HEATED_PLATE
2D Steady State Heat Equation in a Rectangle

HEATED_PLATE is a C++ 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] )

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 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 is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FD2D_HEAT_STEADY, a C++ program which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

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 C++ program which solves the 2D time dependent heat equation on the unit square.

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.

HEAT_MPI, a C++ program which solves the 1D Time Dependent Heat Equation using MPI.

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.

MD, a C++ program which carries out a molecular dynamics simulation, and is intended as a starting point for implementing an OpenMP parallel version.

MXM_SERIAL, a C++ program which sets up a matrix multiplication problem A=B*C, intended as a starting point for implementing a parallel version.

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

SEARCH_SERIAL, a C++ 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.

STOCHASTIC_HEAT2D, a C++ program which implements a finite difference method (FDM) for the steady (time independent) 2D heat equation, with a stochastic heat diffusivity coefficient, using gnuplot to illustrate the results.

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:

• heated_plate.cpp, the source code.
• heated_plate.sh, commands to compile the source code.

Examples and Tests:

The program has a built in grid of M = 500, N = 500. It is only necessary to alter these definitions and recompile in order to run the problem on a different grid. Here are the output files and solution files from two runs.

• output_100x500.txt the output from a run on a 100x500 grid.
• sol_100x500.txt the solution on a 100x500 grid.
• sol_100x500.png a PNG image of the solution on a 100x500 grid.
• output_500x500.txt the output from a run on a 500x500 grid.
• sol_500x500.txt the solution on a 500x500 grid.
• sol_500x500.png a PNG image of the solution on a 500x500 grid.

List of Routines:

• MAIN is the main program for HEATED_PLATE.
• CPU_TIME returns the current reading on the CPU clock.

You can go up one level to the C++ source codes.