An Explicit Divergence-Free DG Method for Incompressible Magnetohydrodynamics

Abstract

We extend the recently introduced explicit divergence-free DG scheme for incompressible hydrodynamics [7] to the incompressible magnetohydrodynamics (MHD). A globally divergence-free finite element space is used for both the velocity and the magnetic field. Highlights of the scheme includes global and local conservation properties, high-order accuracy, energy-stability, pressure-robustness. When forward Euler time stepping is used, we need two symmetric positive definite (SPD) hybrid-mixed Poisson solvers (one for velocity and one for magnetic field) to advance the solution to the next time level.Since we treat both viscosity in the momentum equation and resistivity in the magnetic induction equation explicitly, the method shall be best suited for inviscid or high-Reynolds number, low resistivity flows so that the CFL constraint is not too restrictive.

A Divergence-Conforming Hybridized Discontinuous Galerkin Method for the Incompressible Magnetohydrodynamics Equations

“…Otherwise, the shear layer breaks down into vortices. Like [7,41], we use the Kelvin-Helmholtz instability problem to study the energy stability of our method. In our numerical experiments, we consider the domain Ω = (0, 2) × (0, 1) and initial conditions u 0 = (u x,0 , u y,0 ) and B 0 = (B x,0 , B y,0 ) with…”

“…A recent paper further proved velocity field error estimates for such methods that are independent of both the pressure and the Reynolds number [27]. Divergence-conforming CG and DG methods for the incompressible MHD equations exhibit similar properties [28], though error estimates independent of the Reynolds number and magnetic Reynolds number do not exist yet for such discretizations. The above inspires us to construct an HDG method for the incompressible MHD equations that yields pointwise divergencefree velocity and magnetic fields by extending a divergence-conforming HDG method for the incompressible Navier-Stokes equations [18].…”