= No positive-Courant cancellation for backward Euler diffusion
{title2=$\tau=-(k/2+d^2/12)u_{xxxx}+\cdots$}
For a smooth solution of the <heat equation>, backward Euler with the centered second difference has residual divided by $k$ equal to $-(k/2+d^2/12)u_{xxxx}+O(k^2+kd^2+d^4)$. Both leading terms have the same sign. Under <parabolic mesh refinement> their coefficient is $-d^2(r/2+1/12)$, which cannot vanish for a positive <diffusion Courant number>. Consistency of an update $U^{n+1}-U^n=\alpha(r)\delta^2U^{n+1}$ forces $\alpha(r)=r$ for every fixed positive $r$.
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