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.[8] 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 treat 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.

An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics

“…For simplicity, we consider periodic boundary conditions only. However, the inflow/outflow/wall boundary conditions can be easily included, see [7]. We spefically mention that taking divergence of the equation (1c) yields ∂ t (∇·B) = 0.…”

“…In this paper, we propose a divergence-free DG method for the incompressible MHD equation based on a velocity-magnetic field formulation, extending our previous work on a divergence-free DG scheme for incompressible hydrodynamics [7] to the incompressible MHD setting. In particular, we use a globally divergence-free finite element space for both the velocity and magnetic field.…”

“…The scheme enjoys features such as global and local conservation properties, high-order accuracy, energy-stability, and pressure-robustness. Moreover, the scheme can be efficiently implemented [7] when coupled with explicit time stepping methods. In particular, two hybrid-mixed Poisson solvers (equivalent to the mass matrix inversion of the divergence-free finite elements) is needed to advance solution in time when forward Euler time stepping is used.…”

A divergence‐conforming hybridized discontinuous Galerkin method for the incompressible Reynolds‐averaged Navier‐Stokes equations

“…For example, divergence‐conforming discretizations simultaneously conserve momentum and energy, while discretizations that only weakly satisfy the divergence‐free constraint typically conserve either momentum or energy but not both. The velocity error is also decoupled from the pressure error for divergence‐conforming discretizations, a property commonly referred to as pressure robustness . Finally, mass conservation is considered to be of prime importance for coupled flow transport, so divergence‐free discretizations are particularly attractive for such applications.…”

“…The velocity error is also decoupled from the pressure error for divergence-conforming discretizations, a property commonly referred to as pressure robustness. 28 Finally, mass conservation is considered to be of prime importance for coupled flow transport, 29 so divergence-free discretizations are particularly attractive for such applications. This last point inspires our current work, as the Spalart-Allmaras model may be viewed as a coupled flow-transport problem wherein the eddy viscosity is a scalar transported by the fluid flow.An outline of this paper is as follows.…”