Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations

DECAY OF A LINEAR OSCILLATOR IN A RAREFIED GAS: SPATIALLY ONE-DIMENSIONAL CASE Kazuo Aoki Kyoto University, Mechanical Engineering and Science An infinitely wide plate, subject to an external force in its normal direction obeying Hooke’s law, is placed in an infinite expanse of a rarefied gas. When the plate is displaced from its equilibrium position and released, it starts in general an oscillatory motion in its normal direction. This is the one-dimensional setting of a linear oscillator (or pendulum) considered previously for a collisionless gas and for a special Lorentz gas in our paper [T. Tsuji and K. Aoki, J. Stat. Phys. 146, 620 (2012)]. The motion decays as time proceeds because of the drag force on the plate exerted by the surrounding gas. The long-time behavior of the unsteady motion of the gas caused by the motion of the plate is investigated numerically on the basis of the BGK model of the Boltzmann equation, as well as the compressible Navier-Stokes equation (with the temperature-jump condition), with special interest in the rate of the decay of the oscillatory motion of the plate decays in proportion to an inverse power of time (power -3/2) for large time. This talk contains some results of the works in collaboration. The result provides numerical evidence that the displacement of the plate with Tetsuro Tsuji (Osaka University), Shingo Kosuge (Kyoto University, Japan), and Taiga Fujiwara (Kyoto University). A SPARSE GRID DISCONTINUOUS GALERKIN METHOD FOR HIGH-DIMENSIONAL TRANSPORT EQUATIONS Yingda Cheng Michigan State University, Department of Mathematics In this talk, we present a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG schemes for hyperbolic problems and is proven to be L2 stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. Good performance in accuracy and conservation is verified by numerical tests in up to four dimensions.