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.

在此提案中，研究人员建议开发一种基于高阶网格的数值方法，用于弗拉索夫模拟。在阶段空间中，提出的方法对空间对流的高阶有限元不连续的Galerkin（DG）方法，用于按场方程（Maxwell's或Poisson方程）计算远距离力量（Maxwell's或Poisson的方程），而高阶差异基本上是非振荡（WENO），实质上是非振荡（WENO），用于通过驾驶员启动速度指导中的粒子相互作用。该方法旨在在处理复杂的几何形状方面具有DG方法的优势，以及在解决复杂/不足的解决方案结构方面的稳健性和稳定性中的WENO重建。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对流使用超大的数值时间步骤。设计合适的数值溶液空间，以确保在六维相空间中不同数值方法之间的高阶耦合。提出了光谱/积分递延校正框架，以确保高阶时间准确性。除了空间和时间的高阶精度外，所提出的方案还将设计为质量保守和积极性，这是分析解决方案的两个重要特性。研究者和她的小组将进行收敛研究，并跟踪物理保守数量（例如动量和能量）的时间演变，以衡量所提出的方案的质量。拟议活动的智力优点在于，在半lasangangian框架下的稳健，有效且高度准确的数字algorithm simporcution for vlasov for vlasov for vlasov for vlasov for vlasov for vlasov simporcution simporcution simporcution for。 拟议项目的目的是设计一种高阶数值方法，该方法允许具有准确性和大型数值时间步长且稳定性相对较粗糙的空间网格。同时，将研究所提出的方案在相对简单的设置（例如线性方程）下的理论准确性和稳定性。理论研究将为数字算法设计提供坚实的基础和良好的指导。发达的算法将在融合模拟以及其他应用领域（例如天体物理学，半导体设备模拟）中产生影响。进一步的影响来自拟议研究的多学科性质，以及对本科生和研究生的培训。