 Solution

For the Fourier mode $u_m^n=g^n{e}^{im\xi }$, substitution givesThe denominator is never zero for real $\mu ,\xi$, and numerator and denominator are complex conjugates, so $|g|=1$. Parseval's identity therefore makes each step norm-preserving on square-summable grid data. The method is von Neumann stable for every $\mu >0$, with no upper Courant restriction. It is an implicit nondissipative scheme; unconditional stability does not remove phase error or guarantee accuracy at a large time step.