Five-point heat-reaction stability threshold ($\kappa \le 8h^{-2}{\sin }^2(\pi h/2)$)

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 $2{\pi }^2-{\pi }^4{h}^2/6+O(h^4)$, 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.