A High Order Semi-Lagrangian Approach for the Vlasov Equation

在这个建议中，研究者提出了一个非常高阶的网格为基础的数值方法Vlasov模拟。在相空间中，所提出的方法耦合高阶有限元间断Galerkin（DG）方法的空间平流和计算远程力的场方程（麦克斯韦的或泊松方程）和高阶有限差分加权基本无振荡（韦诺）计划的粒子相互作用的速度方向上通过运营商分裂。该方法的目的是利用DG方法的灵活性和紧凑性，在处理复杂的几何形状，和韦诺重建的鲁棒性和稳定性，解决复杂/欠解决的解决方案结构。为了提高计算效率，研究者建议使用超大的数值时间步长，通过使用半拉格朗日框架的平流。设计了一个合适的数值解空间，保证了六维相空间中不同数值方法之间的高阶耦合。为了保证高阶时间精度，提出了谱/积分延迟校正框架。除了在空间和时间上的高阶精度外，所提出的格式将被设计为质量守恒和正性保持，这是解析解的两个重要性质。研究员及其小组将进行收敛性研究，并追踪物理守恒量（例如动量和能量）的时间演化，以衡量拟议方案的质量。拟议活动的智力价值在于在半拉格朗日框架下为Vlasov模拟开发一个稳健、高效和高精度的数值算法。 该项目的目标是设计一种高阶数值方法，允许相对粗糙的空间网格的精度和超大的数值时间步长的稳定性。同时，在相对简单的设置（如线性方程组）下，所提出的计划的理论精度和稳定性将进行研究。理论研究为数值算法的设计提供了坚实的基础和良好的指导。该算法的发展将对聚变模拟以及天体物理、半导体器件模拟等应用领域产生重要影响。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。

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.