Stability and spectral convergence of Fourier method for nonlinear problems. On the shortcomings of the 2/3 de-aliasing method

TL;DR

This paper analyzes the stability and spectral convergence of Fourier methods for nonlinear PDEs, revealing their stability for smooth solutions and instabilities after shock formation, highlighting shortcomings of the 2/3 de-aliasing method.

Contribution

It proves stability and convergence of Fourier methods for nonlinear problems with smooth solutions and identifies their instabilities post-shock, exposing limitations of the 2/3 de-aliasing approach.

Findings

Fourier methods are stable for solutions with minimal C^{1+eta} regularity.

Spectral convergence is achieved for smooth solutions.

Instabilities and spurious oscillations occur after shock formation.

Abstract

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the 2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal C^{1+\alpha} spatial regularity, then both the spectral and the 2/3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the 2/3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular, after shock formation in inviscid Burgers' equation, the total variation of bounded (pseudo-) spectral Fourier solutions must increase with the number of increasing modes and we stipulate the analogous situation occurs with the 3D incompressible Euler equations: the limiting Fourier solution is shown to enforce L^2-energy conservation, and the contrast with energy dissipating Onsager solutions is reflected through spurious oscillations.