An Asymptotic Preserving Scheme for the ES-BGK model

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme

Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations

We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator, the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally we show some numerical results which confirm the theoretical analysis

A Comparative Study of an Asymptotic Preserving Scheme and Unified Gas-kinetic Scheme in Continuum Flow Limit

Asymptotic preserving (AP) schemes are targeting to simulate both continuum and rarefied flows. Many AP schemes have been developed and are capable of capturing the Euler limit in the continuum regime. However, to get accurate Navier-Stokes solutions is still challenging for many AP schemes. In order to distinguish the numerical effects of different AP schemes on the simulation results in the continuum flow limit, an implicit-explicit (IMEX) AP scheme and the unified gas kinetic scheme (UGKS) based on Bhatnagar-Gross-Krook (BGk) kinetic equation will be applied in the flow simulation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for the comparison of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles are very close to the classical benchmark solutions. However, the IMEX AP scheme seems have difficulty to get these solutions. Based on the analysis and the numerical experiments, it is realized that the dissipation of AP schemes in continuum limit is closely related to the numerical treatment of collision and transport of the kinetic equation. Numerically it becomes necessary to couple the convection and collision terms in both flux evaluation at a cell interface and the collision source term treatment inside each control volume

The BGK approximation of kinetic models for traffic

We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable non-equilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model