Spectral and finite difference solutions of the Burgers equation
Computers and Fluids
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
An analysis of the discontinuous Galerkin method for wave propagation problems
Journal of Computational Physics
A spectral vanishing viscosity method for large-eddy simulations
Journal of Computational Physics
Spectral methods for hyperbolic problems
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes
Journal of Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Variational multiscale turbulence modelling in a high order spectral element method
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
Journal of Scientific Computing
Journal of Computational Physics
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We present de-aliasing rules to be used when evaluating non-linear 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 non-linear 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 Reτ = 395, and compressible flow past a pitching airfoil at Re = 45,000.