Analysis of a fractional-step scheme for the P1 radiative diffusion model

Raphaele Herbin (1) , Thierry Gallouët (1) , Jean-Claude Latché (2) , Aurélien Larcher (3, 2)

1 I2M - Institut de Mathématiques de Marseille
2 IRSN - Institut de Radioprotection et de Sûreté Nucléaire
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We address in this paper a nonlinear parabolic system, which is built to retain the main mathematical difficulties of the P1 radiative diffusion physical model. We propose a finite volume fractional-step scheme for this problem enjoying the following properties. First, we show that each discrete solution satisfies a priori L -estimates, through a discrete maxi- mum principle; by a topological degree argument, this yields the existence of a solution, which is proven to be unique. Second, we establish uniform (with respect to the size of the meshes and the time step) L2 -bounds for the space and time translates; this proves, by the Kolmogorov theorem, the relative compactness of any sequence of solutions obtained through a sequence of discretizations the time and space steps of which tend to zero; the limits of converging subsequences are then shown to be a solution to the continuous problem. Estimates of time translates of the discrete solutions are obtained through the formalization of a generic argument, interesting for its own sake.

Mots clés: P1 radiative transfer model; Finite volumes; Parabolic systems; Convergence analysis

Domaines: Analyse numérique [math.NA]

Raphaele Herbin, Thierry Gallouët, Jean-Claude Latché, Aurélien Larcher. Analysis of a fractional-step scheme for the P1 radiative diffusion model. Computational & Applied Mathematics, 2016, 35 (1), pp.135-151. ⟨10.1007/s40314-014-0186-z⟩. ⟨hal-00477086v2⟩