= Explicit time stepping for bounded reaction diffusion
{title2=$\|(I+kD_h+kV_h)^n\|\leq e^{Ank}$}
Let $D_h$ be the scaled <Dirichlet discrete Laplacian> and $V_h$ a bounded diagonal reaction matrix. If $0\leq k/h_x^2\leq1/2$, the heat update $H_h=I+kD_h$ is contractive in both the maximum <norm> and the mesh-weighted <L2 norm>. For $\|V_h\|\leq A$,
$$
\|(H_h+kV_h)^n\|\leq(1+kA)^n\leq e^{Ank}.
$$
This proves finite-time mesh-uniform <stability> without requiring the full update to have nonnegative entries or to be contractive. Negative reaction terms can destroy nonnegativity at the endpoint $k/h_x^2=1/2$, so it is the heat part alone that is treated as a nonnegative contraction.
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