Modeling and Computational Methods for Kinetic Equations

The aim of this book is to provide a consistent treatment of kinetic equations both from the viewpoint of modelling and of numerics. The book will be divided in two parts. The first part is devoted to the most fundamental kinetic model: the Boltzmann equation of rarefed gas dynamics. Its connections with macroscopic models through hydrodynamic limits and moments closure hierarchies are developed as they play important role in the modelling of fluids. Then, the most widely used numerical methods for the discretization of the Boltzmann equation are reviewed: the Monte-Carlo method, Spectral methods and Finite-Difference methods. The second part is devoted to more speci.c applications: plasma kinetic models with the 'Landau-Fokker-Planck equations' and its numerical discretization, traffc fow modelling, granular media, quantum kinetic models and coagulation-fragmentation problems. In each case, both modelling aspects and numerical methods are discussed. The originality of this book is to provide a consistent treatment of the models, both from the point of view of theory and modelling and from that of the numerical discretization. Most of the existing monographies focus on either one or the other of these two aspects. However, bringing these two aspects together shines light on points which are important but which are very likely to be discarded in more focused approaches. For instance, the development of spectral or multipole methods for kinetic equations was motivated by the search for efcient ways of discretizing the Boltzmann operator while preserving an accurate description of the various conservation laws as well as entropy dissipation. These properties are deeply related with the kind of system the model aims at describing. The same considerations are obviously true for trafc fow modelling, granular media, quantum kinetic models or coagulation-fragmentation problems, which are the four specifc applications the present project intends to develop.