Assignment #3

Particle Mesh Code -Part I

For Friday March 21, your assignment is to create the first part of a two dimensional particle mesh code. This code will convolve a Green's function with a mass distribution using forward and inverse FFT on a regular rectangular grid. This is a moderately difficult assignment, so please pay attention to the hints toward the end of the problem description.

Problem Description:

  1. Map the Green's function on the grid.
  2. Map the Mass on the grid (see the test cases below).
  3. FFT the Green's function.
  4. FFT the mass distribution.
  5. Multiply the fft's components together element by element. (Make sure to multiple the complex numbers correctly!!!!)
  6. Take the inverse FFT of the problem.

Easy as cake! Piece of pie! Ok... there are some tricky details.

Problem Parameters

  1. You do NOT need to worry about mapping particles to the grid or to worry about interpolating the potential for this assignment! These steps come later. For now, manually set the mass grid for some test cases (see below).
  2. Solve the problem on a 64 by 64 grid. All the mass should be in a one 32 by 32 corner of this grid.
  3. The Green's function should be of the form 1/sqrt(r*r + eps*eps) where eps is a softening length for the POTENTIAL. This is slightly different from a softening length for FORCE, but behaves in much the same way.
  4. You may beg, borrow, or share the FFT routines from "Numerical Recipes" or other platform independent libraries.
  5. You can work in a group of two, if you like.
  6. The final program should be a simple driver routine with some subroutines. You will NEED to incorporate the potential solving subroutines into part II of this project.
  7. Use a set of simple distributions to determine if the routine is working. I strongly suggest looking at:
    a. A single point mass in a random location in the grid.
    b. A single point mass at the edge of the grid.
    c. A single point mass at the corner of the grid.
    d. A single point mass at near the center of the grid.
    e. Two points at random locations on the grid.
    All of these tests should give you the correct analytical solutions to the problem. Be sure and check for symmetry and correctness.
  8. The most difficult part of this project will be making a Green's function which will reproduce the potential. Be sure and use the symmetry properties to test your solution!
  9. I STRONGLY recommend writing a prototype code in Mathematica, Matlab, or Maple. FFT routines are simple calls in all these routines, and can be used to CHECK your Green's functions and solutions. The final code must be in C, C++, or Fortran and should be platform independent. On student prototyped this problem using Matlab in 14 lines!!!!
  10. Keep the idea of machine accuracy in mind when you write this code. It is likely that a purely real FFT is not purely real when it is done on a machine.

Graphics Output:

You need to produce some graphical output to document the successfulness of your tests. You don't have to do all the test cases.

Assignment requirements:

1. You must write this program in C, C++, or Fortran. It must compile and run on a Sparc 10 workstation with either the gnu C and g++ compilers or the standard ANSI Fortran. If it compiles on the SGI machines, it should compile on my workstation.

2. You should include instructions on how to compile this code. A "Makefile" would be nice, especially if you have multiple files and subprograms.

3. Graphics output may be created on any machine using any graphics package, and turned into a postscript file. You can also create Web pages and give me the URL.

4. You should write a short 1 page report documenting the code and your results. This doesn't have to be fancy, but it should summarize your work.
5. When you have completed the project, use the Unix tar command to create a single file which contains: 

	documentation file (postscript, html, or text)
	C, C++, or fortran program
	Makefile and/or compilation instructions
	graphics output file (postscript)
	A short README file explaining what each file contains.
Hint: To create the tar file, put all your files into a single directory on a unix machine. Go one level above this directory and type: 
tar -cvf username.tar directory
The file username.tar is the name of the newly created tar file.
The tar file names should be your email account with a number
indicating the assignment number.  To extract the file,
type 
tar -xvf username.tar
THE NAME OF YOUR FILE SHOULD BE YOUR USER ID FOLLOWED BY THE ASSIGNMENT NUMBER!

6. Use anonymous ftp to leave the file in the pub/nbody directory of galileo.gmu.edu