Z. Angew. Math. Phys. 62 (2011), 189–190 c 2010 Springer Basel AG  0044-2275/11/010189-2 published online June 1, 2010 DOI 10.1007/s00033-010-0083-2

Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

Book Review John A. Trangenstein: Numerical solution of hyperbolic partial differential equations Cambridge University Press, ISBN 978-0-521-87727-5, 597 pages, Euro 47.99, 2009 Thomas Sonar

Hyperbolic conservation laws comprise one of the most important class of partial differential equations as far as applications are concerned, whether in compressible fluid dynamics, magnetohydrodynamics, polymer flooding, oil recovery problems, and many other areas: At the end of the modelling process, there is almost always a set of hyperbolic conservation laws. On the other hand is the mathematical theory of these equations quite demanding due to the nonlinearities occurring fairly natural in form of a flux function involved. Substantial progress was achieved during the last decades but questions of existence and uniqueness are still far from being answered for general systems. Since the epoch-making Mathematische Annalen paper of Courant, Friedrichs, and Lewy was published in 1928, numerical methods played a central role in proofs of existence of solutions but with the advent of the computer as early as in the 1940s, numerical methods became the only means of looking at (hopefully accurate) solutions to practical problems like detonations or shocked flows around airfoils. As in many other areas, John von Neumann was a pioneer also in this realm of applied mathematics. Nowadays, there is no car manufacturer or airplane factory which can dispense with numerical methods for conservation laws. In the 1980s, new ideas like Total Variation Diminishing (TVD) and Essentially Non-Oscillatory (ENO) Schemes appeared while Anthony Jameson created a central scheme with artificial dissipation which became the workhorse of Computational Fluid Dynamics (CFD) around the world. From the 1990s onwards, lessons learned from the mathematical theory of the equations lead to convergence theories for nonlinear numerical methods at least under some restrictive conditions for scalar conservation laws. Both areas, the mathematical theory of conservation laws and the numerical analysis of numerical methods to solve them approximately, are still very active today. Hence, textbooks are needed for the novice who is to enter the vast lands of numerical analysis for these nonlinear partial differential equations. Although some textbooks are now already available—notably the books by Charles Hirsch (mainly addressing engineers), by Randy LeVeque, by Dietmar Kr¨ oner, and by Edwige Godlewski and Pierre-Arnaud Raviart (mainly addressing mathematicians) have to be named here alongside others—there is no book to my knowledge than the one under review which connects the mathematics with a variety of applications. Trangenstein succeeds in giving an overview of the numerical analysis of numerical methods while at the same time describing certain ‘case studies’: Traffic flow, gas dynamics, Maxwell’s equations, magnetohydrodynamics, deformation in elastic solids, three-phase flow, and plasticity. The classical theory of Lax-Richtmyer for linear equations is described but also nonlinear stability and the entropy conditions are treated in a fairly rigorous way. Although the book has nearly 600 pages, the author had to steer a course of compromise between the mathematics and the applications, and in my view, this difficult task has been

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achieved quite nicely. Scalar conservation laws are treated as are systems and fairly modern methods like the discontinuous Galerkin methods are treated alongside classical schemes. Even two-dimensional Riemann problems are discussed as is a strategy for grid refinement based on Marsha Berger’s celebrated algorithms. Since the book is intended to serve as a textbook, one would expect snippets of code so that the beginner can also learn the basics of necessary programming. In fact, Trangenstein presents a whole bunch of codes on an enclosed CD that also contains a hyperlinked copy of the book. In order to run the codes, your computer has to be rebooted from the CD, and the user has to be cautious. On my Intel-based Macintosh, this could be done without any problems but I did not check the CD on a PC. Besides some typos which I think are unavoidable in a first edition, I was a bit irritated by a certain misuse of notation. Let me give two examples: in stating the weak form of systems of conservation laws on page 135, the function u is stated to be a vector in Rm (and the flux F is stated to be an array of Rn vectors) which it simply is not. On the other hand, we find statements like ‘Suppose that u(x, t) ∈ Rm satisfies the conservation law ...’ (e.g. on page 143). Here, it is obviously correct that u(x, t) is a vector in Rm but such a vector cannot satisfy a partial differential equation; the solution has to be the function u. I know that many authors nowadays are fairly sloppy in the distinction between functions and their values but in a book where convergence issues are discussed this sloppiness is out of place. However, Trangenstein’s book is in my opinion a very welcome addition to the literature. It is appropriate for the novice in this field to get an overview as for the experienced numerical analyst alike. I am sure that also engineers will be satisfied when studying the book which is very well endowed with good paper and a lasting binding by the publishing house. T. Sonar Institut Computational Mathematics Technische Universit¨ at Braunschweig Pockelsstraße 14 38106 Braunschweig Germany e-mail: [email protected] (Received: January 25, 2010)