High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows

Abstract

In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the disctinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment.Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem.For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H(div)-conforming finite elements to garantuee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.

Very high order discontinuous Galerkin method in elliptic problems

“…The discontinuous Galerkin method was proposed in the seventies of the previous century in [42] and constant development of the method has been observed since the time, see [9,13,19,34,44] and its application to many problems, such as analysing plates and shell [17,36], elastic wave propagation [6,7] or fluid floats [3,32]. The present paper constitutes another step towards developing the DG method, with particular attention to the DGFD method.…”

A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction

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A locally conservative and energy‐stable finite‐element method for the Navier‐Stokes problem on time‐dependent domains

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