Nonlinear stability of finite volume methods for hyperbolic conservation laws and well balanced schemes for sources 1st Edition by François Bouchut ISBN 3764366656 9783764366650

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ISBN 10: 3764366656 
ISBN 13: 9783764366650
Author: François Bouchut

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.

Nonlinear stability of finite volume methods for hyperbolic conservation laws and well balanced schemes for sources 1st Table of contents:

Part I: Theory and Foundations

1. Quasilinear Systems and Conservation Laws

2. Conservative Systems

3. Invariant Domains

4. Entropy

5. Riemann Invariants & Contact Discontinuities

Part II: Schemes — Consistency and Stability

6. Conservative Schemes

7. Consistency Analysis

8. Stability Analysis

9. Explicit Well-Balanced Schemes (Saint-Venant-type)

Part III: Approximate Riemann Solvers

10. Approximate Riemann Solvers Overview

11. Exact Riemann Solvers

12. Simple Solvers

13. Suliciu Relaxation Solver

14. Suliciu Relaxation with Vacuum Handling

15. Suliciu Relaxation–HLLC Solver for Full Gas Dynamics

Part IV: Further Numerical Techniques

16. Kinetic Solvers

    • for isentropic gas dynamics

17. VFRoe Method

18. Passive Transport

19. Second-Order Extensions (time accuracy)

Part V: Handling Source Terms & Well-Balanced Methods

20. Source Terms

21. Invariant Domains & Entropy with Sources

22. Saint-Venant System with Sources

23. Nonconservative Schemes

24. Well-Balanced Schemes

25. Kinetic Solver for Source Terms

26. VFRoe with Source Terms

27. F-Wave Decomposition Method

28. Hydrostatic Reconstruction Scheme

    • Saint-Venant with variable pressure

    • Nozzle problem

29. Extended Source Cases:

    • Coulomb friction

30. Second-Order Well-Balanced Extensions

    • Centered flux and reconstruction operator

Part VI: Multidimensional Extensions

31. Multidimensional Finite Volumes with Sources

32. Nonconservative Finite Volumes

33. Well-Balanced in Multiple Dimensions

34. Additional Source Term Handling

35. 2D Saint-Venant System

36. Numerical Experiments with Source Terms

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Tags: François Bouchut, Nonlinear, finite volume, hyperbolic