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The high-order accuracy of Fourier method makes it the method of choice inmany large scale simulations. We discuss here the stability of Fourier methodfor nonlinear evolution problems, focusing on the two prototypical cases of theinviscid Burgers' equation and the multi-dimensional incompressible Eulerequations. The Fourier method for such problems with quadratic nonlinearitiescomes in two main flavors. One is the spectral Fourier method. The other is the2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portionof the spectrum; this is often the method of choice to maintain the balance ofquadratic energy and avoid aliasing errors. Two main themes are discussed inthis paper. First, we prove that as long as the underlying exact solution has aminimal C^{1+\alpha} spatial regularity, then both the spectral and the 2/3pseudo-spectral Fourier methods are stable. Consequently, we prove theirspectral convergence for smooth solutions of the inviscid Burgers equation andthe incompressible Euler equations. On the other hand, we prove that after acritical time at which the underlying solution lacks sufficient smoothness,then both the spectral and the 2/3 pseudo-spectral Fourier methods exhibitnonlinear instabilities which are realized through spurious oscillations. Inparticular, after shock formation in inviscid Burgers' equation, the totalvariation of bounded (pseudo-) spectral Fourier solutions must increase withthe number of increasing modes and we stipulate the analogous situation occurswith the 3D incompressible Euler equations: the limiting Fourier solution isshown to enforce L^2-energy conservation, and the contrast with energydissipating Onsager solutions is reflected through spurious oscillations.

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