On the stability of the unsmoothedFourier method for hyperbolic equations

It has been a long open question whether the pseudospectral Fourier methodwithout smoothing is stable for hyperbolic equations with variablecoefficients that change signs. In this work we answer this question with adetailed stability analysis of prototype cases of the Fourier method.We show that due to weighted (Formula presented.)-stability,the (Formula presented.)-degree Fourier solutionis algebraically stable in the sense that its (Formula presented.)amplification does not exceed (Formula presented.).Yet, the Fourier method is weakly(Formula presented.)- unstablein the sense that it does experience such (Formula presented.)amplification. The exact mechanism of thisweak instability is due the aliasing phenomenon, which isresponsible for an (Formula presented.) amplification of the Fourier modes atthe 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 toresolve the underlying phenomenon. Otherwise, the lack ofresolution will excite the weak instability which willpropagate 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 highestmodes of the Fourier solution will bedestroyed by their (Formula presented.) amplification. Happily, with enoughresolution nothing worse can happen.