Overview
- The schemes are analyzed regarding their nonlinear stability
- Recently developed entropy schemes are presented
- A formalism is introduced for source terms
Part of the book series: Frontiers in Mathematics (FM)
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Table of contents (6 chapters)
Keywords
About this book
This book is devoted to finite volume methods for hyperbolic systems of conservation laws. It differs from previous expositions on the subject in that the accent is put on the development of tools and the design of schemes for which one can rigorously prove nonlinear stability properties. Sufficient conditions for a scheme to preserve an invariant domain or to satisfy discrete entropy inequalities are systematically exposed, with analysis of suitable CFL conditions.
The monograph intends to be a useful guide for the engineer or researcher who needs very practical advice on how to get such desired stability properties. The notion of approximate Riemann solver and the relaxation method, which are adapted to this aim, are especially explained. In particular, practical formulas are provided in a new variant of the HLLC solver for the gas dynamics system, taking care of contact discontinuities, entropy conditions, and including vacuum. In the second half of the book, nonconservative schemes handling source terms are analyzed in the same spirit. The recent developments on well-balanced schemes that are able to capture steady states are explained within a general framework that includes analysis of consistency and order of accuracy. Several schemes are compared for the Saint Venant problem concerning positivity and the ability to treat resonant data. In particular, the powerful and recently developed hydrostatic reconstruction method is detailed.
Reviews
From the reviews:
“This is a very interesting and useful book which provides a systematic presentation of the theory of finite volume methods and numerical simulations for hyperbolic systems of conservation laws. The author provides a unified approach and notation to the study of nonlinear stability of finite volume methods for hyperbolic systems of conservation laws as the accent is put on the development of tools and design of schemes. The exposition of the book is very clear. It will be a very useful tool for the researchers in this field as well as for engineers.”(ZENTRALBLATT MATH)
Authors and Affiliations
Bibliographic Information
Book Title: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws
Book Subtitle: and Well-Balanced Schemes for Sources
Authors: François Bouchut
Series Title: Frontiers in Mathematics
DOI: https://doi.org/10.1007/b93802
Publisher: Birkhäuser Basel
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eBook Packages: Springer Book Archive
Copyright Information: Birkhäuser Basel 2004
Softcover ISBN: 978-3-7643-6665-0Published: 25 June 2004
eBook ISBN: 978-3-7643-7792-2Published: 02 February 2005
Series ISSN: 1660-8046
Series E-ISSN: 1660-8054
Edition Number: 1
Number of Pages: VIII, 134
Topics: Partial Differential Equations, Numerical Analysis, Mathematical and Computational Biology, Classical and Continuum Physics