Multiscale Radiative Transfer in Cylindrical Coordinates

doi:10.1007/s42967-019-0007-x

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (1): 117-139.doi: 10.1007/s42967-019-0007-x

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Multiscale Radiative Transfer in Cylindrical Coordinates

Wenjun Sun¹, Song Jiang¹, Kun Xu²

1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China;
2. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

收稿日期:2018-06-19 修回日期:2018-09-24 出版日期:2019-03-20 发布日期:2019-05-11
通讯作者: Song Jiang, Wenjun Sun, Kun Xu E-mail:jiang@iapcm.ac.cn;sun_wenjun@iapcm.ac.cn;makxu@ust.hk
基金资助:
The authors wish to thank all referees for their useful suggestions to improve the current paper;The research of Sun is supported by NSFC (Grant nos.11671048,91630310) and CAEP Project (2015B0202041,2015B0202040);Jiang is supported by the National Basic Research Program under Grant 2014CB745002 and NSFC (Grant no.11631008);and Xu is supported by Hong Kong research Grant council (16206617,16207715) and NSFC (Grant nos.11772281,91530319).

Multiscale Radiative Transfer in Cylindrical Coordinates

Wenjun Sun¹, Song Jiang¹, Kun Xu²

1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China;
2. Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

Received:2018-06-19 Revised:2018-09-24 Online:2019-03-20 Published:2019-05-11
Contact: Song Jiang, Wenjun Sun, Kun Xu E-mail:jiang@iapcm.ac.cn;sun_wenjun@iapcm.ac.cn;makxu@ust.hk
Supported by:
The authors wish to thank all referees for their useful suggestions to improve the current paper;The research of Sun is supported by NSFC (Grant nos.11671048,91630310) and CAEP Project (2015B0202041,2015B0202040);Jiang is supported by the National Basic Research Program under Grant 2014CB745002 and NSFC (Grant no.11631008);and Xu is supported by Hong Kong research Grant council (16206617,16207715) and NSFC (Grant nos.11772281,91530319).

摘要/Abstract

摘要： The radiative transfer equations in cylindrical coordinates are important in the application of inertial confinement fusion. In comparison with the equations in Cartesian coordinates, an additional angular derivative term appears in the cylindrical case. This term adds great difficulty for a numerical scheme to keep the conservation of total energy. In this paper, based on weighting factors, the angular derivative term is properly discretized, and the interface fluxes in the radial r-direction depend on such a discretization as well. A unified gas kinetic scheme (UGKS) with asymptotic preserving property for the gray radiative transfer equations is constructed in cylindrical coordinates. The current UGKS can naturally capture the radiation diffusion solution in the optically thick regime with the cell size being much larger than photon's mean free path. At the same time, the current UGKS can present accurate solutions in the optically thin regime as well. Moreover, it is a finite volume method with total energy conservation. Due to the scale-dependent time evolution solution for the interface flux evaluation, the scheme can cover multiscale transport mechanism seamlessly. The cylindrical hohlraum tests in inertial confinement fusion are used to validate the current approach, and the solutions are compared with implicit Monte Carlo result.

关键词: Cylindrical coordinate system, Gray radiative equations, Multiscale transport, Unified gas kinetic scheme

Abstract: The radiative transfer equations in cylindrical coordinates are important in the application of inertial confinement fusion. In comparison with the equations in Cartesian coordinates, an additional angular derivative term appears in the cylindrical case. This term adds great difficulty for a numerical scheme to keep the conservation of total energy. In this paper, based on weighting factors, the angular derivative term is properly discretized, and the interface fluxes in the radial r-direction depend on such a discretization as well. A unified gas kinetic scheme (UGKS) with asymptotic preserving property for the gray radiative transfer equations is constructed in cylindrical coordinates. The current UGKS can naturally capture the radiation diffusion solution in the optically thick regime with the cell size being much larger than photon's mean free path. At the same time, the current UGKS can present accurate solutions in the optically thin regime as well. Moreover, it is a finite volume method with total energy conservation. Due to the scale-dependent time evolution solution for the interface flux evaluation, the scheme can cover multiscale transport mechanism seamlessly. The cylindrical hohlraum tests in inertial confinement fusion are used to validate the current approach, and the solutions are compared with implicit Monte Carlo result.

Key words: Cylindrical coordinate system, Gray radiative equations, Multiscale transport, Unified gas kinetic scheme

中图分类号:

Wenjun Sun, Song Jiang, Kun Xu. Multiscale Radiative Transfer in Cylindrical Coordinates[J]. Communications on Applied Mathematics and Computation, 2019, 1(1): 117-139.

