# POISSON_SERIAL The Poisson Equation in a 2D Rectangle

POISSON_SERIAL is a FORTRAN77 program which computes an approximate solution to the Poisson equation in a 2D rectangular region.

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.

POISSON_SERIAL is intended as a starting point for the implementation of a parallel version, using, for instance, MPI.

### Languages:

POISSON_SERIAL is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FEM2D_POISSON_RECTANGLE, a FORTRAN77 program which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise quadratic triangular elements.

FFT_SERIAL, a FORTRAN77 program which demonstrates the computation of a Fast Fourier Transform, and is intended as a starting point for implementing a parallel version.

FIRE_SERIAL, a FORTRAN77program which simulates a forest fire over a rectangular array of trees, starting at a single random location. It is intended as a starting point for the development of a parallel version.

HEATED_PLATE, a FORTRAN77 program which solves the steady (time independent) heat equation in a 2D rectangular region, and is intended as a starting point for implementing a parallel version.

MD, a FORTRAN77 program which carries out a molecular dynamics simulation, and is intended as a starting point for implementing a parallel version.

MG_SERIAL, a FORTRAN77 program which a serial version of the NAS Parallel Benchmark MG (MultiGrid).

MPI, FORTRAN77 programs which illustrate the use of the MPI application program interface for carrying out parallel computations in a distributed memory environment.

MXM_SERIAL, a FORTRAN77 program which sets up a matrix multiplication problem A=B*C, intended as a starting point for implementing a parallel version.

POISSON_OPENMP, a FORTRAN77 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_SERIAL, a FORTRAN77 program which counts the number of primes between 1 and N, intended as a starting point for the creation of a parallel version.

QUAD_SERIAL, a FORTRAN77 program which approximates an integral using a quadrature rule, and is intended as a starting point for parallelization exercises.

SEARCH_SERIAL, a FORTRAN77 program which searches the integers from A to B for a value J such that F(J) = C. this version of the program is intended as a starting point for a parallel approach.

SUBSET_SUM_SERIAL, a FORTRAN77 program which seeks solutions of the subset sum problem, in which it is desired to find a subset of a set of integers which has a given sum; this version of the program is intended as a starting point for a parallel approach.

### List of Routines:

• MAIN is the main program for POISSON_SERIAL.
• R8VEC_NORM returns the L2 norm of an R8VEC.
• RHS initializes the arrays.
• SWEEP carries out one step of the Jacobi iteration.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.
• 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 FORTRAN77 source codes.

Last revised on 26 October 2011.