Computational Physics Lectures: Partial differential equations

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Jacobi's algorithm extended to the diffusion equation in two dimensions

We assume that we have a square lattice of length

$L$ with equally many mesh points in the

$x$ and

$y$ directions. Setting the diffusion constant

$D=1$ and using the shorthand notation

$u_{xx}={\partial^2 u}/{\partial x^2}$ etc for the second derivatives and

$u_t={\partial u}/{\partial t}$ for the time derivative, we have, with a given set of boundary and initial conditions,

$\begin{equation*} \begin{array}{cc}u_t= u_{xx}+u_{yy}& x, y\in(0,L), t>0 \\ u(x,y,0) = g(x,y)& x, y\in (0,L) \\ u(0,y,t)=u(L,y,t)=u(x,0,t)=u(x,L,t)0 & t > 0\\ \end{array} \end{equation*}$