Numerical Methods for Hyperbolic and Kinetic Problems

Hyperbolic and kinetic equations arise in a large variety of industrial problems. For this reason, the CEMRACS summer research center held at CIRM in Luminy in 2003 was devoted to this topic. During a six-week period, junior and senior researchers worked full time on several projects proposed by industry and academia. Most of this work was completed later on, and the results are now reported in the present book.

The articles address modelling issues as well as the development and comparisons of numerical methods in different situations. The applications include multi-phase flows, plasma physics, quantum particle dynamics, radiative transfer, sprays and aeroacoustics.

Preface v

Kinetic problems

Introduction
by S. Cordier, T. Goudon and E. Sonnendr\"ucker 3

R. Barthelm\´e and C. Parzani
Numerical charge conservation in Particle-In-Cell 7

J.-P. Chehab, A. Cohen, D. Jennequin, J. J. Nieto, Ch. Roland and J. Roche
An adaptive Particle-In-Cell method using multi-resolution analysis 29

M. Campos Pinto and M. Mehrenberger
Adaptive numerical resolution of the Vlasov equation 43

N. Crouseilles and F. Filbet
A conservative and entropic method for the Vlasov--Fokker--Planck--Landau equation 59

C. Besse, N. J. Mauser and H. P. Stimming
Numerical studies for nonlinear Schr\"odinger equations:

the Schr\"odinger--Poisson-X $\alpha$ model and Davey--Stewartson systems 71

C. Besse, J. Claudel, P. Degond, F. Deluzet, G. Gallice and C. Tessieras
Ionospheric plasmas: model derivation, stability analysis and numerical simulations 101

L. Gosse
A case study on the reliability of multiphase WKB approximation for
the one-dimensional Schr\"odinger equation 131

Hyperbolic problems

Introduction by B. Desprès 145

C. Baranger, G. Baudin, L. Boudin, B. Despr\`es, F. Lagouti\`ere, E. Lap\´ebie and T. Takahashi
Liquid jet generation and break-up 149

B. Despr\`es, S. Jaouen, C. Mazeran and T. Takahashi
Numerical study of a conservative bifluid model with interpenetration 177

F. Caro, F. Coquel, D. Jamet and S. Kokh
DINMOD: A diffuse interface model for two-phase flows modelling 209

F. Coquel, D. Diehl, C. Merkle and C. Rohde
Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows 239

J. Cartier and A. Munnier
Geometric Eddington factor for radiative transfer problems 271

M. Dumbser and C.-D. Munz
Arbitrary high order discontinuous Galerkin schemes 295

C.-D. Munz, M. Dumbser and M. Zucchini
The multiple pressure variables method for fluid dynamics and aeroacoustics at low Mach numbers 335