5R12. Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Monographs and Surveys in Pure and Applied Mathematics, Vol 118. - AG Kulikovskii (Dept of Mech, Steklov Math Inst, Russian Acad of Sci, Moscow, Russia), NV Pogorelov (Inst for Problems in Mech, Russian Acad of Sci, Moscow, Russia), and AY Semenov (General Phys Inst, Russian Acad of Sci, Moscow, Russia). Chapman and Hall/CRC, Boca Raton FL. 2001. 540 pp. ISBN 0-8493-0608-6. $94.95.
Reviewed by K Piechor (Inst of Fund Tech Res, Polish Acad of Sci, ul Swietokrzyska 21, Warsaw, 00-049, Poland).
The conservation laws are usually quasilinear partial differential equations of hyperbolic type, therefore equations of this type play an exceptional role in physics, mechanics etc. But equations of this type have their own special “stubborn character,” therefore various methods were invented for numerical treatment of them. The authors of the book undertake the effort to systemize these methods and to point out similarities and differences between them. Also they show, with examples from mechanics, physics, and engineering, the efficiency of the presented methods.
The book consists of seven chapters. In Chapter 1, the basic notions indispensable for the numeric analysis and the basic properties of hyperbolic systems, such as existence of smooth and discontinuous (shock waves) solutions, are briefly discussed. It is worth to note here that they use the notion of a so-called generalized solution to introduce shocks, which is a little bit different from the notion of weak solution used in the western literature. The content of this chapter is strictly limited to what is absolutely necessary for understanding the properties of solutions of hyperbolic systems and the used numeric methods. It is very insufficient from the point of view of a mathematician, for example the name of Glimm is not mentioned even in the references.
Chapter 2 is devoted to a presentation of numerical methods. They are divided into two types: shock-capturing and shock-fitting methods; various specific schemes belonging to each of these groups are discussed. They are presented in an encyclopedic fashion—no formal theorems or their proofs are given; the reader is sent to the suitable references (mainly Russian or Soviet).
In the following four chapters the use, the advantages and disadvantages of these methods are discussed via their applications.
In Chapter 3, applications of these methods to the gas dynamics equations are given. One-dimensional, multidimensional, unsteady, and stationary problems are discussed. Also fixed and moving grids are presented. The chapter ends with some numerical results.
In Chapter 4, the Godunov-type methods designed for computations of many specific problems arising in the shallow water equations are discussed. Exact solutions and the types of discontinuities are given. Finally, numerical results for some one- and two-dimensional problems illustrate the use of the exact and approximate Riemann solvers.
Chapter 5 is devoted to the MHD equations. Since this system of equations consists, roughly speaking, of the gas dynamics equations and the Maxwell equations, the general structure of this overall system is explained first. Next, the types of discontinuities are discussed. How to cope numerically with the enormous variety of problems inherent in magnetohydrodynamics is shown with examples.
Chapter 6 concerns some problems arising in the numeric analysis of solid materials. These include not only the classical elasticity (the Hook law), but also plastic and viscoplastic flows, as well as the dynamics of elastoplastic materials. Much attention, accompanied by examples, is attached to various aspects of the shell theory.
The final part of the book, Chapter 7, is concerned with non-classical hyperbolic systems as opposed to the problems discussed in the preceding part of the book. The variety of non-classical problems is, presumably, unlimited. So, the authors present only some of them, mainly according to their personal interests. These include small-amplitude waves in anisotropic media, electromagnetic waves in ferromagnets, shock waves in composite materials, nonlinear waves in elastic rods, and ionization fronts in a magnetic field. In this chapter, only the mathematical peculiarities of such systems are presented; no numerical schemes are given. The material given in this part of the book covers all non-classical problems, but much could be added, for instance, hyperbolic-elliptic problems. It seems the aim of this chapter was to show the reader that the applications of hyperbolic systems discussed in the previous chapters do not exhaust all possibilities.
The book hardly can be treated as a textbook; it does not answer many naive questions a beginner usually has, but it can be used for graduate-level courses. As a matter of fact, this book, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, is 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.