2,∆t2) but when p = 1 the localtruncation error is one order bigger O(∆x,∆t).

The most important error is the global error though and not the local error. The total errorprovides and estimate of the actual error introduced by the discretisation at the end of thecomputation. We will denote the global error by

eji = U ji − u(xi, t

j) . (7.33)

123

As a first step we consider the total error at the step j + 1, then

ej+1i = U j+1

i − u(xi, tj+1)

= U ji − p(U j

i − U ji−1)− u(xi, t

j+1) .

Expanding u(xi, tj+1) we get

ej+1i = U j

i − p(U ji − U j

i−1)− [u(xi, tj) + ∆tut(xi, t

j) + 12∆t2utt(xi, τ)] .

Again, since ut = −c ux we have

ut(xi, tj) = −cux(xi, tj) = −cu(xi, t

j)− u(xi−1, tj)

∆x− c ∆x uxx(ξ, t

j) , (7.34)

for some ξ ∈ (xi−1, xi). Moreover, utt(xi, τ) = c2∆t2uxx(xi, τ). Therefore,

ej+1i = eji − p(U j

i − u(xi, tj)) + p(U j

i−1 − u(xi−1, tj)) + (c2∆t2uxx(x, τ)− c∆x∆tuxx(ξ, t

j))

= (1− p)eji + peji−1 +∆t c (c∆tuxx(x, τ)−∆xuxx(ξ, tj)) .

TakingEj = max

i|eji | , (7.35)

and observing that

|c (c∆tuxx(x, τ)−∆xuxx(ξ, tj))| ≤ c (c∆t+∆x)maxx,t|uxx(x, t)| ,

we get the inequalityEj+1 ≤ Ej + C ∆t (∆t+∆x) , (7.36)

Since E0 = 0 the by induction we obtain that Ej ≤ C j ∆t(∆t +∆x), but since tj = j ∆twe obtain the estimate

Ej ≤ C tj(∆t+∆x) , (7.37)max

i|U j

i − u(xi, tj)| ≤ Ctj(∆t+∆x) , (7.38)

or even bettermaxi,j

|U ji − u(xi, t

j)| ≤ C(T )(∆t+∆x) . (7.39)

Therefore, we proved the following theorem:

thrm:conv1 Theorem 7.6. If u(x, t) is a solution of the problem (eq:fdtransport7.1) and u ∈ C2(a, b), then the is a

constant C that depends only on a, b, T and maxx,t|uxx(x, t)| such that the upwind solutionU ji satisfies:

maxi,j

|U ji − u(xi, t

j)| ≤ C(∆t+∆x) .

124

This theorem gives an estimate for the global error between the numerical and the analyticalsolution and states that not only the error is of O(∆t,∆x) but also that the numericalsolution U j

i converges to the actual solution u(xi, tj) provided that ∆x,∆t → 0. In general

we say that a finite difference scheme converges iflim

∆x,∆t→0|u(xi, tj)− U j

i | = 0 ,

for all i, j, and for any initial condition u0(x). In the previous analysis we used the discretemaximum norm ∥U∥∞ = maxi,j |U j

i | which ensures that the solution converges for all valuesof i, j. We continue with some further studies of the numerical method.

7.3.4 Von Neumann stability analysis

Although the Theoremthrm:conv17.6 guaranties the convergence and the accuracy of the method, it

does not fully justify the numerical method since in practice the choice of ∆x and ∆t iscrucial for stable numerical solution. With the term stable we mean solutions that are notgrowing in magnitude unconditionally with time.Definition 7.7. A finite difference scheme applied to a linear partial differential equationwritten in the form U j+1

i = ξU ji where ξ is a linear operator, is called stable if there exists

a constant C, which may depend on the final time T such that|U j

i | = |ξjU0i | ≤ C|U0

i | , (7.40)for all i, j and for any initial condition u0(x).

The definition applies only to linear equations as we shall see bellow but if a scheme is stableor unstable for a linear equation then we expect to be unstable for nonlinear equations aswell since nonlinear equations can be considered as perturbations of their linearisation. Itis obvious that in order for |ξ|j to be bounded by a constant C it is required that |ξ| ≤ 1,otherwise it will grow to infinity with time.

