#
poisson

**poisson**,
a MATLAB code 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 /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** is intended as a starting point for the implementation of a
parallel version, using, for instance, MPI.

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Licensing:

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

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Languages:

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

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Related Data and Programs:

fem2d_poisson_rectangle,
a MATLAB code which
solves the 2D Poisson equation on a rectangle,
using the finite element method,
and piecewise quadratic triangular elements.

fft_serial,
a MATLAB code which
demonstrates the computation of a Fast Fourier Transform,
and is intended as a starting point for implementing a parallel version.

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

poisson_test

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

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Source Code:

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Last revised on 25 February 2019.
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