High order methods for some kinetic models

这项建议的目的是发展、分析和数值评估用于不同区域碰撞动力学模型的稳健模拟的高精度渐近保持数值方法，以及准确和高效的数值方法，该方法保留了包括Vlasov-Maxwell方程在内的无碰撞动力学模型的重要物理相关性质。高阶方法因其计算效率高、计算时间长、能够捕捉小尺度特征或多尺度现象而被广泛应用。动力学理论及其相关的数值模拟在稀薄气体动力学、等离子体物理、交通网络和蜂群等广泛的应用中发挥着越来越重要的作用。准确、稳健、高效的动力学模型模拟具有基础性意义。数值挑战在于大多数动力学模型的高维、小尺度、时间和空间的多尺度、非线性耦合、具有多重积分的非线性或奇异碰撞算子以及解的重要守恒性。通过拟议的项目，将改进直接或有可能解决上述某些挑战的数值和分析技术。计算机模拟工具，特别是那些高精度和高成本效益、保留关键物理属性、能够处理空间和时间多尺度的工具，将得到极大的丰富。

The objective of this proposal is to develop, analyze, and numerically evaluate highly accurate asymptotic preserving numerical methods for robust simulations of collisional kinetic models in different regimes, and accurate and efficient numerical methods which preserve important physically relevant properties for the collisionless kinetic model including Vlasov-Maxwell equations. High order methods are widely used for their computational efficiency to achieve expected accuracy, the excellent performance over long time, and their capability of capturing features with small scales or phenomena involving multiple scales. Kinetic theory and its related numerical simulations play an increasingly important role in a broad range of applications, such as rarefied gas dynamics, plasma physics, traffic networking, and swarming. Accurate, robust and efficient simulations of kinetic models are of fundamental significance. The numerical challenges lie in the high dimensionality of most kinetic models, small scales, multiple scales in both time and space, nonlinear coupling, nonlinear or singular collision operators with multi-fold integrals, and important conservation properties of the solutions. Through the proposed projects, both numerical and analytical techniques will be advanced which either directly or have potential to address some of the challenges mentioned above. Computer simulation tools, especially those being highly accurate and cost efficient, preserving key physical properties, and capable of handling both spatial and temporal multiscales, will be greatly enriched.