# POISSON_OPENMP Poisson Equation, Jacobi Iteration Parallelized with OpenMP

POISSON_OPENMP is a C++ program 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:

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.

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

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

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

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

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

### Reference:

1. Michael Quinn,
Parallel Programming in C with MPI and OpenMP,
McGraw-Hill, 2004,
ISBN13: 978-0071232654,
LC: QA76.73.C15.Q55.

### List of Routines:

• MAIN is the main program for POISSON_OPENMP.
• R8MAT_RMS returns the root mean square of data stored as an R8MAT.
• RHS initializes the right hand side "vector".
• SWEEP carries out several steps of the Jacobi iteration.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• U_EXACT evaluates the exact solution.
• UXXYY_EXACT evaluates ( d/dx d/dx + d/dy d/dy ) of the exact solution.

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

Last revised on 14 December 2011.