An explicit divergence-free DG method for incompressible flow

Abstract

We present an explicit divergence-free DG method for incompressible flow based on velocity formulation only. An H(div)-conforming, and 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 identical velocity field as the divergence-conforming DG method of Cockburn et al. (2007) 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 and its interplay with the boundary conditions, it is very hard to construct local bases for our finite element space. Here we present a key result on the efficient implementation of the scheme by identifying the equivalence of the 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. In the actual implementation of this fully discrete scheme, the pressure field is also computed (via the hybrid-mixed Poisson solver). Hence, the scheme can be interpreted as a velocity–pressure formulation that treats the incompressibility constraint and pressure forces implicitly, but the viscous and convective part explicitly. 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.