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Implicit Asymptotic Preserving Method for Linear Transport Equations

Part of: Numerical linear algebra Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  03 May 2017

Qin Li  and
Li Wang
Qin Li*
Affiliation:
Mathematics Department, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53705, USA The Optimization Group, The Wisconsin Institute of Discovery, Madison, WI 53715, USA
Li Wang*
Affiliation:
Departments of Mathematics and Computational Data-Enabled Science and Engineering Program, State University of New York at Buffalo, 244 Mathematics Building, Buffalo, NY 14260, USA
*
*Corresponding author. Email addresses:qinli@math.wisc.edu (Q. Li), lwang46@buffalo.edu (L. Wang)
*Corresponding author. Email addresses:qinli@math.wisc.edu (Q. Li), lwang46@buffalo.edu (L. Wang)

Abstract

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The computation of the radiative transfer equation is expensive mainly due to two stiff terms: the transport term and the collision operator. The stiffness in the former comes from the fact that particles (such as photons) travel at the speed of light, while that in the latter is due to the strong scattering in the optically thick region. We study the fully implicit scheme for this equation to account for the stiffness. The main challenge in the implicit treatment is the coupling between the spacial and angular coordinates that requires the large size of the to-be-inverted matrix, which is also ill-conditioned and not necessarily symmetric. Our main idea is to utilize the spectral structure of the ill-conditioned matrix to construct a pre-conditioner, which, along with an exquisite split of the spatial and angular dependence, significantly improve the condition number and allows a matrix-free treatment. We also design a fast solver to compute this pre-conditioner explicitly in advance. Our method is shown to be efficient in both diffusive and free streaming limit, and the computational cost is comparable to the state-of-the-art method. Various examples including anisotropic scattering and two-dimensional problems are provided to validate the effectiveness of our method.

Keywords

Implicit methodasymptotic preservingpre-conditionerdiffusive regimefree streaming limit

MSC classification

Secondary: 65M06: Finite difference methods 65M30: Improperly posed problems 65F08: Preconditioners for iterative methods 65F35: Matrix norms, conditioning, scaling

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 157 - 181
Copyright
Copyright © Global-Science Press 2017 

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