7R1. Fundamentals of Computational Fluid Dynamics. - H Lomax (Deceased), TH Pulliam (NASA Ames Res Center, Moffett Field CA 94035), DW Zingg (Inst for Aerospace Stud, Univ of Toronto, 4925 Dufferin St, Toronto, ON, M3H 5T6, Canada). Springer-Verlag, Berlin. 2001. 249 pp. ISBN 3-540-41607-2. \$49.95.

Reviewed by TA Kowalewski (Center of Mech and Info Tech, IPPT PAN, Inst of Fund Res, Polish Acad of Sci, Swietokrzyska 21, Rm 211, Warsaw, PL 00-049, Poland).

The book provides an elementary tutorial presentation on computational methods in fluid mechanics. The authors intend this book to serve as a survey of mathematical fundamentals which provide foundation for the computational fluid dynamics. As a result of this focus, the book is suitable as a textbook for a first course in CFD. The underlying philosophy of the authors is that the theory of linear algebra provides a basic mathematical framework for understanding numerical methods. Hence, the material presented emphasizes fundamental concepts of numerical methods in which the partial differential equations are reduced to ordinary differential equations, and finally to the difference equations, ie, to the linear system of algebraic equations.

The book begins with an outline of the basic equations. The authors leading concept is to explain fundamentals of CFD on the basis of simple model equations, isolating physical characteristics of the complete set of equations. Therefore, after a brief introduction to the Euler and Navier-Stokes equations, the two basic models are discussed in more details: diffusion and convection equations. In the following chapter, the concept of finite-difference approximation to partial derivatives is explained. The basic issues in constructing differencing schemes for solving partial differential equations using matrix operators are given. The Fourier error analysis is applied to estimate the accuracy of a finite-difference scheme. An extensive discussion of the difference operators at boundaries is exemplified for two selected model equations, linear convection and diffusion. The two strategies for obtaining finite-difference approximation for time dependent problems are discussed using the basic concept of the biconvection and diffusion model equations, with fundamental analysis of properties of solutions expressed in real and eigenspace.

In Chapter 5, the reader finds the basics of finite-volume approximation and the discussion on advantages of using integral form of the model equations. The next two chapters describe, in detail, the theory and implementation of time-marching discretization methods, including their stability and the accuracy analysis. Chapter 8 gives several practical hints for applying time-marching methods to the specific problem. The concept of the numerical stiffness is used to classify stability of particular discrete approximation. This chapter can be very valuable for the end users of CFD methods, helping them to select appropriate numerical approach to the physical problem studied. In principle, the first eight chapters may appear to be sufficient to explain the basic mathematics of CFD modeling. The three chapters following them form a practical handbook of different CFD approaches, describing methodology for designing, analyzing, and choosing time-marching methods. The description of properties and applications of several classical relaxation methods gives good introduction to iterative techniques accelerating solution of large systems of equations. It is followed by a very brief description of the idea of constructing multigrid computational domain. The important problem of numerical dissipation is elucidated for few typical schemes in Chapter 11. The last two chapters present and analyze split and factored algorithms.

The book is well written and well organized. It can be easily adopted as a textbook for senior or graduate students studying numerical methods of fluid mechanics. Practice exercises are provided at the end of each chapter, some of them expecting the reader to write his own computer codes. This reviewer would regard Fundamentals of Computational Fluid Dynamics as essential to anyone planning to use CFD modeling. The emphasis of the book is on understanding basic mathematics underlying the discretization ideology. However, this book is not a practical handbook of numerical methods. The emphasis of the book is on the mathematical aspects of CFD with limited attention on the physics of the problems solved. Therefore, inexperienced readers may have difficulties in applying knowledge gained here to construct their own codes for solving practical problems of fluid mechanics.

Deceased.