Energy stability for variable-coefficient reaction diffusion ($\Vert U(t){\Vert }_h\le e^{a_{+}t}\Vert U(0){\Vert }_h$)

For a zero-boundary centered semidiscretization $U^{\prime }=D_{h}U+V_{h}U$, where $D_h$ is the scaled Dirichlet discrete Laplacian and $V_h=\mathrm{diag}(a_j)$ with $a_j\le a_{+}$, summation by parts gives $(U,D_{h}U{)}_h\le 0$. Hence $\frac{1}{2}(d/dt)\Vert U{\Vert }_h^2\le a_{+}\Vert U{\Vert }_h^2$. The Gronwall inequality yields $\Vert U(t){\Vert }_h\le e^{a_{+}t}\Vert U(0){\Vert }_h$, a mesh-independent stability bound on every fixed finite interval. Positive reaction coefficients can permit physical growth without violating this form of stability.