Finite Volume Schemes: an introduction First lecturedispense.dmsa.unipd.it/putti/dottorato/finitvol1.pdf · 2009-07-15 · Finite Volume Schemes: an introduction First lecture Annamaria

Finite Volume Schemes: an introductionFirst lecture

Annamaria Mazzia

Dipartimento di Metodi e Modelli Matematici per le Scienze ApplicateUniversita di Padova

[email protected]

Scuola di dottorato inScienze dell’Ingegneria Civile e Ambientale

Corso di Metodi Numericimarzo 2008

Outline

Hyperbolic Conservation LawsClassical and Weak solutionsThe Riemann problemFinite Volume Methods in 1DFinite Volume Methods in 2D

The lectures about Finite Volume and Mixed Hybrid FiniteElement will be on line athttp://dispense.dmsa.unipd.it/mazzia

A. Mazzia (DMMMSA) Finite Volume a.a. 2007/2008 2 / 34

Hyperbolic Conservation LawsFinite Volumes (FV) schemes are most useful for modellinghyperbolic conservation laws, which, in their simplest(scalar) form may be expressed by the PDE

ut + ~∇ ·~f = 0

on a domain Ω, with u(x, t) specified on inflow boundaries.

Example: the equation for conservation of mass ingroundwater contamination problem.Assumptions: no diffusion, no mechanical dispersion,density of the contaminant and the water constant,velocity of the fluid mixture constant,Let ρ(x , t) be the density of the contaminant at point xand time t and v(x , t) be the velocity of the fluidmixture.


The total pollutant mass in two sections 1 and 2 is givenby∫ x2

x1ρ(x , t) dx . If no source or sinks are present, the

mass can change only because of the mixture flowingacross the endsections in x1 or x2.The rate of mass change is given by:

ddt

∫ x2

x1

ρ(x , t) dx = ρ(x1, t)v(x1, t)− ρ(x2, t)v(x2, t)

By integrating in time from tk to tk+1 we obtain theintegral form of the conservation law. Assuming ρ andv differentiable, with simple calculations we obtain thedifferential form of the conservation law:

ρt + (ρv)x = 0


Classical and Weak solutionsSolution of hyberbolic conservation laws may be obtainedif initial and boundary conditions are given.The simplest problem is the initial value problem, orCauchy problem, defined for −∞ < x <∞ and t ≥ 0. Wemust specify initial conditions only:

ut + ~∇ ·~f = 0u(x , 0) = u0(x) −∞ < x <∞.

Classical solutions of this problem are constant along thecharacteristics curves defined by x(t) such that

dxdt

= f ′(u(x(t), t)) t ≥ 0

x(0) = x0


Shock waves

u is constant along these characteristics and thecharacteristics travel at constant velocity which is equal tof ′(u0(x0)).Generally, classical solutions exist only for t ∈ [0, T ∗] where

T ∗ = − 1infx u′0(x)f ′′(u0(x))

.

At the time t = T ∗ the characteristics first cross, the functionu(x , t) has an infinite slope – the wave is said to break byanalogy with waves on a beach – and a shock forms.


Weak solutions

To allow discontinuities, which arise in a natural way in thissituation, we define a weak solution of conservation law.

Definition (Weak solution)A function u(x , t), bounded and measurable, is called aweak solution of the conservation law, if for eachφ ∈ C∞′ (R× R+), the following equality holds:∫ ∞

0

∫ +∞

−∞[φtu + φx f (u)] dx dt = −

∫ +∞

−∞φ(x , 0)u(x , 0) dx .


The Riemann problem

Definition (The Riemann problem)A Riemann problem is the conservation law together withinitial data consisting of two constant states separated bya single discontinuity,

u0(x) =

ul x < 0,ur x > 0.

Example: The Riemann problem in Burgers’equation

ut + (12

u2)x = 0


If ul > ur , there is a unique weak solution,

u(x , t) =

ul x < stur x > st .

s =(ul + ur )

2is the shock speed

If ul < ur , there are infinitely many weak solutions, sincebetween the points ult < x < ur t , there is noinformation available from the characteristics. Todetermine the correct physical behavior we adoptthe vanishing viscosity approach by adding the termεuxx depending on small parameter ε

ut + (12

u2)x = εuxx .


If the initial data is smooth and ε very small then,before the wave begins to break, the εuxx term isnegligible compared to other terms and the solutionsto the two PDEs look nearly identical. As the wavebegins to break, the term uxx grows much faster thanux and at some point the εuxx term is comparable tothe other terms and begins to play a role. This termkeeps the solution smooth for all time, preventing thebreakdown of solutions that occurs for the hyperbolicproblem. The weak solution is called rarefaction waveor expansion fan:

u(x , t) =

ul x < ultx/t ult ≤ x ≤ ur tur x > ur t


ul

ur

0

Shock wave

0

ul

ur

Rarefaction wave


Some common numerical schemes

To solve numerically hyberbolic conservation laws somecommon numerical schemes are:

Finite DifferencesFinite VolumesFinite ElementsDiscontinuous Galerkin· · ·

We will see more in detail (but not too much...) the FiniteVolume approach.


