相对论流体力学方程组的紧致高阶气体动理学格式研究

The relativistic hydrodynamic (RHD) equations or the relativistic.magnetohydrodynamics (RMHD) equations are one of the most important models in the fields such as astrophysics and plasma physics. The relativistic hydrodynamic equations are complicated so that their analytical treatment is extremely difficult. A powerful and primary approach to improve our understanding of the physical mechanisms in the RHD equations or the RMHD equations is through numerical simulations in which constructing high-order numerical schemes is one of the hot research topics. The most commonly used equation of state (EOS) in the existing high-order numerical schemes is a simplified form of the exact EOS derived by Synge for the single-component perfect gas in relativistic regime. The simplified EOS is essentially valid only for the gas of either sub-relativistic or ultra-relativistic temperature. This project will research on the numerical schemes for the RHD equations and the RMHD equations with the exact EOS for the relativistic perfect gas based on the kinetic theory. Firstly, the compact high-order gas kinetic schemes for the RHD equations will be developed. Based on the governing equations describing the particle motion, the high-order accuracy particle distribution functions are calculated at the cell interfaces, and then macroscopic quantities such as high-order accuracy numerical fluxes are obtained,thereby the numerical schemes are developed while the high-order reconstruction and the two-stage fourth order time-accurate discretization are performed. Secondly, the high-order and high resolution methods will be developed for the RMHD equations in which the approximate magnetic field is divergence-free. The developed numerical schemes are not only applicable to the above extreme cases but also get quantitatively correct results in problems involving a transition from non-relativistic temperature to relativistic temperature or vice versa. It is expected to be an effective and reliable tool for accurately solving complex relativistic fluid problems.

相对论（磁）流体力学方程组是研究天体物理和等离子体物理等领域的重要模型之一，其形式复杂，解析处理极其困难，数值求解成为主要的研究手段，构造高精度数值格式是其中的热点研究课题。已有关于相对论（磁）流体力学方程组的高精度格式的研究工作，多考虑相对论完全气体状态方程的简化形式，仅适用于温度极高或温度极低这两种极限情形。本项目拟针对相对论完全气体状态方程，基于相对论气体动理学理论：研究相对论流体力学方程组的紧致高阶气体动理学格式，从描述粒子运动的控制方程出发，计算单元界面处时空高阶的粒子分布函数，进而得到高精度数值通量等宏观信息，利用紧致高阶空间重构与两级四阶时间离散方法建立格式；研究相对论磁流体力学方程组的高精度高分辨率且保持磁场散度为零的动理学方法。所建立的格式不仅适用于上述两种极限情形，而且适用于温度急剧变化的复杂物理过程，以期为复杂的相对论流体现象的高精度数值模拟提供有效可靠的工具。

相对论流体力学方程组在天体物理和等离子体物理等领域中占有十分重要的地位，其形式复杂，解析处理极其困难，数值求解已成为主要的研究手段。本项目综合运用计算数学、气体动理学和流体力学的知识，研究相对论流体力学方程组的高效高精度数值方法。具体内容包括：针对相对论完全气体状态方程，基于相对论气体动理学理论，建立了相对论欧拉方程组的简化气体动理学格式，为相对论流体提供了一种高效的数值模拟工具；基于非结构三角形网格，针对狭义相对论流体力学方程组提出了一种可证明保物理约束性质的高阶紧致的有限体积WENO方法，为复杂区域上低密度低压力及流体速度接近光速的流体模拟问题提供了稳健高效的计算工具；发展了求解偏微分方程的自适应加权物理嵌入神经网络方法，该网络根据解的信息自适应更新不同损失项的权重系数，可以自动平衡不同约束项的贡献进而达到有效降低损失函数误差的目的。本项目将推动计算数学、天体物理及流体力学的交叉发展，为探索复杂区域上极端相对论情形的流体问题提供新的研究手段，为天体物理中重要物理现象的模拟提供有效的数值算法和理论支撑。