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 | |
| State | Published - Jan 1 2019 |
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Keywords
- Anisotropic scattering
- Asymptotic-preserving scheme
- Degenerate collision frequency
- Fractional diffusion
- Heavy tail equilibrium
Cite this
An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering : Heavy tail equilibrium and degenerate collision frequency. / Wang, Li; Yan, Bokai.
In: SIAM Journal on Scientific Computing, Vol. 41, No. 1, 01.01.2019, p. A422-A451.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - An asymptotic-preserving scheme for the kinetic equation with anisotropic scattering
T2 - Heavy tail equilibrium and degenerate collision frequency
AU - Wang, Li
AU - Yan, Bokai
PY - 2019/1/1
Y1 - 2019/1/1
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.
AB - 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.
KW - Anisotropic scattering
KW - Asymptotic-preserving scheme
KW - Degenerate collision frequency
KW - Fractional diffusion
KW - Heavy tail equilibrium
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U2 - 10.1137/17M1138029
DO - 10.1137/17M1138029
M3 - Article
VL - 41
SP - A422-A451
JO - SIAM Journal of Scientific Computing
JF - SIAM Journal of Scientific Computing
SN - 1064-8275
IS - 1
ER -