混合离子Vlasov-Fokker-Planck与流体电子系统的自适应保守恒低秩IMEX求解器
An adaptive and conservative low-rank IMEX solver for the hybrid ion Vlasov-Fokker-Planck and fluid electron system
中文总结 AI 辅助
该研究针对混合离子Vlasov-Fokker-Planck与流体电子系统，提出结合RAIL、LoMaC等方法的自适应保守恒低秩IMEX求解器，验证了其有效性并证明宏观量守恒的重要性。
中文摘要 AI 辅助
我们提出了一种宏观量守恒、秩自适应的方法，用于求解混合Vlasov-Fokker-Planck（VFP）方程，其中离子被视为动力学量，电子被视为流体。求解该系统面临若干耦合计算挑战：维数灾难、宏观量守恒、模型多尺度特性带来的刚性，以及结构保持。为应对这些挑战，我们将多种针对特定难题的成熟方法整合为求解该非线性系统的统一框架。具体而言，我们采用最新的降维增广隐式低秩（RAIL）方法实现速度空间的秩自适应，以应对维数灾难；采用高阶隐式-显式时间步进处理多尺度刚性；采用局部宏观量守恒（LoMaC）程序对解进行守恒截断。RAIL和LoMaC方法已从笛卡尔坐标扩展到柱坐标。所得框架可实现隐式秩自适应时间积分，同时保持守恒性并利用保结构离散化。我们通过一系列测试问题验证该格式，并模拟驻定激波以证明守恒宏观量的重要性。
英文摘要
We present a macroscopically conservative, rank-adaptive method for solving a hybrid Vlasov-Fokker-Planck (VFP) equation in which the ions are treated kinetically, and the electrons are treated as a fluid. Solving this system poses several coupled computational challenges: the curse of dimensionality, conservation of macroscopic quantities, stiffness arising from the multiscale nature of the model, and structure preservation. To address these challenges, we combine several established methods, each targeting a specific difficulty, into a unified framework for solving this nonlinear system. Specifically, we employ the recent Reduced Augmentation Implicit Low-rank (RAIL) method for rank adaptivity in velocity space to combat the curse of dimensionality, high-order implicit-explicit time stepping to handle the multiscale stiffness, and the Local Macroscopic Conservative (LoMaC) procedure for conservative truncation of the solution. The RAIL and LoMaC methods are extended from Cartesian to cylindrical coordinates. The resulting framework accommodates implicit rank-adaptive time integration while remaining conservative and leveraging structure-preserving discretizations. We verify the scheme on a suite of test problems and simulate a standing shock to demonstrate the importance of conserving the macroscopic quantities.
