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

TL;DR

This paper introduces a high-order hybrid discontinuous Galerkin method for unsteady incompressible flows that ensures exactly divergence-free velocity solutions by combining operator-splitting schemes with specialized spatial discretizations.

Contribution

The paper develops a novel high-order hybrid DG method that separates linear and nonlinear parts, using H(div)-conforming elements for divergence-free velocities and efficient operator-splitting for unsteady Navier-Stokes equations.

Findings

Efficient implementation of the method demonstrated on benchmark problems.

Exact divergence-free velocity solutions achieved with H(div)-conforming elements.

Method shows promising results in 2D and 3D simulations.

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.