HEATED_PLATE
2D Steady State Heat Equation in a Rectangle

HEATED_PLATE is a program which solves the steady state heat equation in a 2D rectangular region, and is intended as a starting point for implementing an OpenMP parallel version.

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.

Usage:

heated_plate epsilon output_file
epsilon
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_file
the name of the file into which the final estimate of the solution will be written, for possible display by GRID_TO_BMP.

Source Code:

Batch Script:

Once you have made a heated_plate executable, you need to copy this script file to the same directory that contains the executable, and issue the command qsub heated_plate.sh. That submits a request to have the executable program run on 4 processors. You can type showq to see if the request has started yet.

You can go up one level to the HPPC 2008 page.

Last revised on 23 July 2008.