Approach

The general solution for a Laplace equation $\nabla^2 \phi(x,y)=0$ is reasonably complex and depends on the specified boundary conditions. For simplicity, the boundary conditions in our numerical experiment are specified only on the ``edges'' and are set to be zero, so that the expected solution of the problem is trivial (zero everywhere), thus making the comparison easy. Two versions of the relaxation algorithm are implemented, one using the average of only the ``old'' values of the neighbors (version 1):

$$\begin{eqnarray*} \textrm{Step 1 of 2:}\hspace{1cm} & \phi^{'}_{i,j} = 0.25(\phi... ...,j} = \phi^{'}_{i,j} & \hspace{0.5cm} \forall \hspace{0.5cm} i,j \end{eqnarray*}$$

and the other using whatever values are stored in the neighboring locations whether or not they are old or new (version 2):

$$\begin{eqnarray*} \textrm{Step 1 of 1:}\hspace{1cm} & \phi_{i,j} = 0.25(\phi_{i-... ...i,j-1}+\phi_{i,j+1}) & \hspace{0.5cm} \forall \hspace{0.5cm} i,j \end{eqnarray*}$$

The MATLAB program for the above approaches is attached at the end of the report.