Properties and frequential hybridisation of the multigroupM1 model for radiative transfer

Radiative transfer is a phenomenon that has importance in a wide range of applications from climatology to astrophysics. Depending on the physical regimes involved, a hierarchy of models may be used, each of which having drawbacks and qualities. However, there are still applications for which no model is fully satisfying. An example is ICF (Inertial Confinement Fusion) where dozens of lasers converge on a fuel pellet of the size of a pinehead. This kind of simulation requires a coupling between radiation and other processes and hence one would require a model that is cheap enough in terms of computation cost to carry it out. The M1 model (Dubroca and Klar (2002) [14]) may then seem to be an interesting choice. But the directional complexity of the problem and the fact that the energy is mainly located inside a narrow frequency interval is hardly compatible with this model and one would rather use an (expensive) model such as a kinetic model. In this paper, we prove several important properties of the multigroupM1 model that extends the results of the grey M1 model. Then we introduce a hybrid model that mixes it with a kinetic model. This hybrid model intends to be an extension of both of them and therefore adds degrees of freedom that allows us to properly take into account a wide variety of problems. We also show that it possesses important properties including the correct asymptotic limit in the diffusion regime and the local decreasing of the total entropy. © 2009 Elsevier Ltd. All rights reserved. 0. Introduction The radiative transfer equation (RTE) describes the evolution of the radiative intensity Iν(Ω) = I(t, x,Ω, ν) where Ω is the photons’ direction of propagation and ν is the frequency. This radiative intensity Iν(Ω) is linked to the photons’ distribution function. Assuming local thermal equilibrium and neglecting scattering, the RTE writes: 1 c ∂t Iν(Ω)+Ω.∇xIν(Ω) = σν (Bν(T )− Iν(Ω)) , (0.1) where c is the speed of the light, σν is the emission opacity, T is the material temperature and Bν(T ) is Planck’s blackbody function given by: Bν(T ) = 2hν3 c2 [ exp ( hν kT ) − 1 ]−1 . The constants h and k are respectively Planck and Boltzmann constants. It is to be noted that the energy of B (i.e. its integral over all directions and frequencies) is proportional to T 4. The constant of proportionality is denoted by a ('7.56.10−16 SI). E-mail address: [email protected]. 1468-1218/$ – see front matter© 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.08.008 R. Turpault / Nonlinear Analysis: Real World Applications 11 (2010) 2514–2528 2515 Due to emission and absorption process, the radiative energy is not conserved. To preserve the conservation of the total (radiative + material) energy, the following simplified temperature evolution equation will be used in this paper (for full coupling with hydrodynamics see for example [1–3]):