A High Order Semi-Lagrangian Approach for the Vlasov Equation

Vlasov方程的高阶半拉格朗日方法

• 批准号: 1217008
• 负责人: Jing-Mei Qiu
• 金额: $ 18.55万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217008&HistoricalAwards=false
• 关键词: Order; Semi; Lagrangian; Approach; Vlasov

In this proposal, the investigator proposes to develop a very high order mesh-based numerical method for Vlasov simulations. In the phase space, the proposed methodology couples the high order finite element discontinuous Galerkin (DG) method for spatial advection and for computing long range forces by field equations (Maxwell's or Poisson's equations) and the high order finite difference weighted essentially non-oscillatory (WENO) scheme for particle interactions in velocity directions via operator splitting. The methodology is designed to take advantages of the DG method in its flexibility and compactness in handling complicated geometry, and the WENO reconstructions in their robustness and stability in resolving complicated/under-resolved solution structures. To improve computational efficiency, the investigator proposes to use extra large numerical time steps by using semi-Lagrangian framework for advection. A suitable numerical solution space is designed to ensure high order coupling among different numerical methods in six-dimensional phase space. Spectral/integral deferred correction framework is proposed to guarantee high order temporal accuracy. Besides the high order accuracy in both space and time, the proposed scheme would be designed to be mass conservative and positivity preserving, which are two important properties of the analytical solution. The investigator and her group are going to perform convergence study, as well as track the time evolution of physically conserved quantities (e.g. momentum and energy) as a measurement of the quality of the proposed scheme.The intellectual merit of the proposed activity lies in the development of a robust, efficient and highly accurate numerical algorithm under a semi-Lagrangian framework for Vlasov simulations. The objective of the proposed project is to design a high order numerical approach that allows for relatively coarse spatial mesh with accuracy and extra large numerical time steps with stability. At the same time, theoretical accuracy and stability properties of the proposed scheme under relatively simple setting (e.g. linear equations) will be studied. The theoretical study will provide a solid foundation, as well as a good guidance, to the design of numerical algorithm. The well-developed algorithm will have impact in fusion simulations, as well as other applied fields such as astrophysics, semi-conductor device simulations. Further impact comes from the multidisciplinary nature of the proposed research, as well as the training of undergraduate and graduate students.

在该提案中，研究人员建议开发一种用于 Vlasov 模拟的非常高阶的基于网格的数值方法。在相空间中，所提出的方法将用于空间平流和通过场方程（麦克斯韦或泊松方程）计算长程力的高阶有限元不连续伽辽金（DG）方法与本质上非振荡加权的高阶有限差分（ WENO）通过算子分裂实现速度方向上的粒子相互作用的方案。该方法旨在利用 DG 方法在处理复杂几何形状方面的灵活性和紧凑性，以及 WENO 重建在解决复杂/未解决的解决方案结构方面的鲁棒性和稳定性。为了提高计算效率，研究人员建议通过使用半拉格朗日框架进行平流来使用超大数值时间步长。设计合适的数值解空间，保证六维相空间中不同数值方法之间的高阶耦合。提出了谱/积分延迟校正框架来保证高阶时间精度。除了空间和时间上的高阶精度外，所提出的方案还将被设计为质量保守和正性保持，这是解析解的两个重要属性。研究人员和她的团队将进行收敛性研究，并跟踪物理守恒量（例如动量和能量）的时间演化，作为所提议方案质量的衡量标准。所提议活动的智力价值在于在用于 Vlasov 模拟的半拉格朗日框架下开发稳健、高效且高精度的数值算法。 该项目的目标是设计一种高阶数值方法，该方法允许相对粗糙的精确空间网格和稳定的超大数值时间步长。同时，将研究所提出方案在相对简单的设置（例如线性方程）下的理论精度和稳定性。理论研究将为数值算法的设计提供坚实的基础和良好的指导。成熟的算法将对聚变模拟以及天体物理学、半导体器件模拟等其他应用领域产生影响。进一步的影响来自拟议研究的多学科性质，以及本科生和研究生的培训。