A RKDG Method for 2D Lagrangian Ideal Magnetohydrodynamics Equations with Exactly Divergence-Free Magnetic Field (Communications in Computational Physics, 2022)
In this paper, we present a Runge-Kutta Discontinuous Galerkin (RKDG)method for solving the two-dimensional ideal compressible magnetohydrodynamics(MHD) equations under the Lagrangian framework. The fluid part of the ideal MHDequations along with $z$-component of the magnetic induction equation are discretizedusing a DG method based on linear Taylor expansions. By using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation ofthe ideal MHD, an exactly divergence-free numerical magnetic field can be obtained.The nodal velocities and the corresponding numerical fluxes are explicitly calculatedby solving multidirectional approximate Riemann problems. Two kinds of limiter areproposed to inhibit the non-physical oscillation around the shock wave, and the second limiter can eliminate the phenomenon of mesh tangling in the simulations of therotor problems. This Lagrangian RKDG method conserves mass, momentum, andtotal energy. Several numerical tests are presented to demonstrate the accuracy androbustness of the proposed scheme.
Keywords: Lagrangian RKDG method; ideal compressible MHD equations; Taylor basis; exactly divergence-free magnetic field; limiter.