Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit

Pauline Lafitte (1, 2, 3) , Giovanni Samaey (4)

1 SIMPAF - SImulations and Modeling for PArticles and Fluids
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 MAS - Mathématiques Appliquées aux Systèmes - EA 4037
4 UCL - Université Catholique de Louvain = Catholic University of Louvain

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test.

Mots clés: kinetic equation; diffusion; projective integration

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Pauline Lafitte, Giovanni Samaey. Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit. SIAM Journal on Scientific Computing, 2012, 34 (2), pp.A579-A602. ⟨10.1137/100795954⟩. ⟨hal-00485189⟩