Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T05:24:51.436Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected] (Q. Li), [email protected] (L. Wang)
*Corresponding author. Email addresses:[email protected] (Q. Li), [email protected] (L. Wang)
Get access

Abstract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ashby, S., Brown, P., Dorr, M. and Hindmarsh, A., A linear algebraic analysis of diffusion synthetic acceleration for the Boltzmann transport equation, SIAM J. Numer. Anal., 32 (1995), 128–178.Google Scholar
[2] Adams, M. and Larsen, E., Fast iterative methods for discrete-ordinate particle transport calculations, Prog. Nucl. Eng., 40 (2002), 3–159.Google Scholar
[3] Azmy, Y., Unconditionally stable and robust adjacent-cell diffusive preconditioning of weighted-difference particle transport methods is impossible, J. Comput. Phys., 182 (2002), 213.CrossRefGoogle Scholar
[4] Brantley, P. and Larsen, E., The simplified P3 approximation, Nucl. Sci. Eng., 134 (2001), 1–21.CrossRefGoogle Scholar
[5] Boscarino, S., Pareschi, L. and Russo, G., Implicit-explicit Runge-Kutta scheme for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35 (2013), 22–51.Google Scholar
[6] Brown, P., A linear algebraic development of diffusion synthetic acceleration for three-dimensional transport equations, SIAM J. Numer. Anal., 32 (1995), 179–214.Google Scholar
[7] Fleck, J. A. and Cummings, J. D., An implicit Monte Carlo scheme for calculating time and frequency dependent nonlinear radiation transpor, J. Comput. Phys., 8 (1971), 313–342.Google Scholar
[8] Frank, M., Klar, A., Larsen, E. and Yasuda, S., Time-dependent simplified pn approximation to the equations of radiative transfer, J. Comput. Phys., 226 (2007), 2289–2305.Google Scholar
[9] Frank, M. and Seibold, B., Optimal prediction for radiative transfer: a new perspective on moment closure, Kinet. Relat. Models, 31 (2011), 717–733.Google Scholar
[10] Guthrie, B., Holloway, J. and Patton, B., GMRES as a multi-step transport sweep accelerator, Trans. Theory Stat. Phys., 289(1) (1999), 83–102.Google Scholar
[11] Golse, F., Jin, S. and Levermore, D., The convergence of numerical transfer schemes in diffusive regimes i: The discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333–1369.Google Scholar
[12] Hauck, C., High-order entropy-based closures for linear transport in slab geometries, Commun. Math. Sci., 9 (2011), 187–205.Google Scholar
[13] Hauck, C. and Lowrie, R., Temporal regularization of the pn equations, Multiscale Model. Simul., 7 (2009), 1497–1524.Google Scholar
[14] Jin, S. and Levermore, D., Fully discrete numerical transfer in diffusive regimes, Transp. Theory Stat. Phys., 22 (1993), 739–791.Google Scholar
[15] Jin, S., Pareschi, L. and Toscani, G., Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913–936.Google Scholar
[16] Kupper, K., Frank, M. and Jin, S., An asymptotic preserving 2D staggered grid method for multiscale transport equations, submitted, 2015.Google Scholar
[17] Klar, A., An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35(6) (1998), 1097–1094.CrossRefGoogle Scholar
[18] Larsen, E., Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean free paths, Ann. Nucl. Energy, 7 (1980), 249–255.Google Scholar
[19] Lewis, E. Jr., and Miller, W., Computational Methods of Neutron Transport John Wiley and Sons, 1983.Google Scholar
[20] Lemou, M. and Mieussens, L., New asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334–368.Google Scholar
[21] Larsen, E. and Morel, J., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes ii, J. Comput. Phys., 83 (1989), 212–236.Google Scholar
[22] Mieussens, L., On the asymptotic preserving property o fate unified gas kinetic scheme for the diffusion limit of linear kinetic modle, J. Comput. Phys., 253 (2013), 138–156.Google Scholar
[23] Morel, J., Wareing, T., Lowrie, R. and Parsons, D., Analysis of ray-effect mitigation techniques, Nucl. Sci. Eng., 144(1) (2003), 1–22.Google Scholar
[24] Olbrant, E., Hauck, C. and Frank, M., A realizability-preserving discontinuous Galerkin method for the m1 model of radiative transfer, J. Comput. Phys., 231 (2012), 5612–5639.Google Scholar
[25] Olbrant, E., Larsen, E., Frank, M. and Seibold, B., Asymptotic derivation and numerical investigation of time-dependent simplified PN equations, J. Comput. Phys., 238(1) (2012), 315–336.CrossRefGoogle Scholar
[26] Pomraning, G., Asymptotic and variational derivations of the simplified PN equations, Ann. Nucl. Energy, 20 (1993), 623–637.Google Scholar
[27] Sun, W., Jiang, S. and Xu, K., An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations, J. Comput. Phys., 285(5) (2015), 265–279.CrossRefGoogle Scholar
[28] Tomasevic, D. and Larsen, E., The simplified P2 approximation, Nucl. Sci. Eng., 122 (1996), 309–325.Google Scholar
[29] Wareing, T. A., McGhee, J., Morel, J. and Pautz, S., Discontinuous finite element Sn methods on three-dimensional unstructured grids, Nucl. Sci. Eng., 138(3) (2001), 256–268.Google Scholar
[30] Warsa, J., Wareing, T. and Morel, J., Krylov iterative methods and the degraded effectiveness of diffusion synthetic acceleration for multidimensional Sn calculations in problems with material discontinuities, Nuclear Math. Comput. Sci., 147 (2004), 218–248.Google Scholar