# Discrete Poisson equation

### From CFD-Wiki

In mathematics, the **discrete Poisson equation** is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics.

## Contents |

## On a two-dimensional rectangular grid

Using the finite difference numerical method to discretize
the 2 dimensional Poisson equation (assuming a uniform spatial discretization) on an *m x n* grid gives the following formula:

where and . The preferred arrangement of the solution vector is to use natural ordering which, prior to removing boundary elements, would look like:

This will result in an * mn x mn * linear system:

where

is the * m x m * identity matrix, and , also * m x m *, is given by:

^{[1]}
For each equation, the columns of correspond to the components:

while the columns of to the left and right of correspond to the components:

and

respectively.

From the above, it can be inferred that there are block columns of in . It is important to note that prescribed values of (usually lying on the boundary) would have their corresponding elements removed from and . For the common case that all the nodes on the boundary are set, we have and , and the system would have the dimensions * (m - 2) (n - 2) x (m - 2) (n - 2) *, where and
would have dimensions * (m-2) x (m-2) *.

## Example

For a * 5 x 5 * ( and ) grid with all the boundary nodes prescribed,
the system would look like:

with

and

As can be seen, the boundary 's are brought to the right-hand-side
of the equation. ^{[2]} The entire system is * 9 x 9 *
while and are * 3 x 3 * and given by:

and

## Methods of Solution

Because is block tridiagonal and sparse, many methods of solution have been developed to optimally solve this linear system for . Among the methods are a generalized Thomas algorithm, cyclic reduction, successive overrelaxation, and Fourier transforms. A theoretically optimal solution can be computed using multigrid methods.

## Applications

In Computational fluid dynamics, for the solution of an incompressible flow problem, the incompressiblity condition acts as a constraint for the pressure. There is no explicit form available for pressure in this case due to a strong coupling of the velocity and pressure fields. In this condition, by taking the divergence of all terms in the momentum equation, one obtains the pressure poisson equation. For an incompressible "Flow" this constraint is given by:

where is the velocity in the direction, is velocity in and is the velocity in the direction. Taking divergence of the momentum equation and using the incompressibility constraint, the pressure poisson equation is formed given by:

where is the kinematic viscosity of the fluid and is the velocity vector.
^{[3]}.

## Footnotes

- ↑ Golub, Gene H. and C.F. Van Loan,
*Matrix Computations, 3rd Ed.*, The Johns Hopkins University Press, Baltimore, 1996, pages 177-180. - ↑ Cheny, Ward and David Kincaid,
*Numerical Mathematics and Computing 2nd Ed.*, Brooks/Cole Publishing Company, Pacific Grove, 1985, pages 443-448 - ↑
Fletcher, Clive A. J.,
*Computational Techniques for Fluid Dynamics: Vol I*, 2nd Ed., Springer-Verlag, Berlin, 1991, page 334-339.

## References

- Cheny, Ward and David Kincaid,
*Numerical Mathematics and Computing 2nd Ed.*, Brooks/Cole Publishing Company, Pacific Grove, 1985.

- Golub, Gene H. and C.F. Van Loan,
*Matrix Computations, 3rd Ed.*, The Johns Hopkins University Press, Baltimore, 1996.

- Hoffman, Joe D.,
*Numerical Methods for Engineers and Scientists, 4th Ed.*, McGraw-Hill Inc., New York, 1992.

- Sweet, Roland A.,
*SIAM Journal on Numerical Analysis, Vol. 11, No. 3*, June 1974, 506-520.