Modeling and Computational Methods for Kinetic Equations

Degond, Pierre; Pareschi, Lorenzo; Russo, Giovanni · 2004 · OpenAlex-citations

DOI: 10.1007/978-0-8176-8200-2

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Summary

This text presents the preface and structural overview of the edited volume *Modeling and Computational Methods for Kinetic Equations*, edited by Pierre Degond, Lorenzo Pareschi, and Giovanni Russo. The work addresses the challenge of multiscale modeling by providing a comprehensive overview of kinetic models and their applications across diverse fields, including gas dynamics, semiconductor modeling, granular flows, traffic flows, and plasma physics. The primary motivation is to bridge the gap between theoretical model derivation and the numerical methods required for quantitative prediction, a connection often neglected in existing monographs that focus exclusively on either theory or computation. The book is organized into two main parts. Part I focuses on the Boltzmann equation for rarefied gas dynamics, examining its macroscopic limits, such as the Euler and Navier-Stokes equations, derived via Hilbert and Chapman-Enskog expansions. It also covers moment equations for charged particles in semiconductors, highlighting global existence results for hydrodynamical models. This section reviews three primary numerical approaches for the Boltzmann equation: Monte-Carlo methods, valued for their low computational cost and robustness in non-equilibrium regimes; spectral methods, which offer high accuracy and preservation of conservation laws near equilibrium; and finite-difference methods, noted for their flexibility in handling binary gas mixtures. Part II explores specific applications of kinetic theory. It examines plasma kinetic models through the Fokker-Planck-Landau equation, detailing both its mathematical properties and multipole approximation techniques for numerical solution. The volume further addresses traffic flow modeling, granular media (focusing on inelastic collisions and cooling processes), quantum kinetic theory for Bose-Einstein condensation, and coagulation-fragmentation phenomena. A consistent theme throughout these applications is the development of numerical schemes that preserve essential physical features, such as mass and energy conservation, entropy dissipation, and correct steady-state distributions. The significance of this work lies in its integrated treatment of modeling and discretization. By linking the mathematical structure of kinetic equations to the design of efficient numerical algorithms, the book highlights how preserving physical properties at the discrete level is crucial for accurate simulation. This approach provides a rigorous yet accessible resource for graduate students and engineers, offering insights into the mathematical challenges of high-dimensional, nonlinear, and nonlocal kinetic problems while demonstrating the practical utility of these models in industrial and scientific contexts.

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