# HEATED_PLATE_OPENMP 2D Steady State Heat Equation Using OpenMP

HEATED_PLATE_OPENMP, a C++ code 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
-------------[N-1]
|                  |
J = 0  |                  |  J = N-1
|                  |
[M-1]-----------[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:

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HEATED_PLATE, a C++ code 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.

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

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### 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:

Last revised on 12 March 2020.