Wave Equation in two Dimensions
It is easy to account for different step lengths for \( x \) and \( y \).
The partial derivative is treated in much the same way
as for the one-dimensional case, except that we now have an additional
index due to the extra spatial dimension, viz., we need to compute
\( u_{i,j}^{-1} \) through
$$
\begin{equation*}
u_{i,j}^{-1}=u_{i,j}^0+\frac{\Delta t}{2h^2}\left(u_{i+1,j}^0-4u_{i,j}^0+u_{i-1,j}^0+u_{i,j+1}^0+u_{i,j-1}^0\right),
\end{equation*}
$$
in our setup of the initial conditions.