Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources (Frontiers in Mathematics)

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. "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 that needs very practical advice on how to get such desired stability properties."--Jacket This book is devoted to finite volume methods for hyperbolic systems of conservation laws. The accent is put on the development of tools for analyzing the nonlinear stability of Godunov schemes. Starting from theoretical considerations, the schemes are derived until a very practical level, meeting some required features such as for example the treatment of vacuum in gas dynamics. In the case of sources, the general notions of consistency, order of accuracy and well-balancing are developed, and applied to the construction of effective schemes. front-matter......Page 0 1Quasilinear systems and conservation laws......Page 7 2Conservative schemes......Page 19 3Source terms......Page 71 4Nonconservative schemes......Page 75 5Multidimensional finite volumes with sources......Page 113 6Numerical tests with source......Page 123 back-matter......Page 133

The schemes are analyzed regarding their nonlinear stability

Recently developed entropy schemes are presented

A formalism is introduced for source terms

The schemes are analyzed regarding their nonlinear stability Recently developed entropy schemes are presented A formalism is introduced for source terms Our aim is not to develop here a full theory of the Cauchy problem for hyperbolic systems.