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Numerical solution of hyperbolic partial differential equations. With CD-ROM. (English) Zbl 1187.65088

Cambridge: Cambridge University Press (ISBN 978-0-521-87727-5/hbk). xxi, 597 p. (2009).
This book is a graduate textbook presenting shock-capturing methods, including finite difference, finite volume and finite element methods for solving hyperbolic partial differential equations. It covers also the theory of hyperbolic conservation laws and the theory of numerical methods. A broad range of applications is presented: shallow water, compressible gas dynamics, magnetohydrodynamics, finite deformation in solids, plasticity, polymer flooding and gas/water injection in oil recovery.
The numerical methods involve a variety of important approaches, such as MUSCL and PPM, TVD, wave propagation, Lax-Friedrich (aka central schemes), ENO and discontinuous Galerkin, all in one and multiple space dimensions.
The book is divided in 8 chapters entitled as follows:
{1.} Introduction to partial differential equations (5p.);
{2.} Scalar hyperbolic conservation laws (75 p.);
{3.} Nonlinear scalar laws (54 p.);
{4.} Nonlinear hyperbolic systems (191 p.);
{5.} Methods for scalar laws (106 p.);
{6.} Methods for hyperbolic systems (42 p.);
{7.} Methods in multiple dimensions (70 p.);
{8.} Adaptive mesh refinement (40 p.).
There is an accompanying CD containing a hyperlinked version of the text which provides access to computer codes for all text figures. In this way a reader can see the codes and run them, choosing own input parameters interactively and view the online numerical results as movies. The codes can be downloaded and modified for other applications.
The author says in the introduction that the book grew out of a one-semester course he taught at Duke University. This is hard to believe since there is enough material for several courses. The book has much more the character of a monography than a textbook. The interested reader can find a lot of information on the topic.

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35Lxx Hyperbolic equations and hyperbolic systems
65Y15 Packaged methods for numerical algorithms
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76N15 Gas dynamics (general theory)
76W05 Magnetohydrodynamics and electrohydrodynamics
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
76Mxx Basic methods in fluid mechanics
82D60 Statistical mechanics of polymers
86A20 Potentials, prospecting
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