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by Lale Yurttas, Texas A&M University

Chapter 6 1

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Chapter 6DIFFERENTIAL

EQUATIONS : PARTIAL DIFFERENTIAL EQUATIONS

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Finite Difference: Elliptic Equations

Solution Technique• Elliptic equations in engineering are typically used to

characterize steady-state, boundary value problems.

• For numerical solution of elliptic PDEs, the PDE is transformed into an algebraic difference equation.

• Because of its simplicity and general relevance to most areas of engineering, we will use a heated plate as an example for solving elliptic PDEs.

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by Lale Yurttas, Texas A&M University

The Laplacian Difference Equations/

04

022

2

2

0

,1,1,,1,1

21,,1,

2,1,,1

21,,1,

2

2

2,1,,1

2

2

2

2

2

2

jijijijiji

jijijijijiji

jijiji

jijiji

TTTTTyx

yTTT

xTTT

yTTT

yT

xTTT

xT

yT

xT

Laplacian difference equation.

Holds for all interior points

Laplace Equation

O[(x)2]

O[(y)2]

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by Lale Yurttas, Texas A&M University

4

Figure 29.4

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• In addition, boundary conditions along the edges must be specified to obtain a unique solution.

• The simplest case is where the temperature at the boundary is set at a fixed value, Dirichlet boundary condition.

• A balance for node (1,1) is:

• Similar equations can be developed for other interior points to result a set of simultaneous equations.

754075

04

211211

10

01

1110120121

TTTTT

TTTTT

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6

1504100417545040475450404754

332332

33231322

231312

33322231

2332221221

13221211

323121

22132111

122111

TTTTTTT

TTTTTTT

TTTTTTTTT

TTTTTTT

TTT

• The result is a set of nine simultaneous equations with nine unknowns:

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7

The Liebmann Method• Most numerical solutions of Laplace

equation involve systems that are very large.

• For larger size grids, a significant number of terms will be zero.

• For such sparse systems, most commonly employed approach is Gauss-Seidel, which when applied to PDEs is also referred as Liebmann’s method.

oldnewnew

jijijijiji

jijijiTTT

TTTTT

,,,)1(

41,1,,1,1

,

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Example Use Liebmann’s method to solve for the temperature of the heated plate in figure 1. Employ overrelaxation with a value of 1.5 for the weighting factor and iterate to ɛs=1%

by Lale Yurttas, Texas A&M University

Figure 1

8