Discrete Poisson equation

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.

On a two-dimensional rectangular grid

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

$({\nabla }^{2}u)_{ij}={\frac {1}{dx^{2}}}(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{ij})=g_{ij}$

where $2\leq i\leq m-1$ and $2\leq j\leq n-1$.

This will result in an mn x mn linear system:

${\begin{bmatrix}A\end{bmatrix}}{\begin{bmatrix}U\end{bmatrix}}={\begin{bmatrix}b\end{bmatrix}}$

where

$A={\begin{bmatrix}~D&-I&~0&~0&~0&\ldots &~0\\-I&~D&-I&~0&~0&\ldots &~0\\~0&-I&~D&-I&~0&\ldots &~0\\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\vdots \\~0&\ldots &~0&-I&~D&-I&~0\\~0&\ldots &\ldots &~0&-I&~D&-I\\~0&\ldots &\ldots &\ldots &~0&-I&~D\end{bmatrix}}$

$I$ is the m x m identity matrix, and $D$, also m x m , is given by:

$D={\begin{bmatrix}~4&-1&~0&~0&~0&\ldots &~0\\-1&~4&-1&~0&~0&\ldots &~0\\~0&-1&~4&-1&~0&\ldots &~0\\\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\vdots \\~0&\ldots &~0&-1&~4&-1&~0\\~0&\ldots &\ldots &~0&-1&~4&-1\\~0&\ldots &\ldots &\ldots &~0&-1&~4\end{bmatrix}}$

Example

For a 5×5 ( $m=5$ and $n=5$ ) grid with all the boundary nodes prescribed, the system would look like:

${\begin{bmatrix}U\end{bmatrix}}={\begin{bmatrix}u_{22},u_{32},u_{42},u_{23},u_{33},u_{43},u_{24},u_{34},u_{44}\end{bmatrix}}^{T}$

with

$A={\begin{bmatrix}~4&-1&~0&-1&~0&~0&~0&~0&~0\\-1&~4&-1&~0&-1&~0&~0&~0&~0\\~0&-1&~4&~0&~0&-1&~0&~0&~0\\-1&~0&~0&~4&-1&~0&-1&~0&~0\\~0&-1&~0&-1&~4&-1&~0&-1&~0\\~0&~0&-1&~0&-1&~4&~0&~0&-1\\~0&~0&~0&-1&~0&~0&~4&-1&~0\\~0&~0&~0&~0&-1&~0&-1&~4&-1\\~0&~0&~0&~0&~0&-1&~0&-1&~4\end{bmatrix}}$

and

$b={\begin{bmatrix}-dx^{2}g_{22}+u_{12}+u_{21}\\-dx^{2}g_{32}+u_{31}~~~~~~~~\\-dx^{2}g_{42}+u_{52}+u_{41}\\-dx^{2}g_{23}+u_{13}~~~~~~~~\\-dx^{2}g_{33}~~~~~~~~~~~~~~~~\\-dx^{2}g_{43}+u_{53}~~~~~~~~\\-dx^{2}g_{24}+u_{14}+u_{25}\\-dx^{2}g_{34}+u_{35}~~~~~~~~\\-dx^{2}g_{44}+u_{54}+u_{45}\\\end{bmatrix}}.$

As can be seen, the boundary $u$'s are brought to the right-hand-side of the equation. The entire system is 9 x 9 while $D$ and $I$ are 3 x 3 and given by:

$D={\begin{bmatrix}~4&-1&~0\\-1&~4&-1\\~0&-1&~4\\\end{bmatrix}}$

and

$-I={\begin{bmatrix}-1&~0&~0\\~0&-1&~0\\~0&~0&-1\end{bmatrix}}.$

Methods of Solution

Because ${\begin{bmatrix}A\end{bmatrix}}$ is block tridiagonal and sparse, many methods of solution have been developed to optimally solve this linear system for ${\begin{bmatrix}U\end{bmatrix}}$. Among the methods are a generalized Thomas algorithm, cyclic reduction, successive overrelaxation, and Fourier transforms. A theoretically optimal $O(n)$ solution can be computed using multigrid methods.

Applications

In computational fluid dynamics, for the solution of an incompressible flow problem, the incompressibility 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:

${\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}=0$

where $v_{x}$ is the velocity in the $x$ direction, $v_{y}$ is velocity in $y$ and $v_{z}$ is the velocity in the $z$ direction. Taking divergence of the momentum equation and using the incompressibility constraint, the pressure poisson equation is formed given by:

$\nabla ^{2}p=f(\nu ,V)$

where $\nu$ is the kinematic viscosity of the fluid and $V$ is the velocity vector.

The discrete Poisson's equation arises in the theory of Markov chains. It appears as the relative value function for the dynamic programming equation in a Markov decision process, and as the control variate for application in simulation variance reduction.

References

• 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.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 20.4. Fourier and Cyclic Reduction Methods". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.