# 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] )
where "Central" is the index of the grid point, "North" is the index of its immediate neighbor to the "north", and so on.

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 )
where
• 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.

### Languages:

HEATED_PLATE_PARFOR is available in a MATLAB version.

### 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.

HIGH_CARD_PARFOR, a MATLAB program which uses the parfor statement to compute in parallel the statistics for a card game in which you are required to guess the location of the highest card.

MATLAB_RANDOM_PARALLEL, MATLAB programs which illustrate the use of Matlab's random number generator (RNG) functions when using parallel features such as parfor or spmd.

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.

### Examples and Tests:

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

Last revised on 25 December 2010.