= Five-point heat-reaction stability threshold
{title2=$\kappa\leq8h^{-2}\sin^2(\pi h/2)$}
On a unit-square uniform mesh with $h=1/(M+1)$ and zero <Dirichlet boundary conditions>, the semidiscrete <five-point Laplacian> has eigenvalues $-4h^{-2}[\sin^2(j\pi h/2)+\sin^2(k\pi h/2)]$. Adding reaction $\kappa I$ gives a contractive continuous-time grid evolution exactly under the displayed threshold. The threshold is $2\pi^2-\pi^4h^2/6+O(h^4)$, slightly below the continuum threshold. This is an all-time non-growth criterion; finite-time mesh-uniform bounds also hold for fixed larger reaction rates.
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