High order methods for some kinetic models

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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.