 Gershgorin stability bound for a variable-coefficient diffusion stencil

A symmetric matrix obtained from a conservative five-point discretization of a diffusion equation has row centre $-s_i/h^2$ and radius at most $s_i/h^2$, where $h>0$ is the grid spacing and $s_i$ is the sum of its four nonnegative edge coefficients. Here the matrix represents the diffusion generator, with nonnegative off-diagonal entries; boundary elimination may reduce the row radius. If every coefficient is at most $a_{\max }>0$, then $s_i\le 4a_{\max }$, and the Gershgorin bounds for symmetric matrices place every eigenvalue in $[-8a_{\max }/h^2,0]$.

For the forward Euler method, an eigenvalue $\lambda \le 0$ has amplification factor $1+\Delta t\lambda$. Requiring $|1+\Delta t\lambda |\le 1$ for every such eigenvalue gives the sufficient absolute stability condition $\Delta t/h^2\le 1/(4a_{\max })$. Symmetry makes this also prevent growth of the Euclidean norm. For $a_{\max }=2$ and $h=0.1$, the bound is $\lambda \in [-1600,0]$ and $\Delta t\le 0.00125$. At that step size the interval maps to amplification factors in $[-1,1]$. This illustrates why cheap Gershgorin disc bounds can select a stable time step before any eigenvalues are computed.