Skip to main content Accessibility help

An Implicit Unified Gas Kinetic Scheme for Radiative Transfer with Equilibrium and Non-Equilibrium Diffusive Limits

  • Wenjun Sun (a1), Song Jiang (a1) and Kun Xu (a2)


This paper is about the construction of a unified gas-kinetic scheme (UGKS) for a coupled system of radiative transport and material heat conduction with different diffusive limits. Different from the previous approach, instead of including absorption/emission only, the current method takes both scattering and absorption/emission mechanism into account in the radiative transport process. As a result, two asymptotic limiting solutions will appear in the diffusive regime. In the strong absorption/emission case, an equilibrium diffusion limit is obtained, where the system is mainly driven by a nonlinear diffusion equation for the equilibrium radiation and material temperature. However, in the strong scattering case, a non-equilibrium limit can be obtained, where coupled nonlinear diffusion system with different radiation and material temperature is obtained. In addition to including the scattering term in the transport equation, an implicit UGKS (IUGKS) will be developed in this paper as well. In the IUGKS, the numerical flux for the radiation intensity is constructed implicitly. Therefore, the conventional CFL constraint for the time step is released. With the use of a large time step for the radiative transport, it becomes possible to couple the IUGKS with the gas dynamic equations to develop an efficient numerical method for radiative hydrodynamics. The IUGKS is a valid method for all radiative transfer regimes. A few numerical examples will be presented to validate the current implicit method for both optical thin to optical thick cases.


Corresponding author

*Corresponding author. Email addresses: (W. Sun), (S. Jiang), (K. Xu)


Hide All
[1] Chen, S.Z., Xu, K., Lee, C.B., and Cai, Q.D., A unified gas kinetic scheme with moving mesh and velocity space adaptation, J. Comput. Phys. 231 (2012), pp. 66436664.
[2] Gentile, N.A., Implicit Monte Carlo diffusion - an accerlation method for Monte Carlo time-dependent radiative transfer simulations. J. Comput. Phys., 172 (2001), 543571.
[3] Huang, J.C., Xu, K., and Yu, P.B., A unified gas-kinetic scheme for continuum and rarefied flows II: multi-dimensional cases, Communications in Computational Physics, 12, no. 3 (2012), pp. 662690.
[4] Jin, S. and Levermore, C.D., The discrete-ordinate method in diffusive regimes, Transport Theory Statist. Phys., 20(5-6), (1991), 413439.
[5] Jin, S., Pareschi, L. and Toscani, G., Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38(3) (2000), 913936.
[6] Klar, A., An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal., 35(6) (1998), 10731094.
[7] Larsen, A.W. and Morel, J.E., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II, J. Comp. Phys., 83(1) (1989), 212236.
[8] Larsen, A.W., Morel, J.E. and Miller, W.F. Jr., Asymptotic solutions of numerical transport problems in optically thick, diffusiive regimes. J. Comput. Phys., 69(2) (1987), 283324.
[9] Lee, C.E., The Discrete SN Approximation to Transport Theory, LA-2595, 1962.
[10] 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 (2013), 138156.
[11] van Leer, B., Towards the ultimate conservative difference schemes V. A second-order sequal to Godunov's method. J. Comput. Phys. 32 (1979), 101136.
[12] Xu, K. and Huang, J.C., A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys., 229 (2010), 77477764.
[13] Spitzer, L. and Harm, R., Transport phenomena in a completely ionized gas. Phys. Rev., 89 (1953), 977.
[14] Sun, W.J., Jiang, S., and Xu, K., An Asymptotic Preserving unified gas kinetic scheme for gray radiative transfer equations. J. Comput. Phys. 285 (2015), pp. 265279.
[15] Sun, W.J., Jiang, S., Xu, K. and Li, S., An Asymptotic Preserving unified gas kinetic scheme for frequency-dependent radiative transfer equations. J. Comput. Phys. 302 (2015), pp. 222238.
[16] Mousseau, V.A. and Knoll, D.A., Temporal accuracy of the nonequilibrium radiation diffusion equations applied to two-dimensional multimaterial simulations. Nuclear Science and Engineering, 154 (2006), pp. 174189.
[17] Chang, B., The incorporation of the semi-implicit linear equations into Newton's method to solve radiation transfer equations. J. Comput. Phys., 226 (2007), pp. 852878.
[18] Larsen, E.W., A grey transport acceleration method for time-dependent radiative transfer problems. J. Comput. Phys. 78 (1988), pp. 459480.
[19] Godillon-Lafitte, P. And Goudon, T., A Coupled Model For Radiative Transfer: Doppler Effects, Equilibrium, And Nonequilibrium Diffusion Asymptotics. Multiscale Model. Simul., 4(4) (2005), pp. 1245C1279.
[20] Bueta, C. and Despresa, B., Asymptotic analysis of (uid models for the coupling of radiation and hydrodynamics. Journal of Quantitative Spectroscopy and Radiative Transfer 85 (2004), PP. 385C418.



Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed