Uniform stability of a shifted two-level advection scheme ($0<\mu <1$)

For $U_m^{n+1}=(1-2\mu )(U_m^n-U_{m+1}^n)+U_{m+1}^{n-1}$, write an amplification root as $G=e^{i\theta /2}\lambda$. Its equation is ${\lambda }^2+2i(1-2\mu )\sin (\theta /2)\lambda -1=0$. The roots have modulus one and a positive frequency-uniform separation exactly for $0<\mu <1$, giving L2 norm stability through a uniformly bounded diagonalization. At either endpoint the Nyquist symbol has a repeated unit root and a Jordan block, producing linear growth. The normalized local truncation error is generally second order; at $\mu =1/2$ and $1$ exactly sampled characteristics solve the recurrence, although the latter parameter is unstable for arbitrary perturbations.