Explict scheme for the diffusion equation in two dimensions
The \( 2+1 \)-dimensional diffusion equation, with the diffusion constant \( D=1 \), is given by
$$
\begin{equation*}
\frac{\partial u}{\partial t}=\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right),
\end{equation*}
$$
where we have \( u=u(x,y,t) \).
We assume that we have a square lattice of length \( L \) with equally
many mesh points in the \( x \) and \( y \) directions.
We discretize again position and time using now
$$
\begin{equation*}
u_{xx}\approx \frac{u(x+h,y,t)-2u(x,y,t)+u(x-h,y,t)}{h^2},
\end{equation*}
$$
which we rewrite as, in its discretized version,
$$
\begin{equation*}
u_{xx}\approx \frac{u^{l}_{i+1,j}-2u^{l}_{i,j}+u^{l}_{i-1,j}}{h^2},
\end{equation*}
$$
where \( x_i=x_0+ih \), \( y_j=y_0+jh \) and \( t_l=t_0+l\Delta t \), with \( h=L/(n+1) \) and \( \Delta t \) the time step.