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(1) Develop the finite-difference equations for the above PDE using a uniform mesh in both x and y directions, and write a computer program to solve the resulting algebraic equations using the Jacobi method. Use 41 grids points in each direction. Use an absolute convergence tolerance of 10-6. For your derivation, keep the grid spacing in x and y directions separate, so that you can use this same code later for studying grid aspect ratio effects. a) Make contour plots of the numerical solution and the error between the analytical and numerical solutions. b) Make a plot of the residual (log scale) vs. number of iterations, i.e., a semi-log plot. c) Discuss your results. (2) Repeat Part 1 (including all plots), but now using the Gauss-Seidel method. Make sure you use the same initial guess as Part 1. Plot the residual of the Gauss-Seidel method and the Jacobi method (Part 1) on the same plot. Discuss your results. (3) Repeat both methods (including all plots), but now with 81 grid points in each direction. Plot these two residuals, as well as those plotted in Part (2) on the same plot. Discuss your results, and list the main conclusions of this study.

(1) Develop the finite-difference equations for the above PDE using a uniform mesh in both x and y directions, and write a computer program to solve the resulting algebraic equations using the Jacobi method. Use 41 grids points in each direction. Use an absolute convergence tolerance of 10-6. For your derivation, keep the grid spacing in x and y directions separate, so that you can use this same code later for studying grid aspect ratio effects. a) Make contour plots of the numerical solution and the error between the analytical and numerical solutions. b) Make a plot of the residual (log scale) vs. number of iterations, i.e., a semi-log plot. c) Discuss your results. (2) Repeat Part 1 (including all plots), but now using the Gauss-Seidel method. Make sure you use the same initial guess as Part 1. Plot the residual of the Gauss-Seidel method and the Jacobi method (Part 1) on the same plot. Discuss your results. (3) Repeat both methods (including all plots), but now with 81 grid points in each direction. Plot these two residuals, as well as those plotted in Part (2) on the same plot. Discuss your results, and list the main conclusions of this study.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider the following second order partial differential equation and boundary conditions: d²d² + S = 50000 exp[-50{(1 − x)² + y²}]· [100{(1 − x)² + y²} −2] Ox² dy² p(1, y) = 100(1- y) +500 exp(-50y²) p(0, y) = 500 exp(-50{1+ y²}) p(x,0) = 100x +500 exp(-50(1 − x)²) p(x,1)= 500 exp(-50{(1 − x)² +1}) Note that this is a linear system of equations. The above system has an analytical solution, given by p(x, y) = 500 exp(-50{(1 − x)² + y²})+100x(1-y) (1) Develop the finite-difference equations for the above PDE using a uniform mesh in both x and y directions, and write a computer program to solve the resulting algebraic equations using the Jacobi method. Use 41 grids points in each direction. Use an absolute convergence tolerance of 106. For your derivation, keep the grid spacing in x and y directions separate, so that you can use this same code later for studying grid aspect ratio effects. a) Make contour plots of the numerical solution and the error between the analytical and numerical solutions. b) Make a plot of the residual (log scale) vs. number of iterations, i.e., a semi-log plot. c) Discuss your results. (2) Repeat Part 1 (including all plots), but now using the Gauss-Seidel method. Make sure you use the same initial guess as Part 1. Plot the residual of the Gauss-Seidel method and the Jacobi method (Part 1) on the same plot. Discuss your results. (3) Repeat both methods (including all plots), but now with 81 grid points in each direction. Plot these two residuals, as well as those plotted in Part (2) on the same plot. Discuss your results, and list the main conclusions of this study.
Transcribed Image Text:Consider the following second order partial differential equation and boundary conditions: d²d² + S = 50000 exp[-50{(1 − x)² + y²}]· [100{(1 − x)² + y²} −2] Ox² dy² p(1, y) = 100(1- y) +500 exp(-50y²) p(0, y) = 500 exp(-50{1+ y²}) p(x,0) = 100x +500 exp(-50(1 − x)²) p(x,1)= 500 exp(-50{(1 − x)² +1}) Note that this is a linear system of equations. The above system has an analytical solution, given by p(x, y) = 500 exp(-50{(1 − x)² + y²})+100x(1-y) (1) Develop the finite-difference equations for the above PDE using a uniform mesh in both x and y directions, and write a computer program to solve the resulting algebraic equations using the Jacobi method. Use 41 grids points in each direction. Use an absolute convergence tolerance of 106. For your derivation, keep the grid spacing in x and y directions separate, so that you can use this same code later for studying grid aspect ratio effects. a) Make contour plots of the numerical solution and the error between the analytical and numerical solutions. b) Make a plot of the residual (log scale) vs. number of iterations, i.e., a semi-log plot. c) Discuss your results. (2) Repeat Part 1 (including all plots), but now using the Gauss-Seidel method. Make sure you use the same initial guess as Part 1. Plot the residual of the Gauss-Seidel method and the Jacobi method (Part 1) on the same plot. Discuss your results. (3) Repeat both methods (including all plots), but now with 81 grid points in each direction. Plot these two residuals, as well as those plotted in Part (2) on the same plot. Discuss your results, and list the main conclusions of this study.
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