An explicit divergence-free DG method for incompressible flow

TL;DR

This paper introduces an explicit divergence-free discontinuous Galerkin method for incompressible flow that eliminates pressure and uses a globally divergence-free velocity space, enabling efficient high-Reynolds number flow simulations.

Contribution

The paper develops a novel explicit divergence-free DG method with a globally divergence-free velocity space that simplifies pressure handling and links mass matrix inversion to a Poisson solver.

Findings

Produces identical velocity fields to existing divergence-conforming DG methods.

Efficient implementation via hybrid-mixed Poisson solver.

Suitable for unsteady high-Reynolds number flows.

Abstract

We present an explicit divergence-free DG method for incompressible flow based on velocity formulation only. A globally divergence-free finite element space is used for the velocity field, and the pressure field is eliminated from the equations by design. The resulting ODE system can be discretized using any explicit time stepping methods. We use the third order strong-stability preserving Runge-Kutta method in our numerical experiments. Our spatial discretization produces the {\it identical} velocity field as the divergence-conforming DG method of [Cockburn et al., JSC 2007(31), pp.61-73] based on a velocity-pressure formulation, when the same DG operators are used for the convective and viscous parts. Due to the global nature of the divergence-free constraint, there exist no local bases for our finite element space. We present a key result on the efficient implementation of the scheme by identifying the equivalence of the (dense) mass matrix inversion of the globally divergence-free finite element space to a standard (hybrid-)mixed Poisson solver. Hence, in each time step, a (hybrid-)mixed Poisson solver is used, which reflects the global nature of the incompressibility condition. Since we treat viscosity explicitly for the Navier-Stokes equation, our method shall be best suited for unsteady high-Reynolds number flows so that the CFL constraint is not too restrictive.