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De-aliasing on non-uniform grids: algorithms and applications. (English) Zbl 1161.76534
Summary: We present de-aliasing rules to be used when evaluating nonlinear terms with polynomial spectral methods on non-uniform grids analogous to the de-aliasing rules used in Fourier spectral methods. They are based upon the idea of super-collocation followed by a Galerkin projection of the nonlinear terms. We demonstrate through numerical simulation that both accuracy and stability can be greatly enhanced through the use of this approach. We begin by deriving from the numerical quadrature rules used by Galerkin-type projection methods the number of quadrature points and weights needed for quadratic and cubic non-linearities. We then present a systematic study of the effects of super-collocation when using both a continuous Galerkin and a discontinuous Galerkin method to solve the one-dimensional viscous Burgers equation. We conclude by examining three direct numerical simulation flow examples: incompressible turbulent flow in a triangular duct, incompressible turbulent flow in a channel at Re$$_{\tau}=395$$, and compressible flow past a pitching airfoil at $$\operatorname{Re}=45,000$$.

##### MSC:
 76M22 Spectral methods applied to problems in fluid mechanics 76D99 Incompressible viscous fluids 76F10 Shear flows and turbulence 76N15 Gas dynamics (general theory)
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