8. Numerical Solution to Laplace's Equation

• The Jacobi and Gauss-Seidel methods

• Exercise 7.1: Rate of convergence as a function of grid size and accuracy
  (see also Exercise 6.1).
  Here's a starting point...

• Iteration as a diffusive process

• Using images to visualize 2-D data -- the programs make_image and gifsicle

  Exercise 7.2: Use these tools to animate the convergence of one of the calculations in Exercise 7.1.

  • Modify your program to save a copy of the grid at the end of each iteration in a separate file (you probably want to work in a subdirectory to avoid cluttering up your workspace). Alternatively, simply write all the data to stdout and use the split program to break the stream into separate files.

  • Use make_image to convert these data files into images (use a script!). The syntax is

        make_image -s {width} {height} < {datafile} > {imagefile} 

    The default image format is GIF.

  • Use gifsicle to combine these images into a movie:

        gifsicle {list of image files} > {moviefile} 

  • Use xanim or gifview to show the movie:

        xanim {moviefile} 

  If you can't find these programs, use the versions in ~steve/bin on newton.

• Doing better: successive overrelaxation

  Exercise 7.3: Repeat Exercise 7.1, but using sucessive overrelaxation with

          omega  =  1.5,  1.75,  omega_optimal,  2
  

• Poisson's equation

• Homework #4