= Reaction-diffusion spectral decay threshold
{title2=$\kappa<\pi^2,\quad\kappa_h=4h^{-2}\sin^2(\pi h/2)$}
On the unit interval with zero <Dirichlet boundary conditions>, $u_t=u_{xx}+\kappa u$ has sine-mode growth rates $\kappa-j^2\pi^2$. Every finite-energy initial condition decays exactly when the largest rate is negative, namely $\kappa<\pi^2$. A centered <finite difference> grid with $h=1/(M+1)$ has eigenvectors $\sin(j\pi m/(M+1))$ and growth rates $\kappa-4h^{-2}\sin^2(j\pi/(2(M+1)))$. Its all-data decay threshold is the displayed $\kappa_h$, strictly less than $\pi^2$ and approaching it from below. Equality leaves a stationary lowest mode; exceeding the threshold allows growth.
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