A positivity-preserving Lagrangian Discontinuous Galerkin method for ideal magnetohydrodynamics equations in one-dimension

Abstract

In this paper, we propose a conservative Lagrangian Discontinuous Galerkin (DG) scheme for solving the ideal compressible magnetohydrodynamics (MHD) equations in one-dimensional. This scheme can preserve positivity of physically positive variables such as density and thermal pressure. We first develop a Lagrangian HLLD approximate Riemann solver which can keep positivity-preserving property under some appropriate signal speeds. With this solver a first order positivity-preserving Lagrangian DG scheme can be constructed. Then we design a high order positivity-preserving and conservative Lagrangian DG scheme by using the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter. We adopt TVB minmod limiter with the local characteristic fields for the ideal MHD system to control spurious oscillations around the shock wave. Some numerical examples are presented to demonstrate the accuracy and positivity-preserving property of our scheme.