chapter 8 elliptic equation. 8-1 general remarks the governing equations in fluid mechanics and...

Chapter 8

Elliptic Equation

8-1 General Remarks

The governing equations in fluid mechanics and heat transfer can be reduced to elliptic form for particular applications. Such examples are the steady-state heat conduction equation, velocity potential equation for incompressible, inviscid flow, and the stream function equation.

Typical elliptic equations in a two-dimensional Cartesian system are Laplace’s equations,

and Poisson’s equation

02

2

2

2

y

u

x

u

),(2

2

2

2

yxfy

u

x

u

8-2 Finite Difference Formulations (1)

Five-point formula---central difference

Laplace’s equations:

F.D.

which is accurate to

02

2

2

2

y

u

x

u

0

222

1,,1,

2

,1,,1

y

uuu

x

uuu jijijijijiji

22 , yxo


General form for five-point formula

y

x

uuuuu

uuuy

xuuu

jijijijiji

jijijijijiji

)17(...012

022

,2

1,1,2

,1,1

1,,1,

2

,1,,1

1 yxif

)27(...04 ,1,1,,1,1 jijijijiji uuuuu


From eqs. (1) and (2), we can get

)37(...4

37...12

1,1,,1,1,

2

1,1,2

,1,1,

buuuu

u

auuuu

u

jijijijiji

jijijijiji


In order to explore various solution procedures, first consider a square domain with Dirichlet B.C.s. For instance, a simple 6x6 grid system subject to the following B.C.s. is considered:

x=0 u=u2, y=0 u=u1

x=L u=u4, y=H u=u3


The interior grid points produces sixteen equations with sixteen unknowns. The equations are:


Solution algorithms:

(1) The Gauss-Seidel Iteration Method (point-by-point iteration method):

(a) The finite difference equation is given by

(b) For the computation of the first point,

say (2,2), it follows that

1,1,2

,1,12, 12

1

jijijijiji uuuuu

1,23,22

2,12,321

2,2 12

1uuuuu kkk


(c) For point (3,2), one has

(d) The general formulation is

1,33,221

2,22,421

2,3 12

1uuuuu kkkk

11,1,

21,1,12

1, 12

1

kji

kji

kji

kji

kji uuuuu


(2) SOR Method:

or

What is the optimum value of ω? One such relation, for the solution of elliptic equations in a rectangular domain subject to Dirichlet B.C.s with constant step size, is

kji

kji

kji

kji

kji

kji

kji uuuuuuu ,

211,1,

21,1,12,

1, 12

12

11,1,

21,1,12,

1, 12

1

kji

kji

kji

kji

kji

kji uuuuuu

2

2

2

11

cos1

cos,

122

JMIMa

a

aopt


(3) Line-by-line iteration method: (a) In this formulation, it results in three unknowns at points (i-1,j), (i,j), and (i+1,j). It becomes

(b) This equation, applied to all i at constant j, results in a system of linear equations which, in compact form, has a tridiagonal matrix coefficient. The solution of each row at constant j can be solved by TDMA method.

11,

21,

21,1

1,

21,1 12

k

jik

jik

jik

jik

ji uuuuu


(c) Grid points employed in the line-by line iteration method.


(4) Line-by line SOR method:

1

1,1,2

1,2

1,1

1,

21,1

121

12

k

jik

jik

ji

kji

kji

kji

uuu

uuu

There is no simple way to determine the value of optimum ω.

In practice, trial and error is used to compute ωopt for a particular problem.


(5) The Alternating Direction Implicit (ADI) Method:

(a) An iteration cycle is considered complete once the resulting

tridiagonal system is solved for all rows and then followed

by columns, or vice versa. It follows

1,1

2

1

,11

1,1

,21

1,2

2

1

1,2

1,22

1

,12

1

,22

1

,1

12

12

kji

k

jik

jik

jik

ji

k

jik

ji

k

ji

k

ji

k

ji

uuuuu

and

uuuuu


(b) Grid points used in ADI method:


(c) ADI with SOR Method:

1,1

2

1

,12

1

,2

11,

21,

211,

2

2

1

1,1,2

,2

2

1

,12

1

,22

1

,1

121

12

121

12

kji

k

ji

k

ji

kji

kji

kji

k

jik

jik

ji

k

ji

k

ji

k

ji

uuu

uuu

and

uuu

uuu

Chapter 9

Parabolic Partial Differential Equation

9-1 General Remarks

Equations of motion in fluid mechanics are frequently reduced to parabolic formulations.

