= Solution
Use the discrete norm $\|U\|_h^2=h\sum_{m=1}^N|U_m|^2$, with periodic indices. Two independent arguments prove uniform <stability of a numerical method>.
For <Fourier stability analysis>, a <discrete Fourier mode> $U_m(t)=\widehat U_k(t)e^{im\theta_k}$ with $\theta_k=2\pi k/N$ has semidiscrete eigenvalue
$$
\lambda_h(\theta)=-\frac4{h^2}\sin^2(\theta/2)+i\frac\alpha h\sin\theta.
$$
Its real part is nonpositive, so $|\widehat U_k(t)|\leq|\widehat U_k(0)|$. The unitary <discrete Fourier transform> and <discrete Parseval identity> then give $\|U(t)\|_h\leq\|U(0)\|_h$ for every $N$, $t\geq0$ and fixed real $\alpha$.
For the <energy method>, denote the centered first difference by $D_0$ and the forward difference by $D_+$. Periodic <discrete summation by parts> makes $D_0$ skew-adjoint and gives $\langle U,D_{xx}U\rangle_h=-\|D_+U\|_h^2$. Therefore
$$
\boxed{\frac d{dt}\|U(t)\|_h^2
=2\operatorname{Re}\langle U,D_{xx}U+\alpha D_0U\rangle_h
=-2\|D_+U\|_h^2\leq0.}
$$
The constant mode is preserved rather than strictly damped. Both proofs apply equally to differences of two solutions. \b[The semidiscretization is contractive in the discrete L2 norm.] A later choice of a time integrator would require its own stability analysis; no time-step condition is needed for this continuous-time result.
Back to article page
