Explict scheme for the diffusion equation in two dimensions
We have defined our domain to start \( x(y)=0 \) and end at \( X(y)=L \).
The second derivative with respect to \( y \) reads
$$
\begin{equation*}
u_{yy}\approx \frac{u^{l}_{i,j+1}-2u^{l}_{i,j}+u^{l}_{i,j-1}}{h^2}.
\end{equation*}
$$
We use again the so-called forward-going Euler formula for the first derivative in time. In its discretized form we have
$$
\begin{equation*}
u_{t}\approx \frac{u^{l+1}_{i,j}-u^{l}_{i,j}}{\Delta t},
\end{equation*}
$$
resulting in
$$
\begin{equation*}
u^{l+1}_{i,j}= u^{l}_{i,j} + \alpha\left[u^{l}_{i+1,j}+u^{l}_{i-1,j}+u^{l}_{i,j+1}+u^{l}_{i,j-1}-4u^{l}_{i,j}\right],
\end{equation*}
$$
where the left hand side, with the solution at the new time step, is the only unknown term, since starting with \( t=t_0 \), the right hand side is entirely
determined by the boundary and initial conditions. We have \( \alpha=\Delta t/h^2 \).
This scheme can be implemented using essentially the same approach as we used in Eq.
(7).