# 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.

### Languages:

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

### 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.

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

### Source Code:

Last revised on 25 February 2019.