A 2D cell-centered Lagrangian scheme based on multi-state Riemann solver with exactly divergence-free magnetic fields

Abstract

This paper presents a cell-centered Lagrangian method for the ideal magnetohydrodynamics (MHD) equations in two dimension. In order to compute the nodal velocity and the numerical fluxes through the cell interface, a 2D nodal approximate Riemann solver of HLLD-type is designed. The main new feature of the Riemann solver is two fast waves, two Alfvén waves and one entropy wave are considered for each Riemann problem, and thus the rotational discontinuities can be captured very well. In the Lagrangian scheme, the evolving of the magnetic field is proved to be consistent with the magnetic frozen principle and thus guarantee exactly the divergence-free constraint. In addition, a linear reconstruction method is applied to achieve second order spatial accuracy while the Runge-Kutta method is used to obtain second order temporal accuracy. Various numerical tests are presented to demonstrate the accuracy and robustness of the algorithm.