一种用于Landau动力学方程的结构保持碰撞粒子方法
A structure-preserving collisional particle method for the Landau kinetic equation
AI总结：
本文提出一种结构保持随机粒子方法求解Landau方程，利用球面Brown运动采样避免额外时间离散，实现O(N)复杂度并保持守恒律与熵耗散，且可推广至空间非均匀情形。
摘要：
本文提出并实现了一种用于Landau方程的结构保持随机粒子方法。该方法基于Landau方程的粒子系统，其中两两掠射碰撞被建模为扩散过程。通过利用粒子系统的独特结构以及球面Brown运动采样，该方法避免了对粒子系统进行额外的时间离散化，从而确保离散时间粒子分布与其连续时间对应物精确一致。该方法每个时间步实现O(N)复杂度，并保持基本物理性质，包括质量、动量和能量守恒以及熵耗散。在数值实验中，该方法展现出很强的长时间精度和稳定性。此外，我们还通过Vlasov–Poisson–Landau方程的案例研究，将该方法应用于空间非均匀方程。
英文摘要：
In this paper, we propose and implement a structure-preserving stochastic particle method for the Landau equation. The method is based on a particle system for the Landau equation, where pairwise grazing collisions are modeled as diffusion processes. By exploiting the unique structure of the particle system and a spherical Brownian motion sampling, the method avoids additional temporal discretization of the particle system, ensuring that the discrete-time particle distributions exactly match their continuous-time counterparts. The method achieves $O(N)$ complexity per time step and preserves fundamental physical properties, including the conservation of mass, momentum and energy, as well as entropy dissipation. It demonstrates strong long-time accuracy and stability in numerical experiments. Furthermore, we also apply the method to the spatially non-homogeneous equations through a case study of the Vlasov--Poisson--Landau equation.
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