Properties of the ideal scheme

The ideal scheme for the resolution of hyperbolicconservation laws would reflect the properties of theunderlying physical and mathematical system:

Conservative: for correct capturing of discontinuities;Accurate: obvious....;Free of spurious oscillations: spurious oscillations haveno physical meaning. This goal is remarkably difficult toachieve;Continuous: for convergence to the steady statesolution.


The Finite Volume method

In the Finite Volume method the three main steps to followare:

Partition the computational domain into controlvolumes (or control cells) - wich are not necessarily thecells of the mesh.Discretize the integral formulation of the conservationlaws over each control volume (by applying thedivergence theorem).Solve the resulting set of algebraic equations orupdate the values of the dependent variables.


FV in 1D

We will see two FV schemes in 1D: Godunov’s method andvan Leer’s method.

Godunov’s method uses the exact solution of the localRiemann problem and does not produce oscillationsaround discontinuities, but it is only first order accurateand the solutions display large numerical viscosity.van Leer’s method can be viewed as a generalizationof Godunov’s method, it is a high resolution method,characterized by second order accuracy on smoothsolutions and the absence of spurious oscillations.

Both methods are conservative.


Conservative method

DefinitionGiven a uniform grid with time step ∆t and spatial meshsize ∆x , a numerical method is said to be conservative ifthe corresponding scheme can be written as:

vn+1j = vn

j − λ(gnj+ 1

2− gn

j− 12), j ∈ Z n ≥ 0

where vnj approximates u(xj , tn) at the point

(xj = j∆x , tn = n∆t), λ =∆t∆x

and g : R2k −→ R is a

continuous function, called the numerical flux (function),that defines a (2k + 1)-point scheme.

gnj+ 1

2= g(vn

j−k+1, . . . , vnj+k).


Motivation of the conservative scheme

vnj can be view as an approximation of

unj =

1∆x

∫ xj+1/2

xj−1/2

u(x , tn) dx .

unj is the average of u(·, tn) on the cell [xj−1/2, xj+1/2]

(where xj±1/2 = xj ±∆x2

).

From the integral form of the conservation law, wehave:∫ xj+1/2

xj−1/2

u(x , tn+1) dx =

∫ xj+1/2

xj−1/2

u(x , tn) dx

− [

∫ tn+1

tnf (u(xj+1/2, t)) dt −

∫ tn+1

tnf (u(xj−1/2, t)) dt ].


Dividing by ∆x and using unj we get

un+1j = un

j −1

∆x[

∫ tn+1

tnf (u(xj+1/2, t)) dt−∫ tn+1

tnf (u(xj−1/2, t)) dt ].

The numerical flux function can be considered as anaverage flux through xj+1/2 over the time interval[tn, tn+1],

gnj+1/2 =

1∆t

∫ tn+1

tnf (u(xj+1/2, t)) dt .


Godunov’s scheme1. Projection step Given data vn

j at time tn, construct apiecewise constant function vn

j (x , tn) defined by

vnj (x , tn) = vn

j xj−1/2 ≤ x ≤ xj+1/2.

| | |x_(j−1) x_j x_(j+1)

v_(j−1)

v_j

v_(j+1)


2. Evolution step Solve the local Riemann problem at thecell interfaces, that is, on each subinterval [xj , xj+1] andfor t ≥ tn, solve

∂vnj

∂t+∂f (vn

j )

∂x= 0

vnj (x , tn) =

vn

j , xj < x < xj+1/2

vnj+1, xj+1/2 < x < xj+1

3. Projection step Define the approximation vn+1j at time

tn+1 by averaging the Riemann problem solution vnj at

the time tn+1, so that

vn+1j =

1∆x

∫ xj+1/2

xj−1/2

vnj (x , tn+1) dx .

These values are then used to define new piecewiseconstant data vn+1

j (x , tn+1) and the process repeats.


Example

In the simple case of a linear advection equationut + aux = 0, with a > 0, the third step gives

vn+1j = vn

j − a∆t∆x

(vnj − vn

j−1).

| | |x_(j−1) x_j x_(j+1)

<−>a t

<−>a t

u_(j−1)

u_j

u_(j−1)


Stability condition

The CFL (Courant-Friedrichs-Lewy) condition

λmaxu|f ′(u)| ≤ 1

2,

where λ =∆t∆x

is a necessary stability condition stating that

the domain of dependence of the method includes thedomain of dependence of the PDE.


van Leer’s method

First and third steps of Godunov’s methods are of anumerical nature and can be modified withoutinfluencing the physics, in order to define spatially highorder accurate scheme.van Leer’s scheme is a spatially second orderaccurate scheme (known also as MUSCL (MonotoneUpstream-centered Scheme for Conservation Laws)method).The straightforward replacement of the first-orderscheme by a second-order accurate interpolationleads to the generation of oscillations arounddiscontinuities.


