Backward Euler diffusion scheme

For the centered spatial second difference, backward Euler has amplification factor $G(\theta )=[1+4\mu {\sin }^2(\theta /2){]}^{-1}$ and is stable for every physical Courant number $\mu \ge 0$.

For a smooth solution of the heat equation, backward Euler with the centered second difference has residual divided by $k$ equal to $-(k/2+d^2/12)u_{xxxx}+O(k^2+k{d}^2+d^4)$. 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 $U^{n+1}-U^n=\alpha (r){\delta }^2{U}^{n+1}$ forces $\alpha (r)=r$ for every fixed positive $r$.

On $J$ interior Dirichlet grid points, backward Euler diffusion is stable for $\mu \ge 0$ and also for the nonphysical branch $\mu \le -[2{\sin }^2(\pi /(2(J+1))){]}^{-1}$. The latter makes every amplification denominator at most $-1$.