Stabilisation of spectral/ hp element methods through spectral vanishing viscosity: Application to fluid mechanics modelling

Abstract

In this paper we present a formulation of spectral vanishing viscosity (SVV) for the stabilisation of spectral/ hp element methods applied to the solution of the incompressible Navier–Stokes equations. We construct the SVV around a filter with respect to an orthogonal expansions, and prove that this methodology provides a symmetric semi-positive definite SVV operator. After providing a few simple one- and two-dimensional examples to demonstrate the utility of the SVV, we examine how it can be applied to a spectral/ hp element discretisation of the Navier–Stokes equations using a velocity correction splitting scheme. We provide three fluid flow examples to help illustrate the pros and cons of this approach on stability and accuracy.