# HEATED_PLATE 2D Steady State Heat Equation in a Rectangle

HEATED_PLATE, a MATLAB program which solves the steady state heat equation in a 2D rectangular region, and is intended as a starting point for a 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_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.

### Languages:

HEATED_PLATE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FD1D_HEAT_STEADY, a MATLAB program which uses the finite difference method to solve the 1D Time Dependent Heat Equations.

FD1D_PLOT, a MATLAB program which plots the output from the FD1D program;

FEM_50_HEAT, a MATLAB program which implements a finite element calculation specifically for the heat equation.

FEM1D_HEAT, a MATLAB program which uses the finite element method to solve the 1D Time Dependent Heat Equations.

FEM2D_HEAT, a MATLAB program which solves the 2D time dependent heat equation on the unit square.

FFT_SERIAL, a MATLAB program which demonstrates the computation of a Fast Fourier Transform, and is intended as a starting point for implementing a parallel version.

GRID_TO_BMP, a C++ program which reads a text file of data on a rectangular grid and creates a BMP file containing a color image of the data.

POISSON_SERIAL, a MATLAB program which computes an approximate solution to the Poisson equation in a rectangle, and is intended as the starting point for the creation of a parallel version.

PRIME_SERIAL, a MATLAB program which counts the number of primes between 1 and N, intended as a starting point for the creation of a parallel version.

QUAD_SERIAL, a MATLAB program which approximates an integral using a quadrature rule, and is intended as a starting point for parallelization exercises.

SEARCH_SERIAL, a MATLAB program which searches the integers from A to B for a value J such that F(J) = C. this version of the program is intended as a starting point for a parallel approach.

### Reference:

1. Michael Quinn,
Parallel Programming in C with MPI and OpenMP,
McGraw-Hill, 2004,
ISBN13: 978-0071232654,
LC: QA76.73.C15.Q55.

### Source Code:

Last revised on 29 January 2019.