Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering

Sheng, Qiwei; Hauck, Cory

doi:10.1090/mcom/3670

Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering
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by Qiwei Sheng and Cory D. Hauck;

Math. Comp. 90 (2021), 2645-2669

DOI: https://doi.org/10.1090/mcom/3670

Published electronically: July 19, 2021

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Abstract:

We present an error analysis for the discontinuous Galerkin (DG) method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation with isotropic scattering. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $\varepsilon$ which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.

References

M. L. Adams, Discontinuous finite element transport solutions in thick diffusive problems, Nuclear Sci. Eng. 137 (2001), no. 3, 298–333.
Valeri Agoshkov, Boundary value problems for transport equations, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1638817, DOI 10.1007/978-1-4612-1994-1
Kendall Atkinson and Weimin Han, Theoretical numerical analysis, 3rd ed., Texts in Applied Mathematics, vol. 39, Springer, Dordrecht, 2009. A functional analysis framework. MR 2511061, DOI 10.1007/978-1-4419-0458-4
Alain Bensoussan, Jacques-L. Lions, and George C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci. 15 (1979), no. 1, 53–157. MR 533346, DOI 10.2977/prims/1195188427
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Kenneth M. Case and Paul F. Zweifel, Linear transport theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 225547
Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
J. A. Coakley Jr. and P. Yang, Atmospheric radiation: a primer with illustrative solutions, Wiley-VCH Verlag, Germany, 2014.
Bernardo Cockburn, Bo Dong, and Johnny Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp. 77 (2008), no. 264, 1887–1916. MR 2429868, DOI 10.1090/S0025-5718-08-02123-6
Michael M. Crockatt, Andrew J. Christlieb, C. Kristopher Garrett, and Cory D. Hauck, An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction, J. Comput. Phys. 346 (2017), 212–241. MR 3670635, DOI 10.1016/j.jcp.2017.06.017
James J. Duderstadt and William R. Martin, Transport theory, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 551868
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
François Golse, Shi Jin, and C. David Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal. 36 (1999), no. 5, 1333–1369. MR 1706766, DOI 10.1137/S0036142997315986
Jean-Luc Guermond and Guido Kanschat, Asymptotic analysis of upwind discontinuous Galerkin approximation of the radiative transport equation in the diffusive limit, SIAM J. Numer. Anal. 48 (2010), no. 1, 53–78. MR 2608358, DOI 10.1137/090746938
G. J. Habetler and B. J. Matkowsky, Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation, J. Mathematical Phys. 16 (1975), 846–854. MR 381601, DOI 10.1063/1.522618
Weimin Han, Jianguo Huang, and Joseph A. Eichholz, Discrete-ordinate discontinuous Galerkin methods for solving the radiative transfer equation, SIAM J. Sci. Comput. 32 (2010), no. 2, 477–497. MR 2609327, DOI 10.1137/090767340
Shi Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.) 3 (2012), no. 2, 177–216. MR 2964096
A. D. Kim and J. B. Keller, Light propagation in biological tissue, J. Opt. Soc. Amer. A 20 (2003), no. 1, 92–98.
Edward W. Larsen and Joseph B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys. 15 (1974), 75–81. MR 339741, DOI 10.1063/1.1666510
Edward W. Larsen and J. E. Morel, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II, J. Comput. Phys. 83 (1989), no. 1, 212–236. MR 1010164, DOI 10.1016/0021-9991(89)90229-5
Edward W. Larsen, J. E. Morel, and Warren F. Miller Jr., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes, J. Comput. Phys. 69 (1987), no. 2, 283–324. MR 888058, DOI 10.1016/0021-9991(87)90170-7
E. W. Larsen and J. E. Morel, Advances in discrete-ordinates methodology, Springer Netherlands, Dordrecht, 2010, pp. 1–84.
E. W. Larsen and G. C. Pomraning, The $P_{N}$ theory as an asymptotic limit of transport theory in planar geometry –I: Analysis, Nuclear Sci. Eng. 109 (1991), no. 1, 49–75.
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Academic Press, New York-London, 1974, pp. 89–123. MR 658142
C. L. Leakeas and E. W. Larsen, Generalized Fokker-Planck approximations of particle transport with highly forward-peaked scattering, Nuclear Sci. Eng. 137 (2001), no. 3, 236–250.
E. E. Lewis and W. F. Miller, Computational methods of neutron transport, John Wiley & Sons, New York, 1984.
Jian-Guo Liu and Luc Mieussens, Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit, SIAM J. Numer. Anal. 48 (2010), no. 4, 1474–1491. MR 2684343, DOI 10.1137/090772770
R. B. Lowrie and J. E. Morel, Methods for hyperbolic systems with stiff relaxation, Internat. J. Numer. Methods Fluids 40 (2002), no. 3-4, 413–423. ICFD Conference on Numerical Methods for Fluid Dynamics, Part II (Oxford, 2001). MR 1932991, DOI 10.1002/fld.321
F. Malvagi and G. C. Pomraning, Initial and boundary conditions for diffusive linear transport problems, J. Math. Phys. 32 (1991), no. 3, 805–820. MR 1093826, DOI 10.1063/1.529374
Luc Mieussens, On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models, J. Comput. Phys. 253 (2013), 138–156. MR 3193391, DOI 10.1016/j.jcp.2013.07.002
M. F. Modest, Radiative heat transfer, 3rd ed., Academic Press, 2013.
J. E. Morel, Fokker-Planck calculations using standard discrete ordinates transport codes, Nuclear Sci. Eng. 79 (1981), no. 4, 340–356.
Annamaneni Peraiah, An introduction to radiative transfer, Cambridge University Press, Cambridge, 2002. Methods and applications in astrophysics. MR 1877917
G. C. Pomraning, The Fokker-Planck operator as an asymptotic limit, Math. Models Methods Appl. Sci. 2 (1992), no. 1, 21–36. MR 1159474, DOI 10.1142/S021820259200003X
W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Tech. report, Los Alamos Scientific Lab., N. Mex. (USA), 1973.
G. E. Thomas and K. Stamnes, Radiative transfer in the atmosphere and ocean, Cambridge University Press, 1999.
W. Zdunkowski, T. Trautmann, and A. Bott, Radiation in the atmosphere: A course in theoretical meteorology, Cambridge University Press, 2007.