A Runge-Kutta discontinuous Galerkin method for Lagrangian ideal magnetohydrodynamics equations in two-dimensions

Abstract

In this paper, a Runge-Kutta Discontinuous Galerkin (RKDG) method is proposed for solving the two-dimensional ideal compressible magnetohydrodynamics (MHD) equations under the Lagrangian framework. The method is based on the Lagrangian type scheme designed for the Euler equations of compressible gas dynamics on arbitrary quadrilateral meshes. To deal with the divergence free constraint condition of the magnetic field, we adopt the local divergence free function space to approximate the magnetic field. A HWENO reconstruction limiter is presented for the numerical solution to inhibit the non-physical oscillation for flows with strong discontinuities. This limiter can not only preserve the local divergence free property of the magnetic field but also avoid to calculate the complex eigen-system of the MHD equations. Some numerical examples are presented to demonstrate the accuracy and the non-oscillatory property of the scheme.