International conference in celebration of

Heinz-Otto Kreiss’s 80th Birthday




The Fourier method was introduced in the 70s. I will discuss the role of aliasing in the Fourier method for linear and nonlinear convection problems. It is shown that the Fourier method for linear transport equations is (weakly) unstable. More precisely, the N-degree Fourier solutions experience O(N)-amplification. The exact mechanism of this weak instability is due to aliasing. Stability can be gained by removing the highest 1/3 portion of the spectrum. This is the 2/3 de-aliasing Fourier method.

I will then consider the nonlinear regime, where the 2/3 de-aliasing method is often the method of choice to maintain the balance of quadratic energy. We will prove that this type of de-aliasing is unstable: it is responsible for generating spurious oscillations.

Three practical conclusions emerge from our discussion. First, the Fourier method requires sufficiently many modes to resolve large gradients. Second, independent of whether smoothing is used or not, small scale information contained in the highest modes of the Fourier solution will be lost. Third, a careful tuning of high-order diffusion through smoothing or spectral viscosity is essential for nonlinear stability.


KTH, Stockholm   Sept 13, 2010     Room F3, Lindstedtsvägen 26 (map)