High order methods for some kinetic models

这项建议的目的是开发，分析和数值评估高精度的渐近保持数值方法的鲁棒性模拟的碰撞动力学模型在不同的制度，准确和有效的数值方法，保持重要的物理相关属性的碰撞动力学模型，包括弗拉索夫-麦克斯韦方程组。高阶方法因其计算效率高、能达到预期的精度、长时间内的优良性能以及能捕捉小尺度特征或多尺度现象而被广泛使用。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、交通网络和群集等广泛的应用中发挥着越来越重要的作用。动力学模型的准确、鲁棒和高效的模拟具有重要意义。数值计算的挑战在于大多数动力学模型的高维性、小尺度、时间和空间的多尺度、非线性耦合、多重积分的非线性或奇异碰撞算子以及解的重要守恒性质。通过拟议的项目，数字和分析技术都将得到发展，这些技术或直接或有可能解决上述一些挑战。计算机模拟工具，特别是那些高度准确和成本效益，保留关键的物理特性，并能够处理空间和时间的多尺度，将大大丰富。

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.