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HW3, Math 228A, UC davis, Fall 2010)

Nasser M. Abbasi

Image SOR_h_2_5_abs_residual

Contents

1 Review of Nuemman BC derivation

  1. PDF
  2. HTML

2 problem description

Solve

$\displaystyle \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = - exp(-(x-0.25)^2-(y-0.6)^2) $

on the unit square $ (0,1),(0,1)$ with homogeneous Dirichlet boundary conditions

see PDF

3 residual error animation

Animation of the residual error R as it changes during iterative solution. Tolerance used for these is $ h^2$, stopping criteria used is relative error < tolerance Only SOR was done.

3.1 SOR with $ h=2-{7}$

These animations are large in size. (Click on any to see in actual size, will open in new window)

Image residual_error_animation_SCREEN_SHOT

Image residual_error_animation_SCREEN_SHOT

Image residual_error_animation_SCREEN_SHOT

Image residual_error_animation_SCREEN_SHOT

The above shows the residual changes when using SOR with $ h=2^-7$. Z-axis range is fixed between $ -6h..6h$ (h=grid spacing)

Same as the one on the left, but z-axis is Automatic, full range of residual error, changes automatically

The above shows the residual changes when using SOR with $ h=2^-5$. Z-axis range is fixed between $ -6h..6h$ (h=grid spacing)

Same as the one on the left, but z-axis is Automatic, full range of residual error, changes automatically

3.2 SOR with $ h=2-{5}$

These animations are smaller in size. (Click on any to see in actual size, will open in new window)

Image residual_error_animation_SCREEN_SHOT

Image residual_error_animation_SCREEN_SHOT

Image residual_error_animation_SCREEN_SHOT

Image residual_error_animation_SCREEN_SHOT

The above shows the residual changes when using SOR with $ h=2^-5$. Z-axis range is fixed between -6h..6h (h=grid spacing)

Same as the one on the left, but z-axis is Automatic, full range of residual error, changes automatically

The above shows the residual changes when using SOR with h=2^-5. Z-axis range is fixed between -6h..6h (h=grid spacing)

Same as the one on the left, but z-axis is Automatic, full range of residual error, changes automatically

4 Animation for density plot animations of solvers for problem 1

Animation plots below show solver as it updates each grid point by point in its main loop

Image density_plot_animation_SCREEN_SHOT The above shows each of the solvers (Jacobi, Gauss-Seidel, SOR) in the process of updating the grid during one iteration of the main loop.

Notice the how GS and SOR solvers update the solution immediately (left to right, down to top numbering is used), while Jacobi solver updates the solution only at the end each iterative step.

This one below was done for $ h=2^-3$. Click on it to see the animation. (smaller gif file)

Image density_plot_animation_SCREEN_SHOT

5 Animations of iterative solution

Animations of iterative solution (Click on image to see animation), Stopping criteria used is relative residual method, tolerance is $ h^2$

Jacobi Gauss-Seidel SOR
Image animation_of_iterative_solver_SCREEN_SHOT

Image animation_of_iterative_solver_SCREEN_SHOT

Image animation_of_iterative_solver_SCREEN_SHOT

6 my solution

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me 2012-04-26