Wave Equation in two Dimensions
The situation is rather similar for the
2+1 -dimensional case,
except that we now need to discretize the spatial
y -coordinate as well.
Our equations will now depend on three variables whose discretized versions
are now
\begin{equation*}
\begin{array}{cc} t_l=l\Delta t& l \ge 0 \\
x_i=i\Delta x& 0 \le i \le n_x\\
y_j=j\Delta y& 0 \le j \le n_y\end{array} ,
\end{equation*}
and we will let
\Delta x=\Delta y = h and
n_x=n_y for the sake of
simplicity.
The equation with initial and boundary conditions reads now
\begin{equation*}
\begin{array}{cc} u_{xx}+u_{yy} = u_{tt}& x,y\in(0,1), t>0 \\
u(x,y,0) = g(x,y)& x,y\in (0,1) \\
u(0,0,t)=u(1,1,t)=0 & t > 0\\
\partial u/\partial t|_{t=0}=0 & x,y\in (0,1)\\
\end{array}.
\end{equation*}