Elliptic Partial Differential Equations – Lieberman Method – Part 1 of 2

1. Elliptic Partial Differential Equations – Lieberman Method – Part 1 of 2 http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates 1/2/2020 http://numericalmethods.eng.usf.edu 1

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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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11. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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13. Elliptic Partial Differential Equations – Lieberman Method – Part 2 of 2 http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates 1/2/2020 http://numericalmethods.eng.usf.edu 13

14. For more details on this topic • Go to http://numericalmethods.eng.usf.edu • Click on Keyword • Click on Elliptic Partial Differential Equations

15. You are free • to Share – to copy, distribute, display and perform the work • to Remix – to make derivative works

16. Under the following conditions • Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). • Noncommercial — You may not use this work for commercial purposes. • Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

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 0oC.

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. Iteration #1 i=1 and j=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 Iteration #2 i=1 and j=1

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: Absolute Relative Approximate Error Distribution Temperature Distribution

26. Example: Lieberman Method

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28. Acknowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate

29. For instructional videos on other topics, go to http://numericalmethods.eng.usf.edu/videos/ This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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