One way to study the stability of a finite difference method is by using the von Neumannstability analysis. This method can be applied only to linear equations and it is based onthe method of separation of variables where we assume that the solution consists of linearcombination of simple waves. For example assuming that the solution is given in the form

U ji =

∞∑ℓ=−∞

Aℓeikℓxieωt

j

=∞∑

ℓ=−∞

Aℓeikℓi∆xeωj∆t

=∞∑

ℓ=−∞

Aℓξjeikℓi∆x ,

125

where kℓ is the wavenumber, ω the frequency, ξ = eω∆t and i is the imaginary unit thatsatisfies the identity i2 = −1. Because the advection equation is linear it suffices to studyeach Fourier mode separately , i.e. assume that

U ji = ξjeiki∆x. (7.41) eq:fdtr1

In other words we assume that U ji = ξjU0

i . Since ξ > 0 the term ξj is bounded only if|ξ| ≤ 1. Therefore, in order for the solution (and the method) to be stable we will require

|ξ| ≤ 1 . (7.42)

The term ξ is known to as the amplification factor. Looking more carefully Equation (eq:fdtr17.41)

we observe thatU j+1i = ξj+1eiki∆x = ξ U j

i , (7.43) eq:fdtr2

andU ji+1 = ξjeik(i+1)∆x = eik∆x U j

i . (7.44) eq:fdtr3

Substitution into the Upwind method (eq:upwind7.25) we get

ξU ji = U j

i −c ∆t

∆x(1− eik∆x)U j

i .

Since we work for a general solution we can simplify U ji and by setting p = c ∆t

∆xwe have

ξ = 1− p(1− eik∆x)

= 1− p(1− cos(k∆x)) + ip sin(k∆x) .

In the last relation we used the Euler formula eiθ = cos θ + i sin θ. The modulus square of ξis

|ξ|2 = 1− 2p (1− p) (1− cos(k∆x)) , (7.45)which is |ξ|2 ≤ 1 (the solution remains bounded) as long as 1−p ≥ 0 (since −1 ≤ cos(k∆x) ≤1), or better

p ≡ c ∆t

∆x≤ 1 . (7.46) eq:cfl1

This condition is known to as the CFL condition due to Courant-Friedrichs-Levy and ingeneral is a condition between ∆t and ∆x that must be satisfied in order for the numericalsolutions to be stable. In the specific example one can see that ∆t must be less than∆x/c. This condition ensures that the propagation speed c remains always smaller than thenumerical speed ∆x/∆t since

c = p∆x

∆t≤ ∆x

∆t. (7.47) eq:domdep

Having in mind that c is the characteristic speed, we can relate the stability condition (eq:domdep7.47)

with the domain of dependence of the solution u(xi, tj). We first give the definition of the

numerical domain of dependence:

126

Definition 7.8. For any point (xi, tj) of the grid where we compute the approximation of thesolution u, we define the numerical domain of dependence to be the points (xℓ, t) for t < tj

such that the values U0ℓ determine the value of the approximation U j

i .

Now, using the upwind scheme in order to compute the approximation U ji (going j steps

backwards) we need the solutions from U0i−j until U0

i . These values correspond to the pointsxi− j∆x, · · ·xi and so the numerical domain of dependence is the set xi− j∆x, · · ·xi. Thephysical domain of dependence is the interval (xi − ctj, xi) = (xi − cj∆t, xi). If now we takec < ∆x/∆t then the right boundary of the physical domain of dependence is xi − cj∆t >xi − j∆x which means that the numerical domain of dependence includes the physical one.

Therefore, the CFL condition implies that a finite difference scheme is stable if the numericaldomain of dependence is larger than the physical one. This is shown in Fig.

fig:dependdom31.

Figure 30: The backward differences allow information propagate from left to right. Theshaded area depicts a possible physical domain of dependence while the dashed lines thenumerical domain of dependence of the solution U j+1

i . For a stable scheme the physicaldomain of dependence must be included into the numerical domain of dependence fig:dependdom0

If c < 0 then the scheme we described so far cannot be applied since the waves will travelfrom left to right. Since also the characteristic lines will have a negative slope then thephysical domain of dependence will always lie outside the numerical domain of dependence.For this reason we need to change the scheme appropriately by considering forward finitedifferences for the spatial derivatives of the form (

eq:fdformula17.21). The resulting numerical scheme will

take the form of (eq:fdapprox17.4) and is called the downwind scheme, cf. also Fig.

fig:dependdom31.

