 Centered three-level wave scheme

For the recurrencea Fourier mode with $s={\sin }^2(\theta /2)$ has amplification polynomial of a multilevel finite difference schemeFor real $\rho$ and $\mu \ge 0$, both roots have unit modulus for every $s\in [0,1]$ precisely whenIndeed a real quadratic $G^2+aG+1$ has both roots on the unit circle exactly when $|a|\le 2$. Checking the two endpoints of the affine expression $a=4\mu s-2\rho$ gives these inequalities. For example $\rho =0$, $\mu =1/4$ satisfies this root condition, so unit-modulus roots do not force $\rho =1$.

With fixed $\rho$, the unmodified wave equation selects $\rho =1$ through its consistency requirements. At equality $|a|=2$, a repeated root may produce linear growth for arbitrary two-level starting perturbations. Thus this modal root classification alone is not a full uniform power bound for matrix Fourier symbols.