Chapter 4

Types of Governing equations

4-1 Physical Classification (1)

Equilibrium problems:

(1) They are problems in which a solution of a given PDE is desired in a

closed domain subject to a prescribed set of boundary conditions.

Equilibrium problems are boundary value problems.

(2) Sometimes, equilibrium problems are referred to as jury problems,since

the solution of the PDE at every point in the domain depends upon

the prescribed boundary condition at every point on B.

(3) Mathematically, equilibrium problems are governed by elliptic PDEs.

4-1 Physical Classification (2)

(4) Example 1 of equilibrium problem:

The steady-state temperature distribution in a conducting

medium is governed by Laplace’s equation.

with boundary conditions

T(0,y)=0, T(1,y)=0, T(x,0)=T0, T(x,1)=0

02

2

2

22

y

T

x

TT 10;10 yx

4-1 Physical Classification (3)

(5) Example 2 of equilibrium problem:

The irrotational flow of an incompressible inviscid fluid is

governed by Laplace’s eq.

02

4-1 Physical Classification (4)

Marching Problems:

(1) Marching or propagation problems are transient or transient-like problems where the solution of a PDE is required on an open domain subject to a set of initial conditions and a set of boundary conditions.

(2) Problems in this category are initial value or initial-boundary problems. The solution must be computed by marching outward from the initial data surface while satisfying the boundary conditions. (3) Mathematically, these problems are governed by either hyperbolic or parabolic PDEs.

4-1 Physical Classification (5)

(4)   Example of marching problem:

Determine the transient temperature distribution in a 1-D solid,

as shown in the following figure.

(5) Typical examples of marching problems include unsteady

inviscid flow, steady supersonic inviscid flow, transient heat

conduction and boundary-layer flow.

4-2 Mathematical Classification (1)

The general second-order PDE can be expressed as

where a,b,c,d,e, and f are functions of (x,y), i.e., we consider a linear equation.

The classification of a second-order PDE depends only on the second-derivative terms of the equation, so we may rearrange the above equation as

As in the classification of general second-degree equations in analytic geometry, the PDE is classified as(1) Hyperbolic if b2-4ac > 0(2) Parabolic if b2-4ac = 0(3) Elliptic if b2-4ac < 0

),( yxgfedcba yxyyxyxx

Hgfedcba yxyyxyxx )(

4-2 Mathematical Classification (2)

Example of Hyperbolic PDE:

on the interval

with initial condition u(x,0)=f(x), ut(x,0)=g(x)

Solution:

2

22

2

2

x

uc

t

u

ctx

ctx

dgc

ctxfctxftxu )(

2

1

2

)()(),(

x

4-2 Mathematical Classification (3)

Example of Parabolic PDE:

2

2

y

u

t

u

0),0( yu

0)0,( tUtu

0),( tu

4-2 Mathematical Classification (4)

Parabolic PDEs are associated with diffusion processes. The solutions of parabolic equations clearly show this behavior. While the PDEs controlling diffusion are marching problems, i.e., we solve them starting at some initial data plane and march forward in time or in a time-like direction, they do not exhibit the limited zones of influence that hyperbolic equations have. In contrast, the solution of a parabolic equation at time t1 depends upon the entire physical

domain (t t1), including any side boundary conditions.

4-2 Mathematical Classification (5)

Example of Elliptic PDEs:

Given Laplace’s equation on the unit disk

subject to boundary conditions

1002 ru

)(),1( fr

u

4-2 Mathematical Classification (6)

Domain of Dependence:

4-3 The Well-Posed Problem (1)

In order for a problem involving a PDE to be well-posed, the solution to the problem must exist, must be unique, and must depend continuously upon the initial or boundary data.

Initial and Boundary Conditions(1) In order to obtain a unique solution of a PDE, a set of

supplementary conditions must be provided to determine the arbitrary functions which result from the integration of PDE. The supplementary conditions are classified as boundary or initial conditions. (2) An initial condition is a requirement for which the dependent variable is specified at some initial state. Ex: , for t=0, T=f(x, y)T

t

T 2

4-3 The Well-Posed Problem (2)

(3) A boundary condition is a requirement that the dependent

variable or its derivative must satisfy on the boundary of

the domain of the PDE.

(4) Various types of boundary conditions which will be encountered

are

(a) The Dirichlet boundary condition: the dependent variable

along the boundary is prescribed.

4-3 The Well-Posed Problem (3)

(b) The Neumann boundary condition: the normal gradient of the dependent variable along the boundary is specified.

 

)(yfx

T

10T

x

y0T

)(xfy

T

4-3 The Well-Posed Problem (4)

(c) The Robin (or Mixed or Third) boundary condition:

a combination of the function T and its normal derivative on

the boundary.

)(

TThx

Tk

x

y

4-4 The Ill-Posed Problem (1)

Example 1

0,02 yxforT

using separation of variables yields the solution

)sinh()sin(1

2nynx

nu

However, when n is large,

ysmallforevenn

eu

ny

,2

It violates the third requirement of well-posed problem.

Consequently, it is a ill-posed problem

4-4 The Ill-Posed Problem (2)

02 T Subject to the Neumann condition for all the boundary

The solution of this problem is multiple. Consequently,

it is a ill-posed problem.

Chapter 5

Introduction to Numerical Methods

5-1 Components of a Numerical Solution Method (1)

• Mathematical model:The starting point of any numerical method is the mathematical model, i.e., the set of partial or integro-differential equations and boundary conditions.

• Discretization method: After selecting the mathematical model, one has to choose a suitable discrezation method, i.e., a method of approximating the differential equations by a system of algebraic equations for the variables at someset of discrete locations in space and time. There are many approaches, but the most important of which are: finite difference (FD), finite volume (FV), and finite element (FE) methods. Other methods, like spectral schemes, boundary elements methods are used in CFD but their use is limited to special classes of problems.

5-1 Components of a Numerical Solution Method (2)

• Coordinate and basis vector systems: The conservation equations can be written in many forms, dependingon the coordinate system and the basis vector used. For example, onecan select Cartesian, cylindrical, spherical, curvilinear orthogonal ornon-orthogonal coordinate systems, which may be fixed or moving.

• Numerical grid: the discrete locations at which the variables are to becalculated are defined by the numerical grid which is essentially adiscrete representation of the geometric domain on which the problemis to be solved. It divides the solution domain into a finite number ofsubdomains (elements, control volumes, etc.). Some of the optionsavailable are the following:* Structured (regular) grid* Block-structured grid* Unstructured grid

5-1 Components of a Numerical Solution Method (3)

• 2D, structured, non-orthogonal grid:

5-1 Components of a Numerical Solution Method (4)

• 2D blocked-structured grid:

• 2D unstructured grid:

5-1 Components of a Numerical Solution Method (6)

• Finite approximationsFollowing the choice of grid type, one has to select the approximations to be used in the discretization process. In a finite difference method, approximations for the derivatives at the grid points have to be selected. In a finite volume method, one has to select the methods of approximating surface and volume integrals. In a finite element method, one has to choose the functions (elements) and weighting functions.

• Solution method: Discretization yields a large system of non-linear algebraic equations. The method depends on the problem. For unsteady flows, methods based on those for initial value problems for ODEs (marching in time) are used. At each time step an elliptic problems has to be solved. Steady flow problems are usually solved by pseudo-time marching iteration scheme or an equivalent iteration scheme. These methods use successive linearization of the equations and the resulting linear systems are almost always solved by iterative techniques.

5-2 Discretization Approaches (1)

• Finite Difference (FD) Method:

1. This is the oldest method for numerical solution of PDE’s, believed to

have been introduced by Euler in the 18th century. It is also the

earliest method to use for simple geometry.

2. In principle, the FD method can be applied to any grid type. However,

in all applications of the FD method, it has been applied to structured

grids. The grid lines serve as local coordinate lines.

5-2 Discretization Approaches (2)

4.Taylor series expansion or polynomial fitting is used to obtain

approximations to the first and second derivatives of the variables with

respect to the coordinates. When necessary, these methods are also

used to obtain variable values at locations other than grid points

(interpolation).

5.On structured grids, the FD method is very simple and effective. It is

especially easy to obtain higher-order schemes on regular grids.

6.The disadvantages of FD methods is that the conservation is not

enforced unless special care is taken. Also, the restriction to simple

geometry is a significant disadvantages in complex flows.

5-2 Discretization Approaches (3)

(7) Taylor series formulation:

(i) The idea of finite-difference representation of a derivative can be

introduced by recalling the definition of the derivative of the

function F(x,y) at x = x0, y = y0

Clearly, if the function F(x,y) is continuous, the right-hand side of the above equation can be

reasonable approximation to for a sufficiently small but finite

x

yxFyxxF

x

Fx

),(),(lim 0000

0

xF

x

5-2 Discretization Approaches (4)

(ii) A formal basis for developing finite difference approximation

to derivative is through the use of Taylor series expansion.

Consider Taylor series expansion of a function f(x) about a

point x0 in the forward (i.e., positive x) and backward (i.e.,

negative x) directions given, respectively, by

...

!3!2)()(

...!3!2

)()(

3

0

3

32

0

2

2

000

3

0

3

32

0

2

2

000

x

dx

fdx

dx

fdx

dx

dfxfxxf

x

dx

fdx

dx

fdx

dx

dfxfxxf

5-2 Discretization Approaches (5)

(iii) These two expressions form the basis for developing finite difference approximations for the first derivative df/dx, about

x0. Rearranging the equations in previous page, the forward and backward finite difference approximations for the first derivative, respectively, become

where the “order of “notation characterizes the truncation error associated with the finite-difference approximation. It represents the difference between the derivative and its finite-difference representation.

backwardxox

xxfxf

dx

df

forwardxox

xfxxf

dx

df

)...()()(

)...()()(

00

0

00

0

xo

5-2 Discretization Approaches (6)

• Finite Volume (FV) Method:

1.The FV method can accommodate any type of grid, so it is suitable for

complex geometries.

2.The FV approach is perhaps the simplest to understand and to program.

All terms that need be approximated have physical meaning which is

why it is popular with engineers.

3.The disadvantages of FV methods compared to FD schemes is that

methods of order higher than second are more difficult to develop in

3D.

5-2 Discretization Approaches (7)

4. Control-volume approach:

(i) When applying the control-volume approach to develop the

finite-difference equations, the finite difference nodes must

be established and then the control volume must be identified.

(ii) Example: one-dimensional steady-state heat conduction with

variable thermal conductivity k and energy

generation rate g.

Governing equation:

0

gdx

dTk

dx

d

5-2 Discretization Approaches (8)

To integrate the above equation

S: surface area at the interfaces

V: volume of the control-volume

• Control volume

0

0

2/12/1

VgSdx

dTk

dx

dTk

dVgdx

dTk

dx

d

iii

V