= Monotone half-grid diffusion update
{title2=$k\le h^2/(2\beta)$}
For $0<a(x)\le\beta$, the explicit conservative diffusion update has weights $ka_{m-1/2}/h^2$, $1-k(a_{m-1/2}+a_{m+1/2})/h^2$, and $ka_{m+1/2}/h^2$. 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 $v^TL_hv=-h^{-2}\sum a_{m+1/2}(v_{m+1}-v_m)^2$. 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.
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