A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field (Advances in Applied Mathematics and Mechanics, 2023)
A partial Runge-Kutta Discontinuous Galerkin (RKDG) method whichpreserves the exactly divergence-free property of the magnetic field is proposed inthis paper to solve the two-dimensional ideal compressible magnetohydrodynamics(MHD) equations written in semi-Lagrangian formulation on moving quadrilateralmeshes. In this method, the fluid part of the ideal MHD equations along with $z$-component of the magnetic induction equation is discretized by the RKDG methodas our previous paper [47]. The numerical magnetic field in the remaining two directions (i.e., $x$ and $y$) are constructed by using the magnetic flux-freezing principlewhich is the integral form of the magnetic induction equation of the ideal MHD.Since the divergence of the magnetic field in 2D is independent of its $z$-directioncomponent, an exactly divergence-free numerical magnetic field can be obtained bythis treatment. We propose a new nodal solver to improve the calculation accuracyof velocities of the moving meshes. A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complexeigen-system of the MHD equations. Some numerical examples are presented todemonstrate the accuracy, non-oscillatory property and preservation of the exactlydivergence-free property of our method.