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).