Modelling PDEs - exercises

Modelling the Poisson equation

Adapted from Mark Newman’s book “Computational Physics, p. 412

In one of the tutorials we have used the relaxation method to solve Laplace’s equation for electrostatics:

\begin{equation} \nabla^2\phi = 0 \end{equation}

A more general form of this equation is Poisson’s equation:

\begin{equation} \nabla^2\phi = -\frac{\rho}{\epsilon_0}, \end{equation}

which governs the electric potential in the presence of charge density $\rho$.

Assume, as in the tutorial example for Laplace’s equation, that there is a 1 meter square. However this time all four walls of the square at fixed at zero volts and that there are two 20cm x 20cm charged areas in the box. One charge has a density +1Cm$^{-2}$, the other has a density -1Cm$^{-2}$.

Modify the code for the Laplace’s equation example to solve this problem using the relaxation method.

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