To overcome this limitations, non linear componentsare introduced.This concept was introduced by van Leer under theform of limiters, i.e. functions that control the gradientof the computed solution to prevent the appearanceof unphysical over/under shoots.


1 Reconstruction step: the dependent variable isinterpolated using a piecewise linear function vstarting from the cell averages vn

j , vnj−1, vn

j+1. To this enddefine Sn

j to be the slope on the jth cell calculatedusing vn

j−1 and vnj+1. Then

vn(x) =

vn

j + (x − xj)Sn

j

∆xxj−1/2 < x < xj+1/2,

vn(xj−1/2) x ≤ xj−1/2

vn(xj+1/2) x ≥ xj+1/2.

Note that taking Snj = 0 for all j and n recovers

Godunov’s method;


2 Evolution step: the waves are propagated across cellinterfaces according to an exact or approximatesolution of a local Riemann problem that uses theinterpolated values vn(x) as initial conditions. Onesolves

∂

∂tw +

∂

∂xf (w) = 0 x ∈ R, tn ≤ t ≤ tn+1

w(x , tn) = vn(x).

This step yields w(·, tn+1).3 Cell-averaging step: vn+1

j is obtained by projecting thesolution w(x , tn+1) onto the piecewise constantfunctions

vn+1j =

1∆x

∫ xj+1/2

xj−1/2

w(x , tn+1) dx .


d)

xx xj j+1j-1x x x

j+1jj-1

x x xj-1 j j+1 x x xj-1 j j+1

∆ tv

a) b)

c)

a) reconstruction; b) limiting; c) evolution; d)cell-averaging step.


Slope limiterThe numerical scheme is completely defined afterspecification of the slope limiter Sn

j .Poor choice of slopes may give oscillations in the solution.Generation of oscillations can be prevented by acting ontheir production mechanism and introducing nonlinearcorrection factors, so called the limiters, that force themethod to be total variation diminishing (TVD).

DefinitionA numerical method to solve hyperbolic conservation lawsis called total variation diminishing if

+∞∑j=−∞

|vn+1j+1 − vn+1

j | ≤+∞∑

j=−∞

|vnj+1 − vn

j |


To obtain a slope limiter, let

Snj = Sn

j Φnj

where Snj represents the actual slope approximation and

Φj = Φ(θnj ) is a limiter function, defined in such a way that

the method is TVD and θnj is the ratio of two consecutive

gradients, i.e.:

θnj =

vnj − vn

j−1

vnj+1 − vn

j.


Limiter functionsVarious limiter functions can be found in literature. Forexample:

van Leer’s limiter:

Φ(θ) =|θ|+ θ

1 + |θ|,

minmod limiter that represents the lowest boundary ofthe second-order TVD region:

Φ(θ) =

min(θ, 1) if θ > 00 if θ ≤ 0.

the superbee limiter, that represents the upper limit ofthe second-order TVD region and has beenintroduced by Roe:

Φ(θ) = max[0,min(2θ, 1),min(θ, 2)].


Numerical example with periodic boundaryconditions

ut + ux = 0, with initial condition u0(x) periodic ofperiod 2 defined on the interval [−1, 1] as

u0(x) =

1 −1 ≤ x ≤ −0.750 −0.75 < x < −0.251 −0.25 ≤ x ≤ 0.250 0.25 < x < 0.751 0.75 ≤ x ≤ 1

We solve this problem at time t = 2 s with λ = 0.5, bysetting ∆t = 1× 10−2 s and ∆x = 2× 10−2 m.


−1.0 −0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

(a) Godunov’scheme

−1.0 −0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

(b) van Leer’schemevan Leer limiter

−1.0 −0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

(c) van Leer’s schememinmod limiter

−1.0 −0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

(d) van Leer’s schemesuperbee limiter


Numerical example with non periodicboundary conditions

ut + ux = 0, with the following initial condition:

u(x , 0) = u0(x) =

sin(πx) 0 ≤ x ≤ 20 otherwise.

We solve this problem at time t = 2 s with λ = 0.5 and∆x = 4× 10−2 m.


0.0 2.0 4.0−1.0

−0.5

0.0

0.5

1.0

(a) Godunov’s scheme

0.0 2.0 4.0−1.0

−0.5

0.0

0.5

1.0

(b) van Leer’s schemevan Leer limiter

0.0 2.0 4.0−1.0

−0.5

0.0

0.5

1.0

(c) van Leer’s schememinmod limiter

0.0 2.0 4.0−1.0

−0.5

0.0

0.5

1.0

(d) van Leer’s schemesuperbee limiter


Documents

Finite Volume Schemes: an introduction First lecturedispense.dmsa.unipd.it/putti/dottorato/finitvol1.pdf · 2009-07-15 · Finite Volume Schemes: an introduction First lecture Annamaria