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

2013/08/24 by Claude Bardos, Eitan Tadmor, Bardos, Claude +1 · 1 citation
Engineering · Physics and Astronomy · #Computational Fluid Dynamics and Aerodynamics #Fluid Dynamics and Turbulent Flows #Cosmology and Gravitation Theories

paper · pdf · doi:10.48550/arxiv.1308.5314

Abstract

The high-order accuracy of Fourier method makes it the method of choice in\nmany large scale simulations. We discuss here the stability of Fourier method\nfor nonlinear evolution problems, focusing on the two prototypical cases of the\ninviscid Burgers' equation and the multi-dimensional incompressible Euler\nequations. The Fourier method for such problems with quadratic nonlinearities\ncomes in two main flavors. One is the spectral Fourier method. The other is the\n2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portion\nof the spectrum; this is often the method of choice to maintain the balance of\nquadratic energy and avoid aliasing errors. Two main themes are discussed in\nthis paper. First, we prove that as long as the underlying exact solution has a\nminimal C^1+\α spatial regularity, then both the spectral and the 2/3\npseudo-spectral Fourier methods are stable. Consequently, we prove their\nspectral convergence for smooth solutions of the inviscid Burgers equation and\nthe incompressible Euler equations. On the other hand, we prove that after a\ncritical time at which the underlying solution lacks sufficient smoothness,\nthen both the spectral and the 2/3 pseudo-spectral Fourier methods exhibit\nnonlinear instabilities which are realized through spurious oscillations. In\nparticular, after shock formation in inviscid Burgers' equation, the total\nvariation of bounded (pseudo-) spectral Fourier solutions must increase with\nthe number of increasing modes and we stipulate the analogous situation occurs\nwith the 3D incompressible Euler equations: the limiting Fourier solution is\nshown to enforce L²-energy conservation, and the contrast with energy\ndissipating Onsager solutions is reflected through spurious oscillations.\n

Citations

• The second law of thermodynamics and stability

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