Boundary layer equations are examples of such formulations. In addition, the unsteady heat conduction equation is also parabolic.


A typical parabolic second-order PDE is the unsteady heat conduction equation, which is considered first in one-space dimension. It has the following form

(1) FTCS (forward time/central space) method:

(i) is expressed by a forward difference approximation which

is of order :

2

2

x

u

t

u

t

u

t

)18(...1

tot

uu

t

u ni

ni


(ii) Using the second-order central differencing of order

for the diffusion term , eq. (8-1) can be approximated by

(iii) Eq. (8-2) is also called explicit formulation,

which is of order . It will be

shown that the solution is stable for

2x

)28(...2

2

1121

211

1

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

uuux

tuu

x

uuu

t

uu

2, xt

2

12

x

t


(2) BTCS (backward/central space) method:

(i)

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

Ducubua

uux

tu

x

tu

x

t

x

uuu

t

uu

11

111

112

12

112

2

11

111

1

)38(...21

2

The above equation can be solved by TDMA.


(ii) Eq. (8-3) is defined as being implicit, since more than one

unknown appears in the finite difference equation. As a result,

a set of simultaneous equations needs to be solved, which

require more computation time per time step. Implicit methods

greater advantage on the stability of the finite difference

equations, since most are unditionally stable. Therefore, a

larger step size in time is permitted.


(3) CTCS (central time/central space) method: (The Crank-

Nicolson method)

(i) If the diffusion term of eq. (8-1) is replaced by the average

of the central differences at time levels n and n+1, the

discretized equation would be of the form:

n

ini

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ni

ruurruruurru

x

uuu

x

uuu

t

uu

111

111

1

211

2

11

111

1

22

)48(...22

2

1

Note: The left side of eq. (8-4) is a central difference of

step , i.e., , which is 2/t 2/2

1

t

uu

t

u ni

ni

2to


(ii) The method may be thought of as the addition of two step computations as follows: Using the explicit method,

while using the implicit method,

Adding eqs. (8-5a) and (8-5b), we can get eq. (8-4). (iii) This implicit method is unconditionally stable and is of order , that is a second-order accurate scheme. Example

)58(...

2

2/ 211

2/1

ax

uuu

t

uu ni

ni

ni

ni

ni

)58(...

2

2/ 2

11

111

2/11

bx

uuu

t

uu ni

ni

ni

ni

ni

22 , xt

9-3 Parabolic Equations in Two-Space Dimensions (1)

Consider the model equation

(1) FTCS (or Explicit) method:

which is of order Stability analysis indicates that the method is stable for where If Δx=Δy, i.e., dx=dy=d, then

2

2

2

2

y

u

x

u

t

u

2

1,,1,

2

,1,,1,1

, 22

y

uuu

x

uuu

t

uu nji

nji

nji

nji

nji

nji

nji

nji

22 ,, yxt 2/1 yx dd

22 ,y

td

x

td yx

2/1d


(2) Implicit (BTCS) method:

