2D Steady State Heat Equation Using OpenMP

HEATED_PLATE_OPENMP is a C program which illustrates the use of the OpenMP application program interface by employing an iteration that solves the 2D steady state heat equation.

The C version of this problem has an feature; while FORTRAN OpenMP programs can easily compute a maximum value in parallel using the "reduction" clause, this is not possible in C. This program shows how the maximum can nonetheless be computed, using an approach that combines private variables and a critical section.

The sequential version of this program needs approximately 18/eps 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
               |                  |
        J = 0  |                  |  J = 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.


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


HEATED_PLATE_OPENMP is available in a C version and a C++ version and a FORTRAN90 version.

Related Data and Programs:

DIJKSTRA_OPENMP, a C program which uses OpenMP to parallelize a simple example of Dijkstra's minimum distance algorithm for graphs.

FFT_OPENMP, a C program which demonstrates the computation of a Fast Fourier Transform in parallel, using OpenMP.

FUNCTIONS_OPENMP, a C program which demonstrates the behavior of a few of the OpenMP library functions.

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

HELLO_OPENMP, a C program which prints out "Hello, world!" using the OpenMP parallel programming environment.

IMAGE_DENOISE_OPENMP, a C library which applies simple filtering techniques to remove noise from an image, carrying out the operation in parallel using OpenMP.

JACOBI_OPENMP, a C program which illustrates the use of the OpenMP application program interface to parallelize a Jacobi iteration solving A*x=b.

JULIA_OPENMP, a C program which produces an image of a Julia set, using OpenMP to carry out the computation in parallel.

MD_OPENMP, a C program which carries out a molecular dynamics simulation using OpenMP.

MPI, a library which implements parallel programming in a distributed memory environment, using message passing.

MULTITASK_OPENMP, a C program which demonstrates how to "multitask", that is, to execute several unrelated and distinct tasks simultaneously, using OpenMP for parallel execution.

MXM_OPENMP, a C program which computes a dense matrix product C=A*B, using OpenMP for parallel execution.

OPENMP, C programs which illustrate the use of the OpenMP application program interface for carrying out parallel computations in a shared memory environment.

OPENMP_RCC, C programs which illustrate how a C program, using OpenMP, can be compiled and run in batch mode on the FSU High Performance Computing (HPC) cluster operated by the Research Computing Center (RCC).

POISSON_OPENMP, a C program which computes an approximate solution to the Poisson equation in a rectangle, using the Jacobi iteration to solve the linear system, and OpenMP to carry out the Jacobi iteration in parallel.

PRIME_OPENMP, a C program which counts the number of primes between 1 and N, using OpenMP for parallel execution.

PTHREADS, C programs which illustrate the use of the POSIX thread library to carry out parallel program execution.

QUAD_OPENMP, a C program which approximates an integral using a quadrature rule, and carries out the computation in parallel using OpenMP.

RANDOM_OPENMP, a C program which illustrates how a parallel program using OpenMP can generate multiple distinct streams of random numbers.

SATISFY_OPENMP, a C program which demonstrates, for a particular circuit, an exhaustive search for solutions of the circuit satisfiability problem, using OpenMP for parallel execution.

SCHEDULE_OPENMP, a C program which demonstrates the default, static, and dynamic methods of "scheduling" loop iterations in OpenMP to avoid work imbalance.

SGEFA_OPENMP, a C program which reimplements the SGEFA/SGESL linear algebra routines from LINPACK for use with OpenMP.

ZIGGURAT_OPENMP, a C program which demonstrates how the ZIGGURAT library can be used to generate random numbers in an OpenMP parallel program.


  1. Peter Arbenz, Wesley Petersen,
    Introduction to Parallel Computing - A practical guide with examples in C,
    Oxford University Press,
    ISBN: 0-19-851576-6,
    LC: QA76.58.P47.
  2. Rohit Chandra, Leonardo Dagum, Dave Kohr, Dror Maydan, Jeff McDonald, Ramesh Menon,
    Parallel Programming in OpenMP,
    Morgan Kaufmann, 2001,
    ISBN: 1-55860-671-8,
    LC: QA76.642.P32.
  3. Barbara Chapman, Gabriele Jost, Ruud vanderPas, David Kuck,
    Using OpenMP: Portable Shared Memory Parallel Processing,
    MIT Press, 2007,
    ISBN13: 978-0262533027,
    LC: QA76.642.C49.
  4. OpenMP Architecture Review Board,
    OpenMP Application Program Interface,
    Version 3.0,
    May 2008.

Source code:

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

Last revised on 15 July 2010.