ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
Journal of Scientific Computing
On the rotation and skew-symmetric forms for incompressible flow simulations
Applied Numerical Mathematics - Special issue: Transition to turbulence
The effect of the formulation of nonlinear terms on aliasing errors in spectral methods
Applied Numerical Mathematics
Journal of Computational Physics
Stable and unstable formulations of the convection operator in spectral element simulations
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
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
Spectral Element Methods for Transitional Flows in Complex Geometries
Journal of Scientific Computing
High-Order Algorithms for Large-Eddy Simulation of Incompressible Flows
Journal of Scientific Computing
Comments on "Filter-based stabilization of spectral element methods"
Journal of Computational Physics
Spectral element filtering techniques for large eddy simulation with dynamic estimation
Journal of Computational Physics
De-aliasing on non-uniform grids: algorithms and applications
Journal of Computational Physics
Stabilized spectral element computations of high Reynolds number incompressible flows
Journal of Computational Physics
Spectral Vanishing Viscosity Method for Large-Eddy Simulation of Turbulent Flows
Journal of Scientific Computing
Stabilization Methods for Spectral Element Computations of Incompressible Flows
Journal of Scientific Computing
Variational multiscale turbulence modelling in a high order spectral element method
Journal of Computational Physics
Journal of Computational Physics
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We investigate stability properties of the spectral element method for advection dominated incompressible flows. In particular, properties of the widely used convective form of the nonlinear term are studied. We remark that problems which are usually associated with the nonlinearity of the governing Navier---Stokes equations also arise in linear scalar transport problems, which implicates advection rather than nonlinearity as a source of difficulty. Thus, errors arising from insufficient quadrature of the convective term, commonly referred to as `aliasing errors', destroy the skew-symmetric properties of the convection operator. Recovery of skew-symmetry can be efficiently achieved by the use of over-integration. Moreover, we demonstrate that the stability problems are not simply connected to underresolution. We combine theory with analysis of the linear advection-diffusion equation in 2D and simulations of the incompressible Navier---Stokes equations in 2D of thin shear layers at a very high Reynolds number and in 3D of turbulent and transitional channel flow at moderate Reynolds number. For the Navier---Stokes equations, where the divergence-free constraint needs to be enforced iteratively to a certain accuracy, small divergence errors can be detrimental to the stability of the method and it is therefore advised to use additional stabilization (e.g. so-called filter-based stabilization, spectral vanishing viscosity or entropy viscosity) in order to assure a stable spectral element method.