一种求解一维理想磁流体方程组的保正拉氏方法

拉氏方法在计算流体力学中扮演了一个十分重要的角色，并且十分适合于处理含有强磁场的物理问题，例如 Z 箍缩、托卡马克、惯性约束聚变等等。在这些物理问题中密度和热力学压力总是非负的。然而，运用数值格式对上述方程进行逼近时，得到的近似解并不能总是保持这种正性。为了处理这一问题，首先构建了一种拉氏 HLLD 近似黎曼解，这一近似黎曼解在合适的信号速度下可以保持保正性质。运用这一黎曼解，提出了一种求解一维理想可压缩磁流体方程组的守恒保正拉氏格式。最后，给出一些数值算例来证明方法的保正性。

A Positivity-preserving Lagrangian Method for Ideal Magnetohydrodynamics Equations in One-dimension

Lagrangian methods play a very important role in computational fluid dynamics and especially suitable for dealing with the physical problems related to high-intensity magnetic field such as Z-pinch, Tokamak, ICF, and so on. In these physical problems the density and thermal pressure are always non-negative. However, such positivity property is not always satisfied by approximated solutions which obtained by a numerical scheme. To deal with this problem, the paper develops a Lagrangian HLLD approximate Riemann solver which can keep positivity-preserving property under some appropriate signal speeds. With this solver, a conservative Lagrangian scheme for solving the ideal compressible magnetohydrodynamics equations in one-dimensional is proposed. At last, some numerical examples are presented to demonstrate the positivity-preserving property of our scheme.