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**High-order methods for incompressible fluid flow.**
*(English)*
Zbl 1007.76001

Cambridge Monographs on Applied and Computational Mathematics. 9. Cambridge: Cambridge University Press. xxvii, 499 p. (2002).

High-order numerical methods provide an efficient approach to simulate many physical problems. This interesting book presents the range of certain mathematical, engineering as well as computer science topics that are the foundation of high-order numerical methods for the simulation of incompressible fluid flows in complex domains. The aim of this book is to show the realm of feasibility of high-order methods: accuracy versus efficiency, tractable problems, nonlinearities, complex geometrical configurations, and the influence of computer architectures.

The book contains eight chapters and an appendix divided in two parts. Chapter 1 presents the basic equations of incompressible Newtonian flows, including natural convection through Boussinesq approximation. Chapter 2 presents numerical solutions of one-dimensional elliptic problems, including Galerkin methods in the context of polynomial approximations. Chapter 3 is devoted to the study of hyperbolic and parabolic problems. In the first part of this chapter the authors present numerical schemes for solutions of nonlinear systems of ordinary differential equations. The second part is devoted to steady-state advection and/or advection-diffusion problems as well as to evolutionary problems. Chapter 4 generalizes introductory concepts to multiple space dimensions. Elliptic, parabolic, and hyperbolic problems are systematically studied. Chapter 5 treats the velocity-pressure formulation of steady Stokes and Navier-Stokes equations. The key issues of compatible discretization of velocity and pressure are fully considered. The chapter concludes with the analysis of multidimensional Navier-Stokes problem. Chapter 6 treats unsteady Stokes and Navier-Stokes equations, presenting the state-of-the-art of splitting and projection methods. Here the authors introduce the arbitrary Lagrangian-Eulerian method in order to treat free surface flows. Some relevant applications are also included. Chapter 7 deals with the domain decomposition methods and adaptivity techniques. Chapter 8 is devoted to vectorization and parallelism by studying cases including spectral element and collocation techniques. An appendix contains basic mathematical concepts used in the book. A large bibliography concludes the book.

This excellent textbook is a valuable reference on high-order methods applied to incompressible fluid flow problems. It could be recommended to computer scientists, engineers, and applied mathematicians interested in developing software for solving flow problems. It could be also recommended to graduate students that have a background in fluid mechanics and numerical methods.

The book contains eight chapters and an appendix divided in two parts. Chapter 1 presents the basic equations of incompressible Newtonian flows, including natural convection through Boussinesq approximation. Chapter 2 presents numerical solutions of one-dimensional elliptic problems, including Galerkin methods in the context of polynomial approximations. Chapter 3 is devoted to the study of hyperbolic and parabolic problems. In the first part of this chapter the authors present numerical schemes for solutions of nonlinear systems of ordinary differential equations. The second part is devoted to steady-state advection and/or advection-diffusion problems as well as to evolutionary problems. Chapter 4 generalizes introductory concepts to multiple space dimensions. Elliptic, parabolic, and hyperbolic problems are systematically studied. Chapter 5 treats the velocity-pressure formulation of steady Stokes and Navier-Stokes equations. The key issues of compatible discretization of velocity and pressure are fully considered. The chapter concludes with the analysis of multidimensional Navier-Stokes problem. Chapter 6 treats unsteady Stokes and Navier-Stokes equations, presenting the state-of-the-art of splitting and projection methods. Here the authors introduce the arbitrary Lagrangian-Eulerian method in order to treat free surface flows. Some relevant applications are also included. Chapter 7 deals with the domain decomposition methods and adaptivity techniques. Chapter 8 is devoted to vectorization and parallelism by studying cases including spectral element and collocation techniques. An appendix contains basic mathematical concepts used in the book. A large bibliography concludes the book.

This excellent textbook is a valuable reference on high-order methods applied to incompressible fluid flow problems. It could be recommended to computer scientists, engineers, and applied mathematicians interested in developing software for solving flow problems. It could be also recommended to graduate students that have a background in fluid mechanics and numerical methods.

Reviewer: Mirela Kohr (Cluj-Napoca)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76Mxx | Basic methods in fluid mechanics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |