Advances in Kinetic Theory and Computing: Selected Papers

The following sections are included:

• FOREWORD

• CONTENTS

This paper is the first part of a series of three papers reviewing the mathematical theory of the Child-Langmuir law in the kinetic theory of charged-particle beams and various of its applications.

Eulerian Vlasov codes, using Splitting scheme for the resolution, are presented for the study of electron collisionless plasma in electrostatic and electromagnetic regimes. Many concepts have been investigated as the problem of filamentation in phase space, with the appearance of fine microstructures, followed by their "cleaning" Moreover the mathematical topology of the Vlasov equation, and phase space representation of the distribution function are studied.

Two examples indicating the important role of a minority population of particles, concerning the nonlinear Landau damping and the particle acceleration in the laser-plasma interaction are presented.

In this paper, we consider several plasma physics problems characterized by the existence of boundary layers or sheaths. These problems can all be viewed as singular perturbation problems for the Vlasov-Poisson system. We review some recent results concerning the asymptotic analysis of these problems and introduce new reduced models which lead to nonstandard free boundary problems.

We present the Schrödinger and Wigner-Poisson systems as tools for the modelling of quantum plasmas. For each of them, a numerical code, based on a splitting scheme, is proposed. The codes are used to simulate two typical situations : the quantum Brillouin flow (electron gas confined by a uniform magnetic field), and the one-dimensional anharmonic oscilator (with special concern for the choice of the initial condition and the nature of the classical and quantum phase space).

We present a review of mathematical results available for kinetic models for semiconductors. These models are mainly based on Vlasov-Poisson-Boltzmann system of equations. However the transport of particles (electrons and holes) is not purely classical but takes into account quantum effects due to the periodicity of the semiconductor lattice. Although the mathematical analysis is almost complete from the point of view of existence and uniqueness of solutions, it remains a lot of open problems concerning the connection between quantum, kinetic and fluid levels of modelling.

We investigate the one-dimensional degenerate hyperbolic system of gas dynamics with zero pressure. After a study of a priori estimates, we define a notion of measure solution and solve the system for a few examples of initial data, and especially the Riemann problem. We present a numerical scheme which has all the a priori estimates of our problem.

The present work reviews some technical properties of the Sonine polynomials in the theory of the linearized Boltzmann equation. These properties are used extensively in the Chapman-Enskog approximation as well as in any method aimed at establishing hydrodynamic limits the Boltzmann equation leading to Navier-Stokes equations.

We present theoretical derivations of several random particle methods for solving the Boltzmann equation, including Direct Simulation Monte Carlo methods and a method derived directly from Boltzmann equation. The DSMC methods are presented as discretizations of the master equations.

The following sections are included:

• Series on Advances in Mathematics for Applied Sciences