# POISSON_SERIAL A Program for the Poisson Equation in a Rectangle

POISSON_SERIAL, a C++ 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 /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.

### Licensing:

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

### Languages:

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

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

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

POISSON_OPENMP, a C++ 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.

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

SEARCH_SERIAL, a C++ 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 C++ 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.

### Source Code:

Last revised on 31 March 2020.