Remark 7.9. It is noted that the von Neumann stability analysis provides with a necessarybut not sufficient condition for a stable numerical method. This means that a numericalscheme can be stable for the specific solutions of the Neumann analysis but it can be unstablefor others. Another disadvantage of this analysis is that it can be applied only to linear partial

127

differential equations but as we saw in the first part of this book, most of the mathematicalmodels for waves are nonlinear. For this reason, we usually try to prove stability andconsistency of the numerical scheme. Due to Lax equivalence theorem, consistency andstability is equivalent to convergence. In addition, convergence usually provides with theglobal error estimates that are extremely useful to quantify the accuracy of the numericalmethod.

The upwind scheme depends on the sign of the speed parameter c and thus we can obtainsome satisfactory results only in one-way propagation models. In models such as the waveequation where the waves propagates in both directions the upwind scheme is not a goodchoice and for this reason we present more numerical schemes bellow.

7.4 An unstable scheme

In order to derive a finite difference scheme with better accuracy than the upwind schemethat we will be able to use for any value of the constant c we will make an attempt todiscretise the spatial derivative of the transport equation ut+cux = 0 by the finite differenceformula (

eq:fdformula37.23). This formula has a discretisation error of order O(∆x2) and allows the

propagation of information in both directions since the numerical domain of dependence forthe solution U j+1

i can include characteristics with positive and negative slope, i.e. positiveand negative characteristic speeds, cf. Fig.

fig:dependdom31. Keeping the discretisation of the temporal

Figure 31: The central differences allow information propagate in both directions. Theshaded area depicts a possible physical domain of dependence while the dashed lines thenumerical domain of dependence of the solution U j+1

i . fig:dependdom

derivative the same as before, the resulting numerical scheme can be written in the form:

U j+1i = U j

i −c ∆t

2∆x(U j

i+1 − U ji−1) . (7.48) eq:fdunstable

128

Although this scheme sounds very promising a Fourier stability analysis shows that thismethod is unconditionally unstable, which means that the numerical solution will grow withtime exponentially fast independent of the choice of ∆x and ∆t. Specifically, assumingthat the solution has the form (

eq:fdtr17.41), i.e. U j

i = ξjeiki∆x, and substituting U j+1i = ξ U j

i ,U ji+1 = eik∆x and U j

i−1 = e−ik∆x into (eq:fdunstable7.48) we get:

ξ = 1− c ∆t

2∆x

(eik∆x − e−ik∆x

)= 1− c ∆t

2∆x(cos(k∆x) + i sin(k∆x)− cos(k∆x) + i sin(k∆x))

= 1− ic ∆t

∆xsin(k∆x) .

Therefore,

|ξ|2 = 1 +

(c ∆t

2∆xsin(k∆x)

)2

> 1 ,

which implies that the solution grows in amplitude and the scheme is unstable. Becausethis unconditional instability we will not use the specific scheme but we will try to deriveother schemes that are more accurate and stable at the same time. One such finite differencescheme is the Lax-Friedrichs scheme.

7.5 The Lax-Friedrichs scheme

In order to improve the stability of the previous unstable scheme we modify the discretisationof the temporal derivative based on the following observation: Expanding u(xi+1) and u(xi−1)using first order Taylor polynomials we get

u(xi+1, tj) = u(xi, t

j) + ∆xux(xi, tj) +O(∆x2) ,

u(xi−1, tj) = u(xi, t

j)−∆xux(xi, tj) +O(∆x2) ,

that leads to the first order approximation

u(xi, tj) =

u(xi+1, tj) + u(xi−1, t

j)

2+O(∆x) . (7.49)

Therefore we consider the approximation U ji ≈ (U j

i+1 + U ji−1)/2 in the approximation of the

temporal derivative. In other words we consider the approximation

ut(xi, tj) =

U j+1i − 1

2(U j

i+1 + U ji−1)

