相对论性电子长时间动力学模拟的几何算法研究

Secular dynamics of relativistic electrons has theoretical significance and a wide range of applications. However, conventional algorithms are not applicable to this problem due to the coherent accumulation of numerical errors. To overcome this difficulty, it is desirable to develop advanced algorithms that have both the conservative property and the long term accuracy throughout the entire simulation. Our previous work focus on the mechanics in disregard of relativistic effects, for which we construct a class of advanced geometric algorithms with the phase-space volume-preserving property. These algorithms are proved to be stable for long time. In this program, we plan to further study the advanced geometric algorithms for simulating the dynamics of relativistic electrons. The main contents of the research includes: to develop explicit volume-preserving geometric algorithms that have long term stability, accuracy and fidelity for the dynamics of relativistic electrons, and to provide a systematic constructing procedure; to construct geometric algorithms with higher order of accuracy; to estimate the simulation ability of the newly developed geometric algorithms by analyzing their computing amounts, accuracy, stability and conservation of constants of motion; to apply the geometric algorithms in the design of the simulation program, which can be used to test the effectiveness of the algorithms, and to study the multi-scale problems such as runaway electrons in tokamaks.

相对论性电子在电磁场中的长时间动力学研究涉及加速器技术、磁约束装置中逃逸电子等物理问题。传统算法由于误差积累无法保证长期的数值精度，导致数值结果不可靠。针对这一问题，需要发展可准确描述相对论性电子动力学特征的、具有长期数值精确性与稳定性的先进算法。前期工作中已针对非相对论性电子长时间动力学模拟提出了稳定有效的几何算法。在已有工作基础上，本项目发展适用于相对论性电子动力学的几何算法。具体工作包括：针对相对论性电子的动力学方程，利用其相空间体积不变的特征，发展易于计算、具有长期稳定性的几何算法，并给出系统的构造流程；构造具有高阶精度的几何算法，以适应与相对论性电子动力学相关的不同物理问题的数值模拟研究；对构造的几何算法的计算精度、计算量、稳定性等进行数值分析，评估算法的计算仿真能力和应用价值；将构造的几何算法应用到等离子体数值模拟程序当中，检验算法有效性，并对逃逸电子等物理问题进行模拟研究。

在本项目中，我们使用理论分析与数值实验等手段，系统地发展了相对论电子数值模拟的几何算法。针对相对论洛伦兹力方程解的几何性质，构造了对称显式的保体积算法、保K-辛结构的显式几何算法和保辛几何算法。并且利用分裂技巧提出了几何算法系统的构造流程，应用processing技术，给出了高阶几何算法的构造方法。在此基础上，对所构造的几何算法的计算精度、计算量、长时间稳定性和守恒性质等进行了严格的数值与理论分析，完成了这些算法对相对论性电子动力学模拟的计算仿真能力和应用价值的评估。基于这些具有高度数值精确性和长期稳定性的数值算法，可以更高效、深入地解决加速器技术、磁约束装置中的逃逸电子等物理问题。