Developing Energy-Conserving Deterministic Solvers for Kinetic Electromagnetic Plasma Simulations

在这项提案中，研究人员计划开发和分析一类节能的确定性求解器，用于动态电磁等离子体模拟。所提出的方法克服了许多传统求解方法的困难：它节省了系统的总粒子数和能量；它有一个系统的方法来结合显式或隐式时间步进取决于方程的刚度；它是为物理空间中复杂几何图形的非结构化网格实现而设计的。为了实现这一目标，研究课题包括一系列的分析和计算课题。研究者提出对耦合Vlasov-Maxwell系统设计一种新的节能分裂，使分裂后的方程仍然保持能量守恒，并且可以进行降维计算。研究者计划建立一个综合各种类型的节能时间和空间离散化的总体框架。将探索利用各种局部基、局部时间步进和混合求解方法进一步提高计算效率的方法。将讨论如何加强电荷连续性和正性的问题。分析方面，如数值色散关系，稳定性和误差估计将被考虑。提议的活动介于算法开发、分析和应用之间。所得到的数值格式可广泛应用于等离子体模拟。理论研究将为此类数值方法的设计提供依据和指导。拟议活动的更广泛影响将是其跨学科的推广和教育内容。拟议的研究本质上是多学科的。研究者积极地与物理系、电子工程系的教员进行互动和咨询。为学生和博士后提供培训机会。计算数学课程的发展将被纳入。

In this proposal, the investigator plans to develop and analyze a class of energy-conserving deterministic solvers for kinetic electromagnetic plasma simulations. The proposed methods have several features that overcome the difficulties of many traditional solvers: it conserves the total particle number and energy of the system; it has a systematic way to incorporate explicit or implicit time stepping depending on the stiffness of the equations; and it is designed for implementations on unstructured grids for complex geometries in the physical space. To achieve the objective, the research topics include a range of analytical and computational subjects. The investigator proposes to design a new energy-conserving splitting for the coupled Vlasov-Maxwell system, so that the splitted equations still maintain energy conservation and can be computed in reduced dimensions. The investigator plans to set up a general framework to incorporate various type of energy-conserving temporal and spatial discretizations. Methods to further improve computational efficiency by using various local basis, local time stepping and hybrid solvers will be explored. Issues on how to enforce charge continuity and positivity will be addressed. Analytical aspects such as the numerical dispersion relation, stability and error estimates will be considered.The proposed activity lies between algorithm development, analysis and applications. The resulting numerical schemes can be applied to a wide range of plasma simulations. The theoretical studies will provide foundation and guidance to the design of such numerical methods. The broader impacts of the proposed activity will be its interdisciplinary outreach and educational components. The proposed research is multidisciplinary in its nature. The investigator actively interacts and consults with faculty members in physics, electrical engineering departments. Training opportunities for students and postdocs will be provided. Computational math curriculum development will be incorporated.