HEATED_PLATE_PARFOR
2D Steady State Heat Equation in a Rectangle

HEATED_PLATE_PARFOR is a MATLAB program which solves the steady state heat equation in a 2D rectangular region, using MATLAB's "PARFOR" feature to run in parallel.

The final estimate of the solution is written to a file in a format suitable for display by GRID_TO_BMP.

The sequential version of this program needs approximately 18/epsilon iterations to complete.

The physical region, and the boundary conditions, are suggested by this diagram:

                   W = 0
             +------------------+
             |                  |
    W = 100  |                  | W = 100
             |                  |
             +------------------+
                   W = 100
      

The region is covered with a grid of M by N nodes, and an N by N array W is used to record the temperature. The correspondence between array indices and locations in the region is suggested by giving the indices of the four corners:

                  I = 0
          [0][0]-------------[0][N-1]
             |                  |
      J = 0  |                  |  J = N-1
             |                  |
        [M-1][0]-----------[M-1][N-1]
                  I = M-1
      

The steady state solution to the discrete heat equation satisfies the following condition at an interior grid point:

W[Central] = (1/4) * ( W[North] + W[South] + W[East] + W[West] )

Given an approximate solution of the steady state heat equation, a "better" solution is given by replacing each interior point by the average of its 4 neighbors - in other words, by using the condition as an ASSIGNMENT statement:

W[Central] <= (1/4) * ( W[North] + W[South] + W[East] + W[West] )

If this process is repeated often enough, the difference between successive estimates of the solution will go to zero.

This program carries out such an iteration, using a tolerance specified by the user, and writes the final estimate of the solution to a file that can be used for graphic processing.

Depending on the situation, the function could be executed in parallel:

• interactively, and locally, using the matlabpool command;
• indirectly, and locally, using the batch command;
• indirectly, and on the Ithaca cluster, using the batch command;
• indirectly, and on the FSU HPC cluster, using the fsuClusterMatlab command;

Usage:

heated_plate_fun ( epsilon, output_filename )

• epsilon is the error tolerance used to halt the iteration. This is an absolute error tolerance, and is applied pointwise. A value of 0.1 might be reasonable for the built in problem.
• output_filename is the name of the file into which the final estimate of the solution will be written, for possible display by GRID_TO_BMP.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

COLLATZ_PARFOR, a MATLAB program which seeks the maximum Collatz sequence between 1 and N, running in parallel using MATLAB's "PARFOR" feature.

HEATED_PLATE, a MATLAB program which solves the steady (time independent) heat equation in a 2D rectangular region, and is intended as a starting point for implementing a parallel version.

HEATED_PLATE_OPENMP, a C program which solves the steady (time independent) heat equation in a 2D rectangular region, using OpenMP to run in parallel.

HELLO_PARFOR, a MATLAB program which prints out "Hello, world!" multiple times, using MATLAB's PARFOR command for parallel execution.

MATRIX_ASSEMBLE_PARFOR, a MATLAB program which demonstrates the parfor parallel programming feature by assembling the Hilbert matrix in a parallel loop.

MD_PARFOR, a MATLAB program which carries out a molecular dynamics simulation, running in parallel using MATLAB's "PARFOR" feature.

ODE_SWEEP_PARFOR, a MATLAB program which demonstrates how the PARFOR command can be used to parallelize the computation of a grid of solutions to a parameterized system of ODE's.

PRIME_PARFOR, a MATLAB program which counts the number of primes between 1 and N; running in parallel using MATLAB's "PARFOR" feature.

QUAD_PARFOR, a MATLAB program which estimates an integral using quadrature; running in parallel using MATLAB's "PARFOR" feature.

SATISFY_PARFOR, a MATLAB program which demonstrates, for a particular circuit, an exhaustive search for solutions of the circuit satisfiability problem, running in parallel using MATLAB's "PARFOR" feature.

SPARSE_PARFOR, a MATLAB library which demonstrates how a sparse matrix can be constructed by evaluating individual blocks in parallel with the parfor command, and then assembled (on a single processor) using the sparse() command.

Reference:

The User's Guide for the Parallel Computing Toolbox is available at http://www.mathworks.com/access/helpdesk/help/pdf_doc/distcomp/distcomp.pdf

• Gaurav Sharma, Jos Martin,
  MATLAB: A Language for Parallel Computing,
  International Journal of Parallel Programming,
  Volume 37, Number 1, pages 3-36, February 2009.

Source Code:

• heated_plate_fun.m, the function which performs the computations.
• heated_plate_pool.m a script which uses the MATLABPOOL command to run the function locally and interactively.
• heated_plate_script.m a script which calls the function, necessary for the BATCH command.
• heated_plate_batch_local.m a script which uses the BATCH command to run the function indirectly and locally.
• heated_plate_fsu.m a script which uses the fsuClusterMatlab command to run the function indirectly on the FSU HPC cluster.

Examples and Tests:

• heated_plate_output.txt, the output from a run.
• solution.txt the solution on a 500x500 grid.
• sol_500x500.bmp a BMP image of the solution on a 500x500 grid.

You can go up one level to the MATLAB source codes.