∆t, (7.50)

of the temporal derivative and the central difference formula for the approximation of thespatial derivative:

ux(xi, tj) =

U ji−1 − U j

i−1

2∆x. (7.51)

Using Taylor expansions one can see easily that the approximation

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7.6 The Lax–Wendroff scheme

7.7 The Leapfrog scheme

7.8 Norms and the Lax equivalence theorem

In all the theoretical studies of the numerical methods we used the absolute value of thediscrete solutions. For example we proved for the upwind method that maxij |U j

i −u(xi, tj)| ≤C(∆x + ∆t). In this case we could have defined a solution vector Uj = (U j

0 , Uj1 , · · · , U

jN)

T

and the exact solution vector ≊j = (u(x0, tj), u(x1, t

j), · · · , u(xN , tj))T ; and try to find thedistance between the two vectors. Sometimes in order to define the distance between twovectors, or two matrices (or even functions) we use a function defined as ∥ · ∥ : X → R+ thatmaps the space X into the space real positive numbers. These mappings are called normsand in order to define matrix norms we first need to define vector norms. In what followswe consider the vectors u = (u0, · · · , uN)T and v = (v0, · · · , vN)T .Definition 7.10. The mapping ∥·∥ : Cn → R+ is a norm if and only if satisfies the followingthree properties:

(i) ∥u∥ ≥ 0 and ∥u∥ = 0 if and only if u = 0, for all vectors u ∈ Cn

(ii) ∥cu∥ = |c| ∥u∥, for all c ∈ C and for all vectors u ∈ Cn

(iii) ∥u+ v∥ ≤ ∥u∥+ ∥v∥, for all vectors u, v ∈ Cn.

(Where Cn can be replaced by the space Rn if the vectors are real instead of complex.)

An immidiate consequence of properties (ii) and (iii) is that |∥u∥ − ∥v∥| ≤ ∥u± v∥.

Usually, in numerical analysis we use the following three norms:

∥u∥1 =N∑i=0

|ui| , (ℓ1-norm)

∥u∥2 =

√√√√ N∑i=0

|ui|2 , (ℓ2-norm or Euclidean norm)

∥u∥∞ = maxi=0,··· ,N

|ui| , (ℓ∞-norm or maximum norm)

In the context of finite difference methods the energy norm can be written in the form

∥U j∥2 =

√√√√∆xN∑i=0

|U ji |2 , (7.52)

130

where its square ∥U j∥22 approximates the energy of the solution u(x, tj):

E(u) =

∫ b

a

u2(x, tj) dx ,

which we used in a previous section. It is noted that we didn’t include in this notation thetime tj since these norms are only related to the spatial discretisation.

Using vector norms we can generalise the definitions of consistency, convergence and stabilityin the following form: We consider a general numerical method where the approximation toa smooth solution u(x, t) can be expressed in the following form:

U j+1 = SU j . (7.53)

Here S denotes a (linear) operator and U j = (U j0 , U

j1 , · · ·U

jN)

T . We will also consider thesolution vector uj = (u(x0, t

j), u(x1, tj), · · · , u(xN , tj))T and a norm ∥ · ∥. Then we say that

the numerical scheme

• is convergent iflim

∆t,∆x→0∥uj − U j)∥ = 0 , (7.54)

• is consistent iflim

∆t,∆x→0∥uj+1 − Suj)∥ = 0 , (7.55)

• has order of accuracy p in space and q in time if

∥uj − U j∥ ≤ C(∆xp +∆tq) , (7.56)

• is stable if∥U j∥ ≤ C∥U0∥ , (7.57)

for some generic constant C that might depends on T and the interval [a, b]. We observethat if a numerical method has order of accuracy p ≥ 1 in space and q ≥ 1 in time then itis convergent. Due to Lax equivalence theorem we also have the following result:

Theorem 7.11. A consistent finite different scheme is convergent if and only if it is stable.

Although consistency and stability suffices to prove convergence, it is always very importantto know the order of accuracy of the numerical method and therefore we usually try to findthe constant p and q even by using numerical means.

131

7.9 Finite-difference methods for the two-point boundary-valueproblem

7.10 Finite-difference methods for the wave equation

8 Finite-difference methods for nonlinear and disper-sive wave equations

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