= Power boundedness of a two-level Fourier scheme
For a two-level Fourier recurrence $\widehat u^{n+1}=a(\theta)\widehat u^n+b(\theta)\widehat u^{n-1}$, the amplification <matrix> is
$$
T(\theta)=\begin{pmatrix}a(\theta)&b(\theta)\\1&0\end{pmatrix}.
$$
<Stability> requires its powers to be uniformly bounded in the time index and the mesh frequencies. If its two amplification roots have <modulus> at most one and their separation has a positive mesh-independent lower bound, the <eigenvectors> $(\xi_j,1)^T$ give a uniformly bounded <diagonalization of a matrix>, proving stability. A repeated unit-modulus root of this companion <matrix> instead produces a <Jordan block> and linear growth in time. Thus checking only the <moduli> of the roots is insufficient.
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