Monotone half-grid diffusion update ($k\le h^2/(2\beta )$)

For $0<a(x)\le \beta$, the explicit conservative diffusion update has weights $k{a}_{m-1/2}/h^2$, $1-k(a_{m-1/2}+a_{m+1/2})/h^2$, and $k{a}_{m+1/2}/h^2$. Under the displayed restriction they are nonnegative and sum to one. With zero Dirichlet boundary conditions, each new value is a convex combination of old interior and boundary values, proving contraction in the discrete maximum norm. The spatial matrix is also symmetric with $v^T{L}_{h}v=-h^{-2}\sum a_{m+1/2}(v_{m+1}-v_m{)}^2$. This puts its eigenvalues in $[-4\beta /h^2,0]$, so the same restriction gives L2 norm contraction. A stable recurrence need not be consistent with a different proposed differential equation.