Communications in Mathematical Sciences

Volume 17 (2019)

Number 5

Dedicated to the memory of Professor David Shen Ou Cai

Asymptotic-preserving schemes for two-species binary collisional kinetic system with disparate masses, I: time discretization and asymptotic analysis

Pages: 1257 – 1289



Irene M. Gamba (Department of Mathematics and Oden Institute for Computational Engineering and Sciences, University of Texas, Austin, Tx., U.S.A.)

Shi Jin (School of Mathematical Sciences, Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China)

Liu Liu (Oden Institute for Computational Engineering and Sciences, University of Texas, Austin, Tx., U.S.A.)


We develop efficient asymptotic-preserving time discretization schemes to solve the disparate mass kinetic system of a binary gas or plasma in the “relaxation time scale” relevant to the epochal relaxation phenomenon. Since the resulting model is associated to a parameter given by the square of the mass ratio between the light and heavy particles, we develop an asymptotic-preserving scheme in a novel decomposition strategy using the penalization method for multiscale collisional kinetic equations. Both the Boltzmann and Fokker–Planck–Landau (FPL) binary collision operators will be considered. Other than utilizing several AP strategies for single-species binary kinetic equations, we also introduce a novel splitting and a carefully designed explicit-implicit approximation, which are guided by the asymptotic analysis of the system. We also conduct asymptotic-preserving analysis for the time discretization, for both space homogenous and inhomogeneous systems.


two-species kinetic system, disparate mass, epochal relaxation, asymptotic-preserving method

2010 Mathematics Subject Classification

35Q20, 65M99, 82D10

The first and third authors were supported by funding from the DOE–Simulation Center for Runaway Electron Avoidance and Mitigation. The second author was supported by NSF grants DMS-1522184 and DMS-1107291: RNMS KI-Net and NSFC grant No. 31571071.

Received 25 October 2018

Accepted 4 May 2019

Published 6 December 2019