参考文献

1. Brunner, T.A.:Forms of Approximate Radiation Transport. Technical Report SAND2002-1778, Sandia National Laboratories (2002)
2. Chen, S.Z., Xu, K., Lee, C.B., Cai, Q.D.:A unified gas kinetic scheme with moving mesh and velocity space adaptation. J. Comput. Phys. 231, 6643-6664(2012)
3. Fleck JR, J.A., Cummings, J.D.:An implicit Monte Carlo scheme for calculating time and frequency dependent nonlinear radiation transport. J. Comput. Phys. 8, 313-342(1971)
4. Gentile, N.A.:Implicit Monte Carlo diffusion-an accerlation method for Monte Carlo time-dependent radiative transfer simulations. J. Comput. Phys. 172, 543-571(2001)
5. Huang, J.C., Xu, K., Yu, P.B.:A unified gas-kinetic scheme for continuum and rarefied flows Ⅱ:multidimensional cases. Commun. Comput. Phys. 12, 662-690(2012)
6. Jin, S., Levermore, C.D.:The discrete-ordinate method in diffusive regimes. Transp. Theory Stat. Phys. 20, 413-439(1991)
7. Jin, S., Levermore, C.D.:Fully discrete numerical transfer in diffusive regimes. Transp. Theory Stat. Phys. 22, 739-791(1993)
8. Jin, S., Pareschi, L., Toscani, G.:Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. 38, 913-936(2000)
9. Klar, A.:An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35, 1073-1094(1998)
10. Larsen, A.W., Morel, J.E.:Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. Ⅱ. J. Comput. Phys. 83, 212-236(1989)
11. Larsen, E.W., Pomraning, G.C., Badham, V.C.:Asymptotic analysis of radiative transfer problems. J. Quant. Spectrosc. Radiat. Transf. 29, 285-310(1983)
12. Larsen, A.W., Morel, J.E., Miller Jr., W.F.:Asymptotic solutions of numerical transport problems in optically thick, diffusiive regimes. J. Comput. Phys. 69, 283-324(1987)
13. Lee, C.E.:The Discrete S_N Approximation to Transport Theory, LA-2595(1962)
14. Li, S., Li, G., Tian, D.F., Deng, L.:An implicit Monte Carlo method for thermal radiation transport. Acta Phys. Sin. 62, 249501(2013)
15. Liu, C., Xu, K.:A unified gas kinetic scheme for continuum and rarefied flows V:multiscale and multicomponent plasma transport. Commun. Comput. Phys. 22, 1175-1223(2017)
16. McClarren, R.G., Hauckb, C.D.:Simulating radiative transfer with filtered spherical harmonics. Phys. Lett. A. 374, 2290-2296(2010)
17. Mieussens, L.:On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic model. J. Comput. Phys. 253, 138-156(2013)
18. Morel, J.E., Montry, G.R.:Analysis and elimination of the discrete ordinates flux dip. Transp. Theory Stat. Phys. 13, 615-633(1984)
19. Sun, W.J., Jiang, S., Xu, K.:An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations. J. Comput. Phys. 285, 265-279(2015)
20. Sun, W.J., Jiang, S., Xu, K., Li, S.:An asymptotic preserving unified gas kinetic scheme for frequencydependent radiative transfer equations. J. Comput. Phys. 302, 222-238(2015)
21. Sun, W.J., Zeng, Q.H., Li, S.G.:The asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Ann. Differ. Eqs. 2, 141-165(2016)
22. Sun, W.J., Jiang, S., Xu, K.:An implicit unified gas kinetic scheme for radiative transfer with equilibrium and non-equilibrium diffusive limits. Commun. Comput. Phys. 22, 899-912(2017)
23. Sun, W.J., Jiang, S., Xu, K.:A multidimensional unified gas-kinetic scheme for radiative transfer equations on unstructured mesh. J. Comput. Phys. 351, 455-472(2017)
24. van Leer, B.:Towards the ultimate conservative difference schemes V. A second-order sequal to Godunov's method. J. Comput. Phys. 32, 101-136(1979)
25. Xu, K., Huang, J.C.:A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys. 229, 7747-7764(2010)

Multiscale Radiative Transfer in Cylindrical Coordinates

Multiscale Radiative Transfer in Cylindrical Coordinates

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