# POISSON_OPENMP Poisson Equation, Jacobi Iteration Parallelized with OpenMP

POISSON_OPENMP, a FORTRAN90 code which computes an approximate solution to the Poisson equation in a rectangular region, using OpenMP to carry out the Jacobi iteration in parallel.

The version of Poisson's equation being solved here is

```        - ( d/dx d/dx + d/dy d/dy ) U(x,y) = F(x,y)
```
over the rectangle 0 <= X <= 1, 0 <= Y <= 1, with exact solution
```        U(x,y) = sin ( pi * x * y )
```
so that
```        F(x,y) = pi^2 * ( x^2 + y^2 ) * sin ( pi * x * y )
```
and with Dirichlet boundary conditions along the lines x = 0, x = 1, y = 0 and y = 1. (The boundary conditions will actually be zero in this case, but we write up the problem as though we didn't know that, which makes it easy to change the problem later.)

We compute an approximate solution by discretizing the geometry, assuming that DX = DY, and approximating the Poisson operator by

```        ( U(i-1,j) + U(i+1,j) + U(i,j-1) + U(i,j+1) - 4*U(i,j) ) / dx /dy
```
Along with the boundary conditions at the boundary nodes, we have a linear system for U. We can apply the Jacobi iteration to estimate the solution to the linear system.

OpenMP is used in this example to carry out the Jacobi iteration in parallel. Note that the Jacobi iteration can converge very slowly, and the slowness increases as the matrix gets bigger. Thus, if you must use the Jacobi iteration, parallelism can help you. But you might also find, at some point, that getting a better linear solver (even a non-parallel one!) would help you more.

### Licensing:

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

### Languages:

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

### Related Data and Programs:

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

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

HEATED_PLATE_OPENMP, a FORTRAN90 code which solves the steady (time independent) heat equation in a 2D rectangular region, using OpenMP to run in parallel.

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

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

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

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

openmp_test, FORTRAN90 codes which use the OpenMP application code interface for carrying out parallel computations in a shared memory environment.

POISSON_SERIAL, a FORTRAN90 code 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_OPENMP, a FORTRAN90 code which counts the number of primes between 1 and N, using OpenMP for parallel execution.

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

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

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

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

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

### 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 04 August 2020.