On the stability of the unsmoothed Fourier method for hyperbolic equations

Abstract

It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted \(L^2\)-stability, the \(N\)-degree Fourier solution is algebraically stable in the sense that its \(L^2\) amplification does not exceed \({O}(N)\). Yet, the Fourier method is weakly \(L^2\) -unstable in the sense that it does experience such \({O}(N)\) amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an \({O}(N)\) amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their \({O}(N)\) amplification. Happily, with enough resolution nothing worse can happen.