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Mathematical aspects of numerical solution of hyperbolic systems. (English) Zbl 0965.35001
Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. 118. Boca Raton, FL: Chapman & Hall/CRC. xiii, 540 p. (2001).
This monograph provides a comprehensive description of several mathematical aspects of numerical solution of hyperbolic systems of partial differential equations, which are related to mechanical applications. The authors deal with the well-known problems, such as the Euler equations of gas dynamics, as well as with new nonclassical applications, e.g. in the magnetohydrodynamics (MHD), the shallow water equations, and the mechanics of solids. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. Some selection principles, which lead to physically adequate choices, are presented. The authors give also a collection of recipes for application of high-order nonoscillatory schock-capturing numerical schemes to the above classes of problems.
The book is divided into seven chapters. Chapter 1 introduces the main notations and definitions. Chapter 2 formulates the approaches to numerical solution of quasilinear hyperbolic systems both in conservative as well as nonconservative form. The authors describe the Godunov-type methods, based on the approximate solution to the Riemann problem, the higher-order methods, which include both the application of the generalized Riemann problem and recovery techniques. Chapter 3 is devoted to the equations of gas dynamics, Chapter 4 describes different Godunov-type methods for the shallow water equations, and Chapter 5 deals with the MHD equations. Chapter 6 gives an outline of the problems of solid dynamics, and Chapter 7 introduces the notion of nonclassical discontinuity and formulates evolutionary conditions for them.
This book can be useful to postgraduate students, specialists in pure and applied mathematics, physicists and engineeres. It supplements the existing literature, covers new areas of application and formulates notions of new phenomena, which appear in nonconvex hyperbolic systems.

MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35A35 Theoretical approximation in context of PDEs
35Lxx Hyperbolic equations and hyperbolic systems
76W05 Magnetohydrodynamics and electrohydrodynamics
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65Nxx Numerical methods for partial differential equations, boundary value problems
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