Title: A structure-preserving collisional particle method for the Landau kinetic equation Show Chinese title

Title: 朗道动力学方程的保结构碰撞粒子方法

Abstract: 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. Show Chinese abstract

Abstract: 本文提出并实现了一种保结构的随机粒子法来求解兰道（Landau）方程。 该方法基于一个针对兰道方程的粒子系统，其中成对的碰撞被建模为扩散过程。 通过利用粒子系统的独特结构和球面布朗运动采样技术，该方法避免了对粒子系统进行额外的时间离散化，从而确保离散时间下的粒子分布精确匹配连续时间下的分布。 该方法每步计算复杂度为$O(N)$，并且能够保持质量、动量和能量守恒以及熵耗散等基本物理性质。 数值实验表明该方法具有长期的准确性和稳定性。 此外，我们还通过非均匀空间的 Vlasov--Poisson--Landau 方程的案例研究将此方法应用于非均匀空间方程。