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Numerical methods for fluid dynamics. With applications to geophysics. 2nd ed. (English) Zbl 1214.76001
Texts in Applied Mathematics 32. Berlin: Springer (ISBN 978-1-4419-6411-3/hbk; 978-1-4614-2685-1/pbk; 978-1-4419-6412-0/ebook). xv, 516 p. (2010).
[For the review of the first ed. see (1998; Zbl 0918.76001).]
This book is an introduction to numerical methods for fluid dynamics. The reader can find here numerical examples that illustrate the theoretically derived properties and mathematical properties of numerical methods derived in a style familiar to most scientists. In fact, this book establishes a clear link between the theory and practice and focuses on numerical methods applicable to the simulations of wavelike flows. The methods considered here can be used in atmospheric or ocean models, and they are particularly appropriate for low-viscosity flows, but not for shocks.
After a short introduction, the author derives the equations governing geophysical flows and gives an overview of strategies for numerical approximation. The book covers the classical ODE solvers and describes more recent advances in the design of schemes for stiff equations, with several discussions of numerical stability, consistency, convergence and accuracy. Other chapters examine the behaviour of finite difference schemes for simple problems involving one-dimensional transport equation, and for more general problems with many independent variables.
A large part of the book is devoted to conservation laws in fluid dynamics, and to finite volume schemes that approximate their weak solutions. The author also presents the series expansion methods potentially useful in geophysical fluid dynamics, and spectral, pseudospectral, finite element and discontinuous Galerkin methods. The final part of the book discusses popular specific methods used in global weather forecast models, such as semi-Lagrangian methods, methods for circumventing the time-step constraint imposed by rapidly but physically insignificant waves, and methods with nonreflecting boundary conditions.
The text could be useful to graduate students and scientists working in various branches of applied mathematics and engineering, not only in geophysical fluids. The material is intelligible to readers with a general mathematical background. The book includes exercises and is well illustrated with figures linking theoretical analyses to actual computations.

MSC:
76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics
76Mxx Basic methods in fluid mechanics
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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