1/30/ Elliptic Partial Differential Equations – Lieberman Method – Part 1 of 2 Elliptic Partial Differential Equations – Lieberman Method – Part 1 of 2 Transforming Numerical Methods Education for STEM Undergraduates
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Physical Example of an Elliptic PDE
Discretizing the Elliptic PDE
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.
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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1/30/ Elliptic Partial Differential Equations – Lieberman Method – Part 2 of 2 Elliptic Partial Differential Equations – Lieberman Method – Part 2 of 2 Transforming Numerical Methods Education for STEM Undergraduates
For more details on this topic Go to Click on Keyword Click on Elliptic Partial Differential Equations
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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.
Example: Lieberman Method We can discretize the plate by taking
Example: Lieberman Method We can also develop equations for the boundary conditions to define the temperature of the exterior nodes.
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
Example: Lieberman Method Iteration #1 i=1 and j=2
Example: Lieberman Method After the first iteration the temperatures are as follows. These will be used as the nodal temperatures during the second iteration.
Example: Lieberman Method i=1 and j=1 Iteration #2
Example: Lieberman Method Iteration #2 i=1 and j=2
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
Example: Lieberman Method Node Temperature Distribution in the Plate (°C) Number of Iterations
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This instructional power point brought to you by Numerical Methods for STEM undergraduate Committed to bringing numerical methods to the undergraduate Acknowledgement
For instructional videos on other topics, go to This material is based upon work supported by the National Science Foundation under Grant # 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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