# poisson_openmp

poisson_openmp, a C++ 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.

### Languages:

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

### Related Data and Programs:

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

ZIGGURAT_OPENMP, a C++ 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 31 March 2020.