 Stability of a spatially shifted BDF2 stencil

For the periodic-grid recurrencethe exact von Neumann stability analysis range at fixed $\mu >0$ is $\mu \ge 3$. The Fourier symbol gives $A(\theta ){\xi }^2-4\xi +1=0$, $A=3e^{2i\theta }+8\mu {\sin }^2(\theta /2)$. The root near one obeys $|\xi {|}^2=1+(6-2\mu ){\theta }^2+O({\theta }^4)$, proving instability for $\mu <3$.

For $\mu \ge 3$, $\mathrm{Re}A-3=6(\cos \theta -1{)}^2+4(\mu -3)(1-\cos \theta )\ge 0$. Writing the modal recurrence as $3v^{n+2}-4v^{n+1}+v^n=-(A-3)v^{n+2}$, the BDF2 discrete energy identity proves nonincrease of $|v^{n+1}{|}^2+|2v^{n+1}-v^n{|}^2$. Summing modes proves stability uniformly even when $\mu$ varies within $[3,\infty )$.

This shifted scheme is not a consistent standard heat discretization under $k=\mu h^2$: dividing its extra spatial-shift defect by $2k/3$ produces $3h{u}_x/k+O(h^2/k)$. On a finite interval with Dirichlet boundary conditions it also requires an extra boundary closure. Thus the periodic stability theorem is not a theorem of convergence to the heat equation or of stability for an unspecified boundary treatment.