Shock-capturing methods for free-surface shallow flows.

*(English)*Zbl 0996.76003
Chichester: Wiley. xv, 309 p. (2001).

The book is about modern shock-capturing, finite volume numerical methods of upwind and centered type, specifically tailored for solving time-dependent nonlinear shallow water equations. The methods are applicable to problems involving one and two space dimensions, and to situations including smooth solutions, discontinuous solutions and complex wave interaction. The development of shock-capturing methods of upwind and centered type in the framework of finite volumes has made tremendous progress in the last two to three decades. Applications of these methods have in the main been confined to compressible fluid dynamics; it is only in recent years that this successful numerical technology has started being imported to other distinct areas of application, such as free-surface gravity flows. The experience so far is very encouraging, and this book is an attempt to build upon such experience and to put together the foundations of these methods for the purpose of solving nonlinear shallow water equations for situations of interest to scientists and engineers.

The book contains essentially three parts; the first one is devoted to the mathematical and physical aspects of the governing partial differential equations. The central ingredient of this first part is the Riemann problem, as most of the shock-capturing numerical methods studied later rely on the utilisation of local Riemann problem solutions, exact or approximate, but the role of the Riemann problem is indeed wider. The Riemann problem can be useful in theoretical studies of simple shallow water models; recently, Riemann solvers have also been utilised in conjunction with discontinuous Galerkin finite element methods and with meshless approaches such as the smooth particle hydrodynamics (SPH) methodology. The second part of this book deals with shock-capturing, finite volume numerical methods. Finally, the third part is concerned with illustrative applications of shallow water models and of some of the numerical methods studied in this book: with problems associated with dam-break modelling, and with the occurrence of Mach reflection of bores in two-dimensional shallow water flows.

This book is primarily intended for environmental scientists and engineers in academia, research laboratories, industry and consultancy organisations. Senior undergraduate and postgraduate students exposed to mathematical modelling and computational methods for environmental problems will benefit from studying this book. Lecturers could use most of the material for courses on modern numerical methods for wave propagation problems in hydraulics, oceanography, atmospherics and other environmental disciplines.

The book contains essentially three parts; the first one is devoted to the mathematical and physical aspects of the governing partial differential equations. The central ingredient of this first part is the Riemann problem, as most of the shock-capturing numerical methods studied later rely on the utilisation of local Riemann problem solutions, exact or approximate, but the role of the Riemann problem is indeed wider. The Riemann problem can be useful in theoretical studies of simple shallow water models; recently, Riemann solvers have also been utilised in conjunction with discontinuous Galerkin finite element methods and with meshless approaches such as the smooth particle hydrodynamics (SPH) methodology. The second part of this book deals with shock-capturing, finite volume numerical methods. Finally, the third part is concerned with illustrative applications of shallow water models and of some of the numerical methods studied in this book: with problems associated with dam-break modelling, and with the occurrence of Mach reflection of bores in two-dimensional shallow water flows.

This book is primarily intended for environmental scientists and engineers in academia, research laboratories, industry and consultancy organisations. Senior undergraduate and postgraduate students exposed to mathematical modelling and computational methods for environmental problems will benefit from studying this book. Lecturers could use most of the material for courses on modern numerical methods for wave propagation problems in hydraulics, oceanography, atmospherics and other environmental disciplines.

Reviewer: Leonid B.Chubarov (Novosibirsk)

##### MSC:

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

76M12 | Finite volume methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |