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Fengyan Li : High order asymptotic preserving methods for some kinetic models



Many problems in science and engineering involve parameters in their mathematical models. Depending on the values of the parameters, the equations can differ greatly in nature. Asymptotic preserving (AP) methods are one type of methods which are designed to work uniformly with respect to different scales or regimes of the equations when the parameters vary.

In this talk, I will present our work in developing high order AP methods for some kinetic models, including discrete-velocity models in a diffusive scaling and the BGK model in a hyperbolic scaling. When the Knudson number approaches zero, the limiting equations of the former model can be heat equation, viscous Burgers’ equation, or porous medium equation, while the limiting equations for the latter are the compressible Euler equations. When the Knudson number is very small, the BGK model also leads to compressible Navier-Stokes equations. The proposed methods are built upon a micro-macro decomposition of the equations, high order discontinuous Galerkin (DG) spatial discretizations, and the globally stiffly accurate implicit-explicit Runge-Kutta (IMEX-RK) temporal discretizations. Theoretical results are partially established for uniform stability, error estimates, and rigorous asymptotic analysis. Numerical experiments will further demonstrate the performance of the methods.

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