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Computing Qualitatively Correct Approximations of Balance Laws

Exponential-Fit, Well-Balanced and Asymptotic-Preserving

  • Laurent Gosse

Part of the SIMAI Springer Series book series (SEMA SIMAI, volume 2)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Introduction and Chronological Perspective

  3. Hyperbolic Quasi-Linear Balance Laws

  4. Weakly Nonlinear Kinetic Equations

  5. Back Matter
    Pages 323-341

About this book

Introduction

Substantial effort has been drawn for years onto the development of (possibly high-order) numerical techniques for the scalar homogeneous conservation law, an equation which is strongly dissipative in L1 thanks to shock wave formation. Such a dissipation property is generally lost when considering hyperbolic systems of conservation laws, or simply inhomogeneous scalar balance laws involving accretive or space-dependent source terms, because of complex wave interactions. An overall weaker dissipation can reveal intrinsic numerical weaknesses through specific nonlinear mechanisms: Hugoniot curves being deformed by local averaging steps in Godunov-type schemes, low-order errors propagating along expanding characteristics after having hit a discontinuity, exponential amplification of truncation errors in the presence of accretive source terms... This book aims at presenting rigorous derivations of different, sometimes called well-balanced, numerical schemes which succeed in reconciling high accuracy with a stronger robustness even in the aforementioned accretive contexts. It is divided into two parts: one dealing with hyperbolic systems of balance laws, such as arising from quasi-one dimensional nozzle flow computations, multiphase WKB approximation of linear Schrödinger equations, or gravitational Navier-Stokes systems. Stability results for viscosity solutions of onedimensional balance laws are sketched. The other being entirely devoted to the treatment of weakly nonlinear kinetic equations in the discrete ordinate approximation, such as the ones of radiative transfer, chemotaxis dynamics, semiconductor conduction, spray dynamics of linearized Boltzmann models. “Caseology” is one of the main techniques used in these derivations. Lagrangian techniques for filtration equations are evoked too. Two-dimensional methods are studied in the context of non-degenerate semiconductor models.

Keywords

Asymptotic-Preserving and Well-Balanced schemes Diffusive approximations of kinetic equations Hyperbolic systems of balance laws Kinetic equations and moment approximations Viscosity solutions containing shock-waves

Authors and affiliations

  • Laurent Gosse
    • 1
  1. 1.Istituto per le Applicazioni del Calcolo “Mauro Picone”CNRRomeItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-88-470-2892-0
  • Copyright Information Springer-Verlag Italia 2013
  • Publisher Name Springer, Milano
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-88-470-2891-3
  • Online ISBN 978-88-470-2892-0
  • Series Print ISSN 2199-3041
  • Series Online ISSN 2199-305X
  • Buy this book on publisher's site
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