 Solution

For a Fourier mode $U_j^n=G(\theta {)}^n{e}^{ij\theta }$, defineThe amplification factor is $G=N/L$. To derive its stability condition, use the coefficient abbreviations from the preceding solution and set $q=\cos \theta$. The squared moduli areHere $a+c={\mu }^2-\mu +1$, $a-c=2\mu -1$, and $4ac=K(\mu )$. Subtraction and factorization giveThus $|G|\le 1$ for every frequency exactly when $K(\mu )\ge 0$, provided the implicit denominator does not vanish. For the physically nonnegative Courant number, this givesFor a signed convention the full real range is $(-\infty ,-1]\cup [0,1]\cup [2,\infty )$.

There is no hidden denominator failure in these ranges. If $K>0$ and $L=0$, the identity forces $N=0$ and $\cos \theta =1$; but $N(0)=3$. If $K=0$, then $|L|=|N|$, and $N$ could vanish only when its two real coefficients have equal magnitudes. Their sum is three, so equality requires $\mu =1/2$, which is not a root of $K$. Hence $L$ is nonzero for every frequency in the allowed ranges. On the compact frequency interval its inverse is bounded for each fixed allowed $\mu$.

When $K<0$, choose a small nonzero frequency. The denominator remains close to three, while the boxed identity gives $|N|>|L|$. Its fixed-frequency numerical mode grows geometrically, proving instability in the usual fixed-Courant refinement limit. Conversely, for $K\ge 0$, the Plancherel theorem converts the mode bound into a discrete L2 norm contraction on the whole-line or periodic grid. The large-Courant branch is possible because the scheme is implicit; stability alone does not promise small error at a large step.