= 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 $s_i\leq4a_{\max}$, and the <Gershgorin bounds for symmetric matrices> place every <eigenvalue> in $[-8a_{\max}/h^2,0]$.
For the <forward Euler method>, an <eigenvalue> $\lambda\leq0$ has amplification factor $1+\Delta t\lambda$. Requiring $|1+\Delta t\lambda|\leq1$ for every such <eigenvalue> gives the sufficient <absolute stability> condition $\Delta t/h^2\leq1/(4a_{\max})$. 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 $\Delta t\leq0.00125$. 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.
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