5R37. High-Order Methods for Incompressible Fluid Flow. - MO Deville (Ecole Polytechnique Federale, Lausanne, Switzerland), PF Fischer (Argonne Natl Lab, Argonne IL 60439), EH Mund (Univ Libre, Brussels, Belgium). Cambridge UP, Cambridge, UK. 2002. 499 pp. ISBN 0-521-45309-7. $80.00.

Reviewed by DK Gartling (Comp Fluid Dyn, Sandia Natl Labs, MS 0826, Albuquerque NM 87185-5800).

This book is focused on the development and use of numerical methods for the solution of problems in incompressible fluid dynamics. Though not completely evident from the title, the methods of primary interest are spectral and spectral element methods and the closely associated orthogonal collocation techniques. In terms of flow problems, the text is limited in scope to isothermal, laminar flows, which are sufficient to discuss and demonstrate the essentials of the numerical approach. The book is designed as a graduate-level text and uses a standard progression of increasingly complex equations and algorithms to develop the topic. There are no problem sets or exercises for the student, though this is not really a deficit for a high-level text of this type. The authors are acknowledged experts in this field, and the text certainly demonstrates the breadth of their experience. The book is very well written and produced, with no discernible typographical errors.

The first chapter is primarily comprised of standard introductory material on viscous, incompressible, fluid mechanics and the Navier-Stokes equations. The summary is well done and includes some discussion of turbulence. The last section discusses computational issues, such as computing hardware and software, and makes the argument for the use of high-order methods in simulation. The second chapter is an introduction to variational methods including standard finite elements, spectral elements, and orthogonal collocation. The topics are developed using one-dimensional elliptic equations as the model problem. The chapter is nicely completed with sections on solution methods for algebraic systems, matrix conditioning and preconditioning, and a few numerical examples. Chapter 3 continues with the development of methods for time-dependent, one-dimensional equations, including algorithms for both parabolic and hyperbolic systems. The discussion of linear multistep, predictor-corrector, Runge-Kutta and Taylor-Galerkin methods is well done; the section on splitting methods is a welcome addition to a text. The advection equation, advection-diffusion equation, and Burgers equation are all used to illustrate spatial discretization via spectral techniques and the coupling with time integration methods.

Multidimensional problems form the main part of Chapter 4, starting with elliptic equations (diffusion and Helmholtz) and concluding with parabolic (unsteady advection-diffusion) and hyperbolic (unsteady advection) equations. Initially, rectangular domains that can be represented by tensor products of one-dimensional basis functions are considered, followed by mapping methods for distorted geometries. Spectral element methods are introduced for general geometries and orthogonal collocation is covered in several sections. The next two chapters concentrate on the incompressible flow equations, with the steady Stokes and Navier-Stokes equations covered in Chapter 5 and the unsteady equations discussed in Chapter 6. Essential to these chapters are sections on weak forms and the LBB condition, spectral element and orthogonal collocation choices for single or staggered grids, and solution algorithms including pressure Poisson and projection methods. Also discussed are divergence free bases and methods for free surface flows and simulations requiring ALE procedures. Both chapters conclude with several multidimensional examples of isothermal flow computations.

Chapter 7 is devoted to the important topic of domain decomposition. The main areas covered include preconditioning methods for solving large matrix problems, mortar element methods for joining subdomains, and the treatment of singularities. These topics are of current research interest, but are introduced and well summarized in this chapter. The last chapter also covers topics of current concern in terms of method implementation on vector and parallel computers. A good general description of parallel programming issues is followed by detailed discussion of spectral element implementation. The text is completed by two extensive appendices covering function spaces and orthogonal polynomials. The bibliography is also quite extensive and complete.

As a graduate text on spectral methods for fluid dynamics, this is a very complete and well-written book. The development of the numerical methods featured in the book are well organized and sufficiently detailed to allow the reader to implement the algorithms. High-Order Methods for Incompressible Fluid Flow is certainly recommended for use in both the classroom and as a self-study text for the postgraduate.