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.

本文提出了一种用于二维理想磁流体动力学 (MHD) 方程的以细胞为中心的拉格朗日方法。为了计算通过单元界面的节点速度和数值通量，设计了一个 HLLD 型二维节点近似黎曼求解器。黎曼求解器的主要新特点是两个快速波，每个黎曼问题都考虑了两个阿尔文波和一个熵波，因此可以很好地捕捉到旋转不连续性。在拉格朗日方案中，证明了磁场的演化与磁冻结原理是一致的，从而准确地保证了无散约束。此外，线性重建方法用于实现二阶空间精度，而龙格-库塔方法用于获得二阶时间精度。