Numerical approximation of hyperbolic systems of conservation laws.

*(English)*Zbl 0860.65075
Applied Mathematical Sciences. 118. New York, NY: Springer. viii, 509 p. (1996).

This work is devoted to the theory and approximation of nonlinear hyperbolic systems of conservation laws in one or two space variables. It follows directly a previous publication on hyperbolic systems of conservation laws by the same authors [Hyperbolic systems of conservation laws, MathÃ©matiques et Applications, Ellipses, Paris, (1991; Zbl 0768.35059)] in which the scalar case is covered. The authors mainly concentrate on some theoretical aspects that are needed in the applications, such as the solution of the Riemann problem, with occasional insights into more sophisticated problems.

This book is divided into six chapters, including an introductory chapter. In this introduction the authors resume the notions that are necessary for a self-sufficient understanding of this book (definitions of hyperbolicity, weak solution, and entropy), present practical examples of systems of conservation laws in several space variables that arise in continuum physics and recall the main results concerning the scalar case.

Chapter I is devoted to the resolution of the Riemann problem for a general hyperbolic system in one space dimension, and introduces the classical notions of Riemann invariants and simple waves, the rarefaction and shock curves, and characteristics and entropy conditions. These notions are illustrated on the examples of the \(p\)-systems and the gas dynamics equations. In Chapter II the authors prove that the Riemann problem for the one-dimensional system of gas dynamics is always solvable. These two basic chapters present the results with detailed proofs. In Chapter III the authors develop the most usual numerical scheme for one-dimensional hyperbolic systems with special emphasis on the application to gas dynamics. A short account on the kinetic theory and the kinetic schemes are given. Chapter IV is devoted to the study of finite volume methods for bidimensional systems, preceded by some theoretical considerations on multidimensional systems. An introduction to the boundary conditions is given in Chapter V. Some numerical boundary treatment is presented.

The bibliography is rather long and the authors have not cited again all the references of their first book. Some references are also added in the notes at the end of each chapter.

This book is divided into six chapters, including an introductory chapter. In this introduction the authors resume the notions that are necessary for a self-sufficient understanding of this book (definitions of hyperbolicity, weak solution, and entropy), present practical examples of systems of conservation laws in several space variables that arise in continuum physics and recall the main results concerning the scalar case.

Chapter I is devoted to the resolution of the Riemann problem for a general hyperbolic system in one space dimension, and introduces the classical notions of Riemann invariants and simple waves, the rarefaction and shock curves, and characteristics and entropy conditions. These notions are illustrated on the examples of the \(p\)-systems and the gas dynamics equations. In Chapter II the authors prove that the Riemann problem for the one-dimensional system of gas dynamics is always solvable. These two basic chapters present the results with detailed proofs. In Chapter III the authors develop the most usual numerical scheme for one-dimensional hyperbolic systems with special emphasis on the application to gas dynamics. A short account on the kinetic theory and the kinetic schemes are given. Chapter IV is devoted to the study of finite volume methods for bidimensional systems, preceded by some theoretical considerations on multidimensional systems. An introduction to the boundary conditions is given in Chapter V. Some numerical boundary treatment is presented.

The bibliography is rather long and the authors have not cited again all the references of their first book. Some references are also added in the notes at the end of each chapter.

Reviewer: K.Najzar (Praha)

##### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

76M20 | Finite difference methods applied to problems in fluid mechanics |

76N15 | Gas dynamics, general |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |