u_{m}^{n} = g^{n} e^{im ξ}

, substitution gives

(1 - \frac{i μ}{2} sin ξ) g = 1 + \frac{i μ}{2} sin ξ, g = \frac{1 + i μ sin ξ /2}{1 - i μ sin ξ /2} .

The denominator is never zero for real

μ, ξ

, 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 $μ > 0$ , 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.