引力场下流体力学方程的保辛气体动力学格式
A Symplecticity-preserving Gas-kinetic Scheme for Hydrodynamic Equations under Gravitational Field
摘要：
本文定义了一个能精确保持流体静力学等温解的重力流体动力学系统的平衡格式。在本文中，我们将构造一个良好平衡的气体动力学保辛BGK（SP-BGK）格式。为了发展这样一个计划，我们模型的引力势作为一个分段的阶梯函数与潜在的跳跃在细胞界面。同时，利用刘维尔定理和哈密顿流的保辛性质描述了粒子通过势垒的穿透、反射和变形。使用保辛性质的哈密顿流是至关重要的气体分布函数的高阶矩的评估时，通过潜在的跳跃。SP-BGK方法是目前第一个具有良好平衡性的激波捕捉Navier-Stokes流场求解方法。一些定理将证明这个计划，其中包括使用一个精确的麦克斯韦保持流体静力学状态的必要性，总质量和能量（动能，热和引力的总和）守恒，以及在粒子传输和碰撞过程中保持流体静力学状态的良好平衡的属性。数值算例验证了SP-BGK格式的有效性。
英文摘要：
A well-balanced scheme for a gravitational hydrodynamic system is defined as a scheme which could precisely preserve a hydrostatic isothermal solution. In this paper, we will construct a well-balanced gas-kinetic symplecticity-preserving BGK (SP-BGK) scheme. In order to develop such a scheme, we model the gravitational potential as a piecewise step function with a potential jump at the cell interface. At the same time, the Liouville's theorem and symplecticity preserving property of a Hamiltonian flow have been used in the description of particles penetration, reflection, and deformation through a potential barrier. The use of the symplecticity preserving property for a Hamiltonian flow is crucial in the evaluation of the high-order moments of a gas distribution function when crossing through a potential jump. As far as we know, the SP-BGK method is the first shock capturing Navier-Stokes flow solver with well-balanced property for a gravitational hydrodynamic system. A few theorems will be proved for this scheme, which include the necessity to use an exact Maxwellian for keeping the hydrostatic state, the total mass and energy (the sum of kinetic, thermal, and gravitational ones) conservation, and the well-balanced property to keep a hydrostatic state during particle transport and collision processes. Many numerical examples will be presented to validate the SP-BGK scheme.
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