Wave Equation in two Dimensions
We have now the following discretized partial derivatives
$$
\begin{equation*}
u_{xx}\approx \frac{u_{i+1,j}^l-2u_{i,j}^l+u_{i-1,j}^l}{h^2},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{yy}\approx \frac{u_{i,j+1}^l-2u_{i,j}^l+u_{i,j-1}^l}{h^2},
\end{equation*}
$$
and
$$
\begin{equation*}
u_{tt}\approx \frac{u_{i,j}^{l+1}-2u_{i,j}^{l}+u_{i,j}^{l-1}}{\Delta t^2},
\end{equation*}
$$
which we merge into the discretized \( 2+1 \)-dimensional wave equation
as
$$
\begin{equation}
\tag{25}
u_{i,j}^{l+1}
=2u_{i,j}^{l}-u_{i,j}^{l-1}+\frac{\Delta t^2}{h^2}\left(u_{i+1,j}^l-4u_{i,j}^l+u_{i-1,j}^l+u_{i,j+1}^l+u_{i,j-1}^l\right),
\end{equation}
$$
where again we have an explicit scheme with \( u_{i,j}^{l+1} \) as the only
unknown quantity.