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Mathematical problems in the numerical solution of hyperbolic systems of equations. (Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenij.) (Russian) Zbl 1007.65058
Moscow: Fiziko-Matematicheskaya Literatura. 608 p. (2001).
This book presents a comprehensive description of various mathematical aspects of problems arising in numerical solution of systems of hyperbolic partial differential equations. The authors present the material in the context of important mechanical application of such systems, including the Euler equation of gas dynamics, magnetohydrodynamic (MHD), shallow water and rigid deformative body, and a number of nonclassical problems, such as propagation of shocks in composite materials, ionization fronts in plasma, electromagnetic shock waves in magnets, etc. The book provides a collection of recipes for applying high order total variation diminishing (TVD) schemes to various problems in mechanic and physics, especially in MHD.
The book is divided into 7 chapters. The first describes the basic theory for hyperbolic systems. The second chapter is to present basic ideas of numerical methods for hyperbolic systems. A collection method, not only traditional but also new such as Godunov scheme, Courant, Lax, Lax-Wendroff, TVD, TVB, ENO, WENO, Roe, Osher, and others are presented.
The last 4 chapters present various schemes applied to hydrodynamics (Chapter 3), shallow water (Chapter 4), magnetohydrodynamic (Chapter 5), rigid deformative body (Chapter 6), a number of nonclassical problem (Chapter 7).
The authors’ treatment systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic system. Its unique focus on applications, both traditional and new, makes mathematical problems of the numerical solution of hyperbolic system particularly valuable not only to those interested in the development of numerical methods, but to physicists and engineers who are interested to solve complicated nonlinear problems.

MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 74Bxx Elastic materials 74S20 Finite difference methods applied to problems in solid mechanics 35L45 Initial value problems for first-order hyperbolic systems 35L65 Hyperbolic conservation laws 76M20 Finite difference methods applied to problems in fluid mechanics 76N15 Gas dynamics (general theory) 76W05 Magnetohydrodynamics and electrohydrodynamics 76S05 Flows in porous media; filtration; seepage