等离子体物理中的模型方程及其高效算法的理论和数值研究

This project focuses on the construction, theory and the application in plasma physics of structure-preserving numerical methods. Specifically, the stochastic differential equations related to the physical model is studied, and the corresponding numerical methods and numerical theory are developed. The relationship between the stochastic differential equation and the linear Boltzmann equation is established. Based on the stochastic differential theory, the study of the Boltzmann equation is provided with the help of stochastic differential equations. For the ideal magnetic fluid, its structure and conservation characteristics are studied, and the corresponding efficient numerical methods are developed based on these analysis. In the implementation of corresponding numerical methods, the appropriate preconditioners and the corresponding numerical simulation are provided. The stability theory for dynamical system is established. Combined the developed stability theory, the physical problem in plasma physics such as the two-fluid instability problem is analyzed and simulated. The structure-preserving numerical methods are applied to simulate the evolutionary behavior of runaway electrons. To simulate a large number of charged particle, the parallel computing technique is developed for this problem.

本项目关注保结构算法的构造、理论研究和结合等离子体物理相关问题的应用发展。 项目主要研究物理模型相关的随机微分方程，发展相应的数值方法和数值理论，并结合随机微分方程理论建立与线性Boltzmann方程之间的关系，通过随机微分方程的相关研究为Boltzmann方程的研究提供相应思路；对理想磁流体, 研究其结构和守恒特征并建立相应的数值离散，发展高效的数值计算方法，在数值计算中发展合适的预条件子并进行相应的数值模拟；结合等离子体模型方程发展相应的动力系统和数值算法的稳定性分析理论，并对等离子体物理中如双流不稳定性等物理问题进行动力学行为的分析和相应的数值模拟；对逃逸电子的动力学行为发展和应用相应的保结构数值方法，对大量样本采样的逃逸电子进行并行计算并在机群上进行实现。

本项目的研究背景是基于对等离子体物理问题中带电粒子动力学行为的研究。由于所研究的物理问题具有多尺度效应，物理问题本身又具有很多守恒特征，因此数值模拟要求可以构造能够保持系统特征并长时间稳定的数值方法。 本项目通过研究模型系统的结构特征，如Poisson结构构造了相应的保结构算法，并模拟了Landau damping、 Bernstein波及Weibel不稳定、双流不稳定等不稳定现象。通过理论分析和数值模拟结果验证了算法的有效性。同时发展了并行技术提高了算法的高效性，这帮助对更多物理问题的应用和研究。

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