9R47. Systems of Conservation Laws: Two-Dimensional Riemann Problems. Progress in Nonlinear Differential Equations and Their Applications, Vol 38. - Yuxi Zheng (Dept of Math, Indiana Univ, Bloomington IN 47405). Birkhauser Boston, Cambridge MA. 2001. 317 pp. ISBN 0-8176-4080-0. $69.95.
Reviewed by TH Moulden (Dept of Aerospace Eng, Univ of Tennessee Space Inst, BH Goethert Pkwy, Tullahoma TN 37388-8897).
The text under review is based upon the contents of a course on partial differential equations offered at Indiana University. The subtitle, Two-Dimensional Riemann Problems, indicates the main thrust of the book. However, the early chapters (1–4) are restricted to discussing the basic phenomena in the one-dimensional case. These phenomena include the notion of characteristics and Riemann invariants and, of course, shock waves. Numerical methods are introduced, but not in detail. Glimm’s scheme is also introduced at this point. Difficulties in finding solutions (as stated in the von Neumann paradoxes) are noted, but the possible resolution via a viscous solution is not discussed.
Chapter 5 introduces the two-dimensional scalar Riemann problem and reviews entropy conditions as well as the Rankine-Hugoniot relations in multidimensions. These are some of the concepts needed for the rest of the text. Chapter 6 continues the discussion of this problem and also introduces the pseudo-characteristics. Axisymmetric problems are introduced in Chapter 7, and their many distinct flow cases discussed. This is a long chapter (over 70 pages), and the material is treated in detail. The appendix to Chapter 7 proves some theorems relevant to the material in the chapter.
In Chapter 8, attention is turned to the two-dimensional Euler equations, and some possible flow fields are discussed (some of these examined by numerical experiments). Some of the diagrams shown here are for transonic flows (but transonic only appears three times in the index as does mixed problem). Possibly, more should have been made of such transonic flows.
One of the strong points of the text is the detailed discussion of the many possible solutions of the sets of equations considered. Many of these solutions are shown diagrammatically, some from numerical computation. Thus, as mentioned above, Chapter 7 does this for the axisymmetric solutions, while Chapter 8 provides the same service for the full two-dimensional Euler equations. Some of these solutions have regions of flow at zero density and so are not of physical interest. Chapters 9 and 10 consider a flux splitting of the two-dimensional Euler equations. The final chapter contains a short discussion of the basic numerical methods suitable for the problems under study in the text.
The engineer may find the text less satisfactory than would the applied mathematician since there is little physical motivation for the problems discussed. There is also little physical interpretation of the results obtained. The problems discussed are, in the physical sense, those of inviscid motion. Only in a few places are the viscous equations discussed in the text and then only to provide a limiting condition for an entropy criterion used to select appropriate solutions. Many of the difficulties in selecting solutions would not be in evidence for the viscous flow equations. In the terminology of aerodynamics, the Euler limit is not (necessarily) benign due to the change in boundary conditions between the two types of equation. The limit is a singular one in the sense of boundary layer theory. This is not discussed in Zheng’s book.
Systems of Conservation Laws: Two-Dimensional Riemann Problems provides a good list of references and commentary on this literature at the ends of the appropriate chapters. There are also exercises at the end of chapters (but more would be welcome). The book is recommended for its intended audience.