A BGK‐penalization‐based asymptotic‐preserving scheme for the multispecies Boltzmann equation

Abstract

An asymptotic-preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK-penalization-based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625-7648), to its multispecies counterpart. For the multispecies Boltzmann equation, the new difficulties arise due to: (1) the breaking down of the conservation laws for each species and (2) different convergence rates to equilibria for different species in disparate masses systems. To resolve these issues, we find a suitable penalty function-the local Maxwellian that is based on the mean velocity and mean temperature and justify various asymptotic properties of this method. This AP scheme does not contain any nonlinear nonlocal implicit solver, yet it can capture the fluid dynamic limit with time step and mesh size independent of the Knudsen number. Numerical examples demonstrate the correct asymptotic-behavior of the scheme.

A Successive Penalty-Based Asymptotic-Preserving Scheme for Kinetic Equations

“…One can simply choose α = This can be seen as an approximation of the Dimarco-Pareschi method, with easier extension to more complicated problems. (42) can be applied to both the Boltzmann equation and the Landau equation, with the penalization P to be the BGK operator (8) or the Fokker-Planck operator (24), and the penalization weight β given by (17) or (26), respectively.…”

“…11 (3) and P (f ) the Fokker-Planck operator (24), (44) gives a first order strongly AP scheme. Here β is given by (26).…”

“…[31,15]. O(N log N ) by the spectral method in [16]; while the cost of inverting the Fokker-Planck operator (24) is O(N ), with a conjugate-gradient method (see [25] for details). In practice the computational cost does not increase significantly.…”

“…A similar relaxed AP property was obtained. This method was also extended to the quantum cases [13,20] and multispecies case [24], and also in diffusive limit of linear transport equation [8]. A rigorous analysis on the application of this method to hyperbolic system was given in [17].…”

A numerical scheme for a kinetic model for mixtures in the diffusive limit using the moment method

“…A reasonable attempt to derive an AP numerical scheme for Boltzmann equations for mixtures is to use a penalty method by a linear BGK‐operator as in , which has been extended for mixtures in . This operator would then be defined for each species 1 ≤ i ≤ p as P i : f i ↦ β i ( M i − f i ), where M i is the global Maxwellian equilibrium state defined by and β i is some constant to be specified.…”