A Stable and High-Order Accurate Discontinuous Galerkin Based Splitting Method for the Incompressible Navier-Stokes Equations

摘要

In this paper we consider discontinuous Galerkin (DG) methods for theincompressible Navier-Stokes equations in the framework of projection methods.In particular we employ symmetric interior penalty DG methods within thesecond-order rotational incremental pressure correction scheme. The major focusof the paper is threefold: i) We propose a modified upwind scheme based on theVijayasundaram numerical flux that has favourable properties in the context ofDG. ii) We present a novel postprocessing technique in the Helmholtz projectionstep based on $H(\text{div})$ reconstruction of the pressure correction that iscomputed locally, is a projection in the discrete setting and ensures that theprojected velocity satisfies the discrete continuity equation exactly. As aconsequence it also provides local mass conservation of the projected velocity.iii) Numerical results demonstrate the properties of the scheme for differentpolynomial degrees applied to two-dimensional problems with known solution aswell as large-scale three-dimensional problems. In particular we addresssecond-order convergence in time of the splitting scheme as well as itslong-time stability.