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Leibman Method

The document describes Liebmann's method and the successive over-relaxation (SOR) method for solving the discrete Laplace equation numerically. Liebmann's method rearranges the approximation to set each interior point in terms of its neighbors. SOR improves on this by introducing an over-relaxation parameter ω to relax the residual term, allowing the solution to converge faster. An optimal value of ω can be derived, which increases with coarser grids. Neumann boundary conditions are handled by including boundary points in the scheme and approximating derivatives at boundaries.

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Anbazhagan Selvanathan
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1K views6 pages

Leibman Method

The document describes Liebmann's method and the successive over-relaxation (SOR) method for solving the discrete Laplace equation numerically. Liebmann's method rearranges the approximation to set each interior point in terms of its neighbors. SOR improves on this by introducing an over-relaxation parameter ω to relax the residual term, allowing the solution to converge faster. An optimal value of ω can be derived, which increases with coarser grids. Neumann boundary conditions are handled by including boundary points in the scheme and approximating derivatives at boundaries.

Uploaded by

Anbazhagan Selvanathan
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Liebmanns method

Rearrange the approximation to give ui,j in terms of neighbouring values 1 1 k k+1 k k ui1,j + uk 1 4 1 ui,j ui,j = i+1,j + ui,j1 + ui,j+1 4 1 1 4 1 4 20 4 ui,j 1 4 1 1 = 20
k k k 4uk i1,j + 4ui+1,j + 4ui,j1 + 4ui,j+1 + k k k uk i1,j1 + ui1,j+1 + ui+1,j1 + ui+1,j+1

ui,j

Incorporate boundary conditions as before Creates iteration scheme

EMAT33040 ANA 2001-02

3.7

Department of Engineering Mathematics

Rening Liebmanns method


Compute uk+1 in sequence i,j Some terms on right hand side are known at k + 1 time step previously computed Improve speed of convergence by using them 1 1 k+1 k+1 k+1 k ui1,j + uk 1 4 1 ui,j ui,j = i+1,j + ui,j1 + ui,j+1 4 1 1 4 1 4 20 4 ui,j 1 4 1 1 = 20
k+1 k 4uk+1 + 4uk i+1,j + 4ui,j1 + 4ui,j+1 + i1,j k k k uk+1 i1,j1 + ui1,j+1 + ui+1,j1 + ui+1,j+1

ui,j

Disadvantage: slow convergence

EMAT33040 ANA 2001-02

3.8

Department of Engineering Mathematics

SOR method
SOR = Successive Over Relaxation Consider standard central difference based iteration uk+1 = i,j 1 k+1 k+1 k ui1,j + uk i+1,j + ui,j1 + ui,j+1 4 1 k+1 k+1 k k ui1,j + uk = uk + i,j i+1,j 4ui,j + ui,j1 + ui,j+1 4

Second term is residual; relaxes to zero at convergence Multiply residual term by factor to overrelax uk+1 = uk + i,j i,j k+1 k+1 k k ui1,j + uk i+1,j 4ui,j + ui,j1 + ui,j+1 4

EMAT33040 ANA 2001-02

3.9

Department of Engineering Mathematics

Using SOR
Solve Laplaces equation using SOR uxx + uyy = 0, 0 < x < 20, 0 < y < 10 u(x, 0) = 0 u(x, 10) = 0 u(0, y) = 0 u(20, y) = 100 Note that = 1 is Liebmanns method Stop when maximum change in u is < 103 Find an optimal value of increases with grid spacing
21 interior points (mx 105 interior points (mx

= 3, my = 7)
No. of iterations 20 15 13 12 15 18 23

= 15, my = 7)
No. of iterations 70 58 46 35 29 26 28 36

1.00 1.10 1.20 1.30 1.40 1.50 1.60

1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70

EMAT33040 ANA 2001-02

3.10

Department of Engineering Mathematics

Choosing for SOR


Consider a rectangular region with Dirichlet boundary conditions Optimal value of is the smaller root of cos Can solve to give opt = Previous example mx
7 15
2

mx + 1

+ cos

my + 1

2 16 + 16 = 0

4 , 2 2+ 4c

c = cos

mx + 1

+ cos

my + 1

my
3 7

opt
1.2668 1.5325

Non Dirichlet boundary conditions use results from rst few iterations see textbooks

EMAT33040 ANA 2001-02

3.11

Department of Engineering Mathematics

Neumann boundary conditions


Modication for Neumann boundary conditions follows parabolic case Consider only one edge u (0, y) = ln (x) x Boundary grid points included in numerical scheme On the boundary 1 1 2 u 2 1 4 1 u0,j 1

u1,j is included via approximation for derivative u u1,j u1,j (0, yj ) = = ln (yj ) x 2 Change of +1-diagonal entry, and extra components of right hand side vector

EMAT33040 ANA 2001-02

3.5

Department of Engineering Mathematics

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