Skip to main content
    • Aa
    • Aa

Implicit Asymptotic Preserving Method for Linear Transport Equations

  • Qin Li (a1) (a2) and Li Wang (a3)

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.

Corresponding author
*Corresponding author. Email addresses: (Q. Li), (L. Wang)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] S. Ashby , P. Brown , M. Dorr and A. Hindmarsh , A linear algebraic analysis of diffusion synthetic acceleration for the Boltzmann transport equation, SIAM J. Numer. Anal., 32 (1995), 128178.

[2] M. Adams and E. Larsen , Fast iterative methods for discrete-ordinate particle transport calculations, Prog. Nucl. Eng., 40 (2002), 3159.

[3] Y. Azmy , Unconditionally stable and robust adjacent-cell diffusive preconditioning of weighted-difference particle transport methods is impossible, J. Comput. Phys., 182 (2002), 213.

[4] P. Brantley and E. Larsen , The simplified P3 approximation, Nucl. Sci. Eng., 134 (2001), 121.

[5] S. Boscarino , L. Pareschi and G. Russo , Implicit-explicit Runge-Kutta scheme for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35 (2013), 2251.

[6] P. Brown , A linear algebraic development of diffusion synthetic acceleration for three-dimensional transport equations, SIAM J. Numer. Anal., 32 (1995), 179214.

[7] J. A. Fleck and J. D. Cummings , An implicit Monte Carlo scheme for calculating time and frequency dependent nonlinear radiation transpor, J. Comput. Phys., 8 (1971), 313342.

[8] M. Frank , A. Klar , E. Larsen and S. Yasuda , Time-dependent simplified pn approximation to the equations of radiative transfer, J. Comput. Phys., 226 (2007), 22892305.

[11] F. Golse , S. Jin and D. Levermore , The convergence of numerical transfer schemes in diffusive regimes i: The discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 13331369.

[12] C. Hauck , High-order entropy-based closures for linear transport in slab geometries, Commun. Math. Sci., 9 (2011), 187205.

[13] C. Hauck and R. Lowrie , Temporal regularization of the pn equations, Multiscale Model. Simul., 7 (2009), 14971524.

[14] S. Jin and D. Levermore , Fully discrete numerical transfer in diffusive regimes, Transp. Theory Stat. Phys., 22 (1993), 739791.

[15] S. Jin , L. Pareschi and G. Toscani , Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913936.

[17] A. Klar , An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35(6) (1998), 1097–1094.

[18] E. Larsen , Diffusion theory as an asymptotic limit of transport theory for nearly critical systems with small mean free paths, Ann. Nucl. Energy, 7 (1980), 249255.

[20] M. Lemou and L. Mieussens , New asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334368.

[21] E. Larsen and J. Morel , Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes ii, J. Comput. Phys., 83 (1989), 212236.

[22] L. Mieussens , On the asymptotic preserving property o fate unified gas kinetic scheme for the diffusion limit of linear kinetic modle, J. Comput. Phys., 253 (2013), 138156.

[23] J. Morel , T. Wareing , R. Lowrie and D. Parsons , Analysis of ray-effect mitigation techniques, Nucl. Sci. Eng., 144(1) (2003), 122.

[24] E. Olbrant , C. Hauck and M. Frank , A realizability-preserving discontinuous Galerkin method for the m1 model of radiative transfer, J. Comput. Phys., 231 (2012), 56125639.

[25] E. Olbrant , E. Larsen , M. Frank and B. Seibold , Asymptotic derivation and numerical investigation of time-dependent simplified PN equations, J. Comput. Phys., 238(1) (2012), 315336.

[26] G. Pomraning , Asymptotic and variational derivations of the simplified PN equations, Ann. Nucl. Energy, 20 (1993), 623637.

[27] W. Sun , S. Jiang and K. Xu , An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations, J. Comput. Phys., 285(5) (2015), 265279.

[29] T. A. Wareing , J. McGhee , J. Morel and S. Pautz , Discontinuous finite element Sn methods on three-dimensional unstructured grids, Nucl. Sci. Eng., 138(3) (2001), 256268.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 10 *
Loading metrics...

Abstract views

Total abstract views: 24 *
Loading metrics...

* Views captured on Cambridge Core between 3rd May 2017 - 22nd May 2017. This data will be updated every 24 hours.