= Stability of a spatially shifted BDF2 stencil
For the periodic-grid recurrence
$$
U_{m+2}^{n+2}-\frac43U_m^{n+1}+\frac13U_m^n
=\frac23\mu(U_{m-1}^{n+2}-2U_m^{n+2}+U_{m+1}^{n+2}),
$$
the exact <von Neumann stability analysis> range at fixed $\mu>0$ is $\mu\geq3$. 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 $\mu\geq3$, $\operatorname{Re}A-3=6(\cos\theta-1)^2+4(\mu-3)(1-\cos\theta)\geq0$. 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 $\lvert v^{n+1}\rvert^2+\lvert2v^{n+1}-v^n\rvert^2$. 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 $3hu_x/k+O(h^2/k)$. 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.
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