Communications in Mathematical Sciences

Volume 10 (2012)

Number 1

Special Issue on the Occasion of C. David Levermore’s Sixtieth Birthday

Simulation of fluid–particles flows: heavy particles, flowing regime, and asymptotic-preserving schemes

Pages: 355 – 385



Thierry Goudon (INRIA Lille Nord Europe Research Centre, Villeneuve d’Ascq, France)

Shi Jin (Department of Mathematics, University of Wisconsin at Madison)

Bokai Yan (Department of Mathematics, University of Wisconsin at Madison)


We are interested in an Eulerian–Lagrangian model describing particulate flows. The model under study consists of the Euler system and a Vlasov-Fokker-Planck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, and lead to a limiting model of purely hydrodynamic type. We design a numerical scheme which is able to capture this asymptotic behavior without requiring prohibitive stability conditions. The construction of this Asymptotic Preserving scheme relies on an implicit discretization of the stiff terms which can be treated by efficient inversion methods. This method is a natural coupling of a kinetic solver for the particles with a kinetic scheme for the hydrodynamic Euler equations. Numerical experiments are conducted to study the performance of this scheme in various asymptotic regimes.


fluid–particles flows, hydrodynamic regimes, asymptotic preserving schemes, kinetic schemes

2010 Mathematics Subject Classification

35L65, 35Q35, 65M06, 76M20, 76N15, 82C40, 82C80

Published 14 October 2011