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

了解等离子体的行为在热核聚变、卫星放大器和计算机芯片制造等现代科学和工程中发挥着越来越重要的作用。等离子体物理中的一个基本模型是Vlasov-Maxwell系统，它是一个描述带电粒子在自洽电磁力作用下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性仿真工具，高效、可靠、准确的输运方案具有基础性意义。这类研究的主要数值挑战在于高维、非线性耦合以及在空间和时间上固有的多尺度特性。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。目前的全球气候模型包括数百种示踪剂，以充分表示复杂的物理和化学过程，这导致了计算机模拟的巨大计算成本。PI将开发和分析一类高效、可靠和高精度的数值方法来解决等离子体物理和大气科学中的输运问题。利用高阶不连续Galerkin空间离散化的特点，设计了一种半拉格朗日框架，该框架具有灵活性、紧凑性和良好的多尺度特征分辨能力等优点。通过对方案公式的精心设计，所提出的方案没有分裂误差，并且能够保持系统的总质量。在PI致力于开发一种能够恢复静磁极限的快速渐近保持Maxwell求解器的工作的激励下，将解决与Vlasov-Maxwell模拟相关的时间尺度分离问题。对于在全球化学-气候模拟中的应用，新格式可以方便地适用于基于立方球几何的球面输送模拟。将调查包括稳定性分析和误差估计在内的理论问题。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.