迈向超高效动力学格式。第一部分：BGK方程基础
Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation
AI总结：
本文提出一种基于分裂技术的超高效数值方法求解BGK方程，通过精确求解传输线性部分避免分布函数重构，显著降低计算成本，首次在单处理器笔记本上实现全六维动力学方程求解。
摘要：
本文提出了一种新的超高效数值方法用于求解动力学方程。在这项初步工作中，我们针对BGK松弛算子给出了该格式。该格式基于输运和碰撞之间的分裂技术，可以轻松扩展到其他碰撞算子，如玻尔兹曼碰撞积分，或其他动力学方程，如弗拉索夫方程。该方法所依赖的关键思想是在网格上求解碰撞部分，然后通过沿时间向后追踪特征线来精确求解线性输运部分。所提方法与半拉格朗日方法的主要区别在于，这里我们不需要在每个时间步重构分布函数。这极大地降低了计算成本，并且据作者所知，首次允许在单处理器笔记本电脑上计算完整的六维动力学方程的解。给出了直至全三维情况的数值算例，验证了该方法并评估了其在1D、2D和3D中的效率。
英文摘要：
In this paper we present a new ultra efficient numerical method for solving kinetic equations. In this preliminary work, we present the scheme in the case of the BGK relaxation operator. The scheme, being based on a splitting technique between transport and collision, can be easily extended to other collisional operators as the Boltzmann collision integral or to other kinetic equations such as the Vlasov equation. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport linear part by following the characteristics backward in time. The main difference between the method proposed and semi-Lagrangian methods is that here we do not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost of the method and it permits for the first time, to the author's knowledge, to compute solutions of full six dimensional kinetic equations on a single processor laptop machine. Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D.
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