An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering: Heavy tail equilibrium and degenerate collision frequency

Li Wang; Bokai Yan

doi:10.1137/17M1138029

An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering: Heavy tail equilibrium and degenerate collision frequency

Li Wang, Bokai Yan

Mathematics

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1 Scopus citations

Abstract

We design an asymptotic preserving (AP) scheme for the linear kinetic equation with anisotropic scattering that leads to a fractional diffusion limit. This limit may be attributed to two reasons: A heavy tail equilibrium or a degenerate collision frequency, both of which are considered in this paper. Our scheme builds on the ideas developed in [L. Wang and B. Yan, J. Comput. Phys., 312 (2016), pp. 157-174] but with two major variations. One is a new splitting of the system that accounts for the anisotropy in the scattering cross section by introducing two extra terms. We then showed, via detailed calculation, that the scheme enjoys a relaxed AP property as opposed to the one step AP for the isotropic scattering. Another contribution is for the degenerate collision frequency case, which brings in additional stiffness. We propose to integrate a "body" term, which appears to be the main component in the diffusion limit. This term is precomputed once with a prescribed accuracy, via a change of variable that alleviates the stiffness. Numerical examples are presented to validate its efficiency in both kinetic and fractional diffusion regimes.

Original language	English (US)
Pages (from-to)	A422-A451
Journal	SIAM Journal on Scientific Computing
Volume	41
Issue number	1
DOIs	https://doi.org/10.1137/17M1138029
State	Published - 2019

Bibliographical note

Funding Information:
\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section July 10, 2017; accepted for publication (in revised form) October 31, 2018; published electronically February 5, 2019. http://www.siam.org/journals/sisc/41-1/M113802.html \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The first author's work was partially supported by NSF grants DMS-1620135 and DMS-1903420. \dagger School of Mathematics, University of Minnesota Twin Cities, 206 Church St SE, Minneapolis, MN 55455 (wang8818@umn.edu). \ddagger Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095 (yanbokai@gmail.com).

Publisher Copyright:
©2019 Society for Industrial and Applied Mathematics.

Keywords

Anisotropic scattering
Asymptotic-preserving scheme
Degenerate collision frequency
Fractional diffusion
Heavy tail equilibrium

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title = "An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering: Heavy tail equilibrium and degenerate collision frequency",

abstract = "We design an asymptotic preserving (AP) scheme for the linear kinetic equation with anisotropic scattering that leads to a fractional diffusion limit. This limit may be attributed to two reasons: A heavy tail equilibrium or a degenerate collision frequency, both of which are considered in this paper. Our scheme builds on the ideas developed in [L. Wang and B. Yan, J. Comput. Phys., 312 (2016), pp. 157-174] but with two major variations. One is a new splitting of the system that accounts for the anisotropy in the scattering cross section by introducing two extra terms. We then showed, via detailed calculation, that the scheme enjoys a relaxed AP property as opposed to the one step AP for the isotropic scattering. Another contribution is for the degenerate collision frequency case, which brings in additional stiffness. We propose to integrate a {"}body{"} term, which appears to be the main component in the diffusion limit. This term is precomputed once with a prescribed accuracy, via a change of variable that alleviates the stiffness. Numerical examples are presented to validate its efficiency in both kinetic and fractional diffusion regimes.",

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author = "Li Wang and Bokai Yan",

note = "Funding Information: \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section July 10, 2017; accepted for publication (in revised form) October 31, 2018; published electronically February 5, 2019. http://www.siam.org/journals/sisc/41-1/M113802.html \bfF \bfu \bfn \bfd \bfi \bfn \bfg : The first author's work was partially supported by NSF grants DMS-1620135 and DMS-1903420. \dagger School of Mathematics, University of Minnesota Twin Cities, 206 Church St SE, Minneapolis, MN 55455 (wang8818@umn.edu). \ddagger Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095 (yanbokai@gmail.com). Publisher Copyright: {\textcopyright}2019 Society for Industrial and Applied Mathematics.",

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N2 - We design an asymptotic preserving (AP) scheme for the linear kinetic equation with anisotropic scattering that leads to a fractional diffusion limit. This limit may be attributed to two reasons: A heavy tail equilibrium or a degenerate collision frequency, both of which are considered in this paper. Our scheme builds on the ideas developed in [L. Wang and B. Yan, J. Comput. Phys., 312 (2016), pp. 157-174] but with two major variations. One is a new splitting of the system that accounts for the anisotropy in the scattering cross section by introducing two extra terms. We then showed, via detailed calculation, that the scheme enjoys a relaxed AP property as opposed to the one step AP for the isotropic scattering. Another contribution is for the degenerate collision frequency case, which brings in additional stiffness. We propose to integrate a "body" term, which appears to be the main component in the diffusion limit. This term is precomputed once with a prescribed accuracy, via a change of variable that alleviates the stiffness. Numerical examples are presented to validate its efficiency in both kinetic and fractional diffusion regimes.

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An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering: Heavy tail equilibrium and degenerate collision frequency

Abstract

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Keywords

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