A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

By Francis Filbet, Shi Jin. Journal of Computational Physics, 2009

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In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.

Subfields: Applied Mathematics, Computational Mechanics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/7cf106778d1bc477590a2db1a0a3f9dcdaa2c324

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Citations in the corpus

Numerical methods for kinetic equations* (2014, methodology): “Recently they have been designed to achieve asymptotic preservation for the Boltzmann equation in the fluid limit without requiring the inversion of the collision operator: see Filbet and Jin (2010) and Dimarco and Pareschi (2012, 2013).”
A unified gas-kinetic scheme for continuum and rarefied flows (2011, background): “As defined in [11], a kinetic scheme is AP if (1) it preserves the discrete analogy of the Chapman–Enskog expansion when the Knudsen numbers go to 0.”
A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases (2012, methodology): “Intensive research has been conducting in the further improvement of DOM methods [3, 8, 9, 21, 22].”