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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.