Discrete Poisson equation: Difference between revisions

d and partials -> Delta
(d and partials -> Delta)
==On a two-dimensional rectangular grid==
Using the [[finite difference]] numerical method to discretize
the 2 dimensional Poisson equation (assuming a uniform spatial discretization, <math>\partialDelta x=\partialDelta y</math>) on an ''m''&nbsp;&times;&nbsp;''n'' grid gives the following formula:<ref>{{citation|title=Numerical Methods for Engineers and Scientists|edition=2nd|first=Joe|last=Hoffman|year=2001|chapter=Chapter 9. Elliptic partial differential equations|publisher=McGraw&ndash;Hill|isbn=0-8247-0443-6}}.</ref>
 
:<math>
b =
\begin{bmatrix}
-dx\Delta x^2 g_{22} + u_{12} + u_{21} \\
-dx\Delta x^2 g_{32} + u_{31} ~~~~~~~~ \\
-dx\Delta x^2 g_{42} + u_{52} + u_{41} \\
-dx\Delta x^2 g_{23} + u_{13} ~~~~~~~~ \\
-dx\Delta x^2 g_{33} ~~~~~~~~~~~~~~~~ \\
-dx\Delta x^2 g_{43} + u_{53} ~~~~~~~~ \\
-dx\Delta x^2 g_{24} + u_{14} + u_{25} \\
-dx\Delta x^2 g_{34} + u_{35} ~~~~~~~~ \\
-dx\Delta x^2 g_{44} + u_{54} + u_{45}
\end{bmatrix}.
</math>