Asymptotic-preserving & well-balanced schemes for radiative transfer and the Rosseland approximation

Summary.We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow the Well-Balanced approach’s canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable continuous. These steady-state equations are of Fredholm type. The resulting upwind schemes are proved to be stable under a reasonable parabolic CFL condition of the type Δt≤O(Δx2) among other desirable properties. Some numerical results demonstrate the realizability and the efficiency of this process.

[1]  Lorenzo Pareschi,et al.  Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations , 1998 .

[2]  Geoffrey Ingram Taylor,et al.  Diffusion by Continuous Movements , 1922 .

[3]  A. V. Cardona,et al.  A semi-analytical numerical method for time-dependent radiative transfer problems in slab geometry with coherent isotropic scattering , 2002 .

[4]  Shi Jin,et al.  Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations , 2000, SIAM J. Numer. Anal..

[5]  Guillaume Bal,et al.  Radiative transport limit for the random , 2001 .

[6]  James Paul Holloway,et al.  One-dimensional Riemann solvers and the maximum entropy closure , 2001 .

[7]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[8]  Shaoqiang Tang,et al.  Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems , 2004, Math. Comput..

[9]  Blake Temple,et al.  A Bound on the Total Variation of the Conserved Quantities for Solutions of a General Resonant Nonlinear Balance Law , 2004, SIAM J. Appl. Math..

[10]  Axel Klar,et al.  Half-moment closure for radiative transfer equations , 2002 .

[11]  Alexis Vasseur.  Well-posedness of scalar conservation laws with singular sources , 2002 .

[12]  S. Goldstein.  ON DIFFUSION BY DISCONTINUOUS MOVEMENTS, AND ON THE TELEGRAPH EQUATION , 1951 .

[13]  G. Guerra,et al.  Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws , 2002 .

[14]  B. Perthame,et al.  The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation , 1988 .

[15]  K. Stamnes,et al.  Radiative Transfer in the Atmosphere and Ocean , 1999 .

[16]  T. Carleman,et al.  Problèmes mathématiques dans la théorie cinétique des gaz , 1957 .

[17]  Laurent Gosse,et al.  Localization effects and measure source terms in numerical schemes for balance laws , 2002, Math. Comput..

[18]  Laurent Gosse,et al.  Space Localization and Well-Balanced Schemes for Discrete Kinetic Models in Diffusive Regimes , 2003, SIAM J. Numer. Anal..

[19]  Laurent Gosse,et al.  An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations , 2002 .

[20]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[21]  Guillaume Bal,et al.  Radiative transport limit for the random Schrödinger equation , 2001 .

[22]  C. Schmeiser,et al.  NUMERICAL PASSAGE FROM RADIATIVE HEAT TRANSFER TO NONLINEAR DIFFUSION MODELS , 2001 .

[23]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[24]  Laurent Gosse.  Time-Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System , 2003 .

[25]  Pierre-Louis Lions,et al.  Diffusive limit for finite velocity Boltzmann kinetic models , 1997 .

[26]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[27]  G. F. Dell'Antonio.  Large time, small coupling behaviour of a quantum particle in a random field , 1983 .

[28]  S. Mancini,et al.  DIFFUSION LIMIT OF THE LORENTZ MODEL: ASYMPTOTIC PRESERVING SCHEMES , 2002 .

[29]  E. Larsen,et al.  Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II , 1989 .

[30]  Laurent Gosse,et al.  Godunov-type approximation for a general resonant balance law with large data , 2004 .

[31]  Ramaz Botchorishvili,et al.  Equilibrium schemes for scalar conservation laws with stiff sources , 2003, Math. Comput..

[32]  François Golse,et al.  The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method , 1999 .

[33]  A. Tzavaras,et al.  Representation of weak limits and definition of nonconservative products , 1999 .

[34]  Giovanni Russo,et al.  Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation , 1997 .

[35]  Charles S. Zender,et al.  Radiative Transfer in the Earth System , 2001 .

[36]  A. Klar.  An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit , 1998 .

[37]  G. C. Pomraning,et al.  Linear Transport Theory , 1967 .

[38]  Herbert Spohn,et al.  Derivation of the transport equation for electrons moving through random impurities , 1977 .

[39]  G. C. Pomraning.  Radiative transfer and transport phenomena in stochastic media , 1998 .

[40]  Albert Fannjiang,et al.  High Frequency Behavior of the Focusing Nonlinear Schr[o-umlaut]dinger Equation with Random Inhomogeneities , 2003, SIAM J. Appl. Math..