(i) Consider an implicit formulation for which the FDE is

2

11,

1,

11,

2

1,1

1,

1,1,

1, 22

y

uuu

x

uuu

t

uu nji

nji

nji

nji

nji

nji

nji

nji

nji

njiy

njiy

njiyx

njix

njix uudududdudud ,

11,

11,

1,

1,1

1,1 )122(


(ii) The 2-D FDEs in the ADI formulation are

2

1,,1,

2

2/1,1

2/1,

2/1,1,

2/1, 22

2/ y

uuu

x

uuu

t

uu nji

nji

nji

nji

nji

nji

nji

nji

2

11,

1,

11,

2

2/1,1

2/1,

2/1,1

2/,

1, 22

2/ y

uuu

x

uuu

t

uu nji

nji

nji

nji

nji

nji

nji

nji

and


This method is of order and is unconditionally

stable. The above equations are written in the tridiagonal form as

222 ,, yxt

nji

nji

nji

nji

nji

nji udududududud 1,2,21,2

2/1,11

2/1,1

2/1,11 )21()21(

2/1,11

2/1,1

2/1,11

11,2

1,2

11,2 )21()21(

n

jin

jin

jin

jin

jin

ji udududududud

where

2221 2

1

2

1,

2

1

2

1

y

tdd

x

tdd yx

CHAPTER 10

Stability Analysis

10-1 Stability considerations (1)

At a starting point for stability analysis, consider the simple explicit approximation to the heat equation

)1(2 112

1

2

nj

nj

nj

nj

nj uuu

xt

uu

x

u

t

u

This may be solved for unj to yield

)2(2 112

1

n

jnj

nj

nj

nj uuu

x

tuu


Let D be the exact solution of this equation(2), N the numerical solution of equation(1) and A the analytical solution of the PDE :

2x

u

t

u

Then ,we may write

Discretization error=A-D

Round-off error=N-D


The equation of stability of a numerical method examine the error growth while computations are being performed.

stabilitygrowNor

yinstabilitGrowerroroffround

The equation of stability is usually answered by using a Fourier analysis. This method is also referred to as a von Neumann analysis.

10-2 Fourier or von Neumann analysis (1)

Consider eq.(1) and let εbe the round-off error. The numerical solution actually computed may be written N=D+ε----(3)

N must satisfy eq.(1). Substituting eq.(3) into eq.(1), yields

2

1111

11

11 22

x

DDD

t

DD nj

nj

nj

nj

nj

nj

nj

nj

nj

nj


since the exact solution must satisfy the difference eq.(i.e. Eq(1)), therefore

2

111

1 2

xt

nj

nj

nj

nj

nj


In this case, the exact solution D and the error εmust both satisfy the same difference equation. This means that the numerical error and exact numerical solution both posses the same growth property in time and either could be used to examine stability.


Any perturbation of the input values at the nth time level will either be prevented from growing without bound for a stable system or will grow large for an unstable system


Consider a distribution of error at any time in a mesh. We choose to view this distribution a time t=0 for convenience. This error distribution is shown

ε (x,0)

x


We assume the error ε (x,t) can be written as a series of the form

):(,sincos

)4()(,

numberwavekxkixke

etbtx

mmmikmx

m

xikm

m


Since the difference equation is linear, superposition may be used, and we may examine the behavior of a single term of the series given in eq.(4).

consider the term )5()(, ikmx

m etbtx


For an assessment of numerical stability, we are interested in the variation of with time. Therefore, we external eq.(2) by assuming the amplitude bm is a function of time. Moreover, it is reasonable to assume an exponential variation with time; error tend to grow or diminish exponentially with time. Therefore, we write

)6(, xikat meetx


where km is real, but “a” may be complex.

If eq(6) is substituted into eq(1), we obtain

xxikatxikatxxikat

xikatxiktta

mmm

mm

eeeeeer

eeee

2

……(7)

2x

trwhere


If we divide by eateikmx and utilize the relation

2

cos1

2sin,

)1(cos21

7.2

cos

2

Employingxkwhere

re

becomeseq

ee

m

ta

ii


We can get:

tanj

nj

ta

e

re

1

2

2sin41


∴the error will not grow from one time step to the next, if

2

10)2()1(

2

1

2

1

2sin1

2sin41),2(

002

sin42

sin411),1(

12

sin411

12

sin41..,1

22

22

2

21

r

rrr

rrr

r

reinj

nj

chapter 8 elliptic equation. 8-1 general remarks the governing equations in fluid mechanics and...

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