Development and Application of Efficient High-order Semi-Lagrangian Schemes

了解等离子体的行为在现代科学和工程中起着越来越重要的作用，如热核聚变、卫星放大器和计算机芯片制造。等离子体物理学中的基本模型是弗拉索夫-麦克斯韦系统，其是描述由于自洽电磁力引起的带电粒子的动力学的非线性动力学输运模型。作为研究复杂动力学系统的预测模拟工具，高效、可靠、准确的输运方案具有重要意义。在这些研究中的主要数值挑战在于高维，非线性耦合，在空间和时间的固有的多尺度性质。该项目涉及的另一个应用是大气科学。一个例子是研究平流层臭氧和许多其他化学成分演变的化学气候模型。目前的全球气候模式包括数百种示踪物质，以充分代表复杂的物理和化学过程，导致巨大的计算成本在计算机模拟。 PI将为等离子体物理和大气科学中的传输问题开发和分析一类高效，可靠和高精度的数值方法。一个半拉格朗日框架将设计采用高阶不连续的伽辽金空间离散化，以利用其许多有吸引力的属性，如灵活性，紧凑性，以及出色的能力，解决涉及多个尺度的功能。通过对格式的精心设计，该格式既避免了分裂误差，又能保持系统的总质量。受PI开发能够恢复静磁极限的快速渐近保持麦克斯韦求解器的工作的启发，将解决与Vlasov-麦克斯韦模拟相关的时间尺度分离问题。对于全球化学气候模拟的应用，新方案可以方便地适用于基于立方球几何的球形输运模拟。理论问题，包括稳定性分析和误差估计将进行调查。

Understanding behaviors of plasmas plays an increasingly important role in modern science and engineering such as thermo-nuclear fusion, satellite amplifier, and computer chip manufacturing. A fundamental model in plasma physics is the Vlasov-Maxwell system, which is a nonlinear kinetic transport model describing the dynamics of charged particles due to the self-consistent electromagnetic forces. As predictive simulation tools in studying the complex kinetic system, efficient, reliable and accurate transport schemes are of fundamental significance. The main numerical challenges in such studies lie in the high dimensionality, nonlinear coupling, and inherent multi-scale nature in both space and time. Another application concerned in this project is in atmospheric science. One example is the chemistry-climate model in the study of evolution of stratospheric ozone and many other chemical constituents. The present generation of global climate models include hundreds of tracer species in order to adequately represent complex physical and chemical processes, resulting in huge computational cost in computer simulations. The PI will develop and analyze a class of efficient, reliable and highly accurate numerical methods for transport problems in plasma physics and atmospheric science. A semi-Lagrangian framework will be devised by employing a high order discontinuous Galerkin spatial discretization to take advantage of its many attractive properties, such as flexibility, compactness, and excellent ability to resolve features involving multiple scales. By a careful design in the scheme formulation, the proposed scheme is free of splitting error and able to conserve total mass of the system. Motivated by the work of PI on developing a fast asymptotic preserving Maxwell solver that is capable of recovering the magneto-static limit, the temporal scale separation issue associated with the Vlasov-Maxwell simulations will be addressed. For the applications in the global chemistry-climate modeling, the new schemes can be conveniently adapted to the spherical transport simulations based on the cubed-sphere geometry. Theoretical issues including the stability analysis and error estimates will be investigated.