= Uniform stability of a shifted two-level advection scheme
{title2=$0<\mu<1$}
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.
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