# HEATED_PLATE_OPENMP 2D Steady State Heat Equation Using OpenMP

HEATED_PLATE_OPENMP is a FORTRAN90 program which illustrates the use of the OpenMP application program interface by employing an iteration that solves the 2D steady state heat equation.

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
[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.

### Languages:

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

### Related Data and Programs:

DIJKSTRA_OPENMP, a FORTRAN90 program which uses OpenMP to parallelize a simple example of Dijkstra's minimum distance algorithm for graphs.

FFT_OPENMP, a FORTRAN90 program which demonstrates the computation of a Fast Fourier Transform in parallel, using OpenMP.

HEATED_PLATE, a FORTRAN90 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.

HEATED_PLATE_WORKSHARE, a FORTRAN90 program which solves the steady (time independent) heat equation in a 2D rectangular region, using OpenMP's WORKSHARE directive to run in parallel (however, the results suggest that WORKSHARE is not supported by the GFORTRAN and IFORT compilers!)

HELLO_OPENMP, a FORTRAN90 program which prints out "Hello, world!" using the OpenMP parallel programming environment.

JACOBI_OPENMP, a FORTRAN90 program which illustrates the use of the OpenMP application program interface to parallelize a Jacobi iteration solving A*x=b.

MANDELBROT_OPENMP, a FORTRAN90 program which generates an ASCII Portable Pixel Map (PPM) image of the Mandelbrot fractal set, using OpenMP for parallel execution.

MD_OPENMP, a FORTRAN90 program which carries out a molecular dynamics simulation using OpenMP.

MULTITASK_OPENMP, a FORTRAN90 program which demonstrates how to "multitask", that is, to execute several unrelated and distinct tasks simultaneously, using OpenMP for parallel execution.

MXM_OPENMP, a FORTRAN90 program which computes a dense matrix product C=A*B, using OpenMP for parallel execution.

OPENMP, FORTRAN90 programs which illustrate the use of the OpenMP application program interface for carrying out parallel computations in a shared memory environment.

OPENMP_RCC, FORTRAN90 programs which illustrate how a FORTRAN90 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 FORTRAN90 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 FORTRAN90 program which counts the number of primes between 1 and N, using OpenMP for parallel execution.

QUAD_OPENMP, a FORTRAN90 program which approximates an integral using a quadrature rule, and carries out the computation in parallel using OpenMP.

RANDOM_OPENMP, a FORTRAN90 program which illustrates how a parallel program using OpenMP can generate multiple distinct streams of random numbers.

SATISFY_OPENMP, a FORTRAN90 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 FORTRAN90 program which demonstrates the default, static, and dynamic methods of "scheduling" loop iterations in OpenMP to avoid work imbalance.

SGEFA_OPENMP, a FORTRAN90 program which reimplements the SGEFA/SGESL linear algebra routines from LINPACK for use with OpenMP.

ZIGGURAT_OPENMP, a FORTRAN90 program which demonstrates how the ZIGGURAT library can be used to generate random numbers in an OpenMP parallel program.

### Reference:

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 FORTRAN90 source codes.

Last revised on 04 September 2018.