Overview

This important new book sets forth a comprehensive description of various mathematical aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important mechanical applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics (MHD), shallow water, and solid dynamics equations. This treatment provides-for the first time in book form-a collection of recipes for applying higher-order non-oscillatory shock-capturing schemes to MHD modelling of physical phenomena. The authors also address a number of original ""nonclassical"" problems, such as shock wave propagation in rods and composite materials, ionization fronts in plasma, and electromagnetic shock waves in magnets. They show that if a small-scale, higher-order mathematical model results in oscillations of the discontinuity structure, the variety of admissible discontinuities can exhibit disperse behavior, including some with additional boundary conditions that do not follow from the hyperbolic conservation laws. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. The authors formulate several selection rules, which in some cases easily allow a correct, physically realizable choice. This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes Mathematical Aspects of Numerical Solution of Hyperbolic Systems particularly valuable not only to those interested the development of numerical methods, but to physicists and engineers who strive to solve increasingly complicated nonlinear equations.

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9780849306082

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Reviews

"""The book is a substantial addition to the existing literature It will be of interest to students and researchers in fluid dynamics and continuum mechanics in various field of physics."" -European Mathematical Society Newsletter, No. 41 (September 2001)"" this bookis as a sort of encyclopedia on numerical techniques applied to hyperbolic systems. Being free of, although important, mathematical and physical details, it allows the authors to focus the reader's attention on the core of numerics. The book is worthy of being in the library of everyone interested not only in numerical methods, but also in applied mathematics, mechanics, physics, and engineering, since the hyperbolic conservation laws are the basis of these areas of research."" -Applied Mathematics Review, Vol. 55, no. 3, May 2002"

The book is a substantial addition to the existing literature It will be of interest to students and researchers in fluid dynamics and continuum mechanics in various field of physics. -European Mathematical Society Newsletter, No. 41 (September 2001) this bookis as a sort of encyclopedia on numerical techniques applied to hyperbolic systems. Being free of, although important, mathematical and physical details, it allows the authors to focus the reader's attention on the core of numerics. The book is worthy of being in the library of everyone interested not only in numerical methods, but also in applied mathematics, mechanics, physics, and engineering, since the hyperbolic conservation laws are the basis of these areas of research. -Applied Mathematics Review, Vol. 55, no. 3, May 2002

Author Information

Kulikovskii, A.G.; Pogorelov, N.V.; Semenov, A. Yu.

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