1. 1/30/2016 http://numericalmethods.eng.usf.edu 1 Elliptic Partial Differential Equations – Lieberman Method – Part 1 of 2 Elliptic Partial Differential Equations – Lieberman Method – Part 1 of 2 http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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5. Physical Example of an Elliptic PDE

6. Discretizing the Elliptic PDE

7. The Gauss-Seidel Method Recall the discretized equation This can be rewritten as For the Gauss-Seidel Method, this equation is solved iteratively for all interior nodes until a pre-specified tolerance is met.

8. The Lieberman Method Recall the equation used in the Gauss- Siedel Method If the Guass-Siedel Method is guaranteed to converge, we can accelerate the process by using over- relaxation. In this case,

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17. Example: Lieberman Method Consider a plate that is subjected to the boundary conditions shown below. Find the temperature at the interior nodes using a square grid with a length of. Use a weighting factor of 1.4 in the Lieberman method. Assume the initial temperature guess at all interior nodes to be 0 o C.

18. Example: Lieberman Method We can discretize the plate by taking

19. Example: Lieberman Method We can also develop equations for the boundary conditions to define the temperature of the exterior nodes.

20. Example: Lieberman Method Solve for the temperature at each interior node using the rewritten discretized Laplace equation from the Gauss-Siedel method. Apply the over relaxation equation using temperatures from previous iteration. i=1 and j=1 Iteration #1

21. Example: Lieberman Method Iteration #1 i=1 and j=2

22. Example: Lieberman Method After the first iteration the temperatures are as follows. These will be used as the nodal temperatures during the second iteration.

23. Example: Lieberman Method i=1 and j=1 Iteration #2

24. Example: Lieberman Method Iteration #2 i=1 and j=2

25. Example: Lieberman Method The figures below show the temperature distribution and absolute relative error distribution in the plate after two iterations: Temperature Distribution Absolute Relative Approximate Error Distribution

26. Example: Lieberman Method Node Temperature Distribution in the Plate (°C) Number of Iterations 129 43.750052.2813 73.7832 41.562551.313392.9758 40.796987.0125119.9378 145.5289160.9353173.3937 32.812554.178977.5449 26.031357.9731103.3285 23.3898122.0937138.3236 164.1216215.6582198.5498 63.984469.145882.9805 66.505576.1516104.3815 66.4634155.0472131.2525 220.7047181.4650182.4230

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