A Program for the Poisson Equation in a Rectangle

**poisson**,
a MATLAB program which
computes an approximate solution to the Poisson equation in a 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 /dyAlong 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** is intended as a starting point for the implementation of a
parallel version, using, for instance, MPI.

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

**poisson** is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version.

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

FFT_SERIAL, a MATLAB 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 MATLAB program 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 MATLAB 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.

MATLAB_PARALLEL, examples which illustrate parallel programming using MATLAB's Parallel Computing Toolbox.

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

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

- poisson.m, the source code.
- timestamp.m, prints the YMDHMS date as a timestamp.