Wave Equation in two Dimensions
If we assume that all values at times \( t=j \) and \( t=j-1 \) are known, the only unknown variable is \( u_{i,j+1} \) and the last equation yields thus an explicit
scheme for updating this quantity. We have thus an explicit finite difference
scheme for computing the wave function \( u \). The only additional complication
in our case is the initial condition given by the first derivative in time,
namely \( \partial u/\partial t|_{t=0}=0 \).
The discretized version of this first derivative is given by
$$
\begin{equation*}
u_t\approx \frac{u(x_i,t_j+\Delta t)-u(x_i,t_j-\Delta t)}{2\Delta t},
\end{equation*}
$$
and at \( t=0 \) it reduces to
$$
\begin{equation*}
u_t\approx \frac{u_{i,+1}-u_{i,-1}}{2\Delta t}=0,
\end{equation*}
$$
implying that \( u_{i,+1}=u_{i,-1} \).