 Solution

For energy stability for variable-coefficient reaction diffusion, let $h_x=1/(M+1)$, let $D_h$ be the Dirichlet discrete Laplacian, and set $V_h=\mathrm{diag}(a_m)$. Use the mesh-weighted inner product $(v,w{)}_h=h_x{\sum }_{m=1}^M{v}_m{w}_m$ with the corresponding discrete L2 norm. For zero boundary values, summation by parts givesFor a solution, or a difference of two solutions, the energy method givesThe Gronwall inequality therefore provesThe constant is independent of $M$, so this is stability of a numerical method on every fixed finite time interval. The bound allows physical growth if $a_{+}>0$; stability here does not mean uniform boundedness as $t\to \infty$.

There is also a uniform maximum-norm bound. At a positive spatial maximum the second difference is nonpositive. Applying this observation to $e^{-a_{+}t}v$ and to its negative gives $\Vert v(t){\Vert }_{\infty }\le e^{a_{+}t}\Vert v(0){\Vert }_{\infty }$. With a perturbing source $r(t)$, the Duhamel principle gives the corresponding initial-data bound plus ${\int }_0^t{e}^{a_{+}(t-s)}\Vert r(s)\Vert ds$ in either norm. Thus the estimate controls accumulated residuals as well as initial perturbations.