An explicit divergence-free DG method for incompressible magnetohydrodynamics
This provides a novel numerical method for simulating incompressible MHD flows with exact divergence-free constraints, benefiting computational fluid dynamics researchers.
The authors extend an explicit divergence-free DG scheme from hydrodynamics to incompressible magnetohydrodynamics, achieving global divergence-free fields, high-order accuracy, and energy stability. The method requires two SPD Poisson solvers per time step and is best suited for high-Reynolds, low-resistivity flows.
We extend the recently introduced explicit divergence-free DG scheme for incompressible hydrodynamics [arXiv:1808.04669]. 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.