 Gershgorin stability bound for a variable-coefficient diffusion stencil

A symmetric conservative five-point diffusion matrix has row center $-s/h^2$ and radius at most $s/h^2$, where $s$ is the sum of its four nonnegative edge coefficients. If every coefficient is at most $a_{\max }$, its eigenvalues lie in $[-8a_{\max }/h^2,0]$. Forward Euler is therefore stable when $\Delta t/h^2\le 1/(4a_{\max })$.