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**Spectral methods for incompressible viscous flow.**
*(English)*
Zbl 1005.76001

Applied Mathematical Sciences. 148. New York, NY: Springer. xi, 432 p. (2002).

The importance of spectral methods in obtaining the solutions of boundary value problems in a variety of situations of physical and engineering interest is well-known. The present book deals with spectral methods with a special emphasis on the study of flows of incompressible viscous fluid. The book is divided into three parts. The first part presents the basic spectral methods. The solutions of Navier-Stokes equations are given in the second part, and the third part discusses special topics. In all, there are nine chapters and three appendices.

The first chapter introduces the general principles on which the spectral methods are based. Fourier method and its applications to differential equations with constant and variable coefficients are discussed in chapter 2. The pseudospectral technique and the associated problem of aliasing are given in detail. In chapter 3, the author presents in detail the basic properties of Chebyshev polynomials and series, and Galerkin-tau method and collocation method for obtaining approximate solutions of boundary value problems associated with second-order elliptic equations. Chapter 4, which ends the first part, describes the time discretization of unsteady equations. One-step and multistep methods are considered in various situations in explicit, semi-implicit and fully implicit schemes. The stability of time discretization is investigated for Fourier and Chebyshev methods.

Efficient methods for computation of complex flows governed by Navier-Stokes equations are the subject of the entire second part. Two classical formulations, namely velocity-pressure and vorticity-stream function formulations for studying the flows of incompressible viscous fluids are dealt with in chapter 5. The Boussinesq approximation which is commonly used in convection problems is also presented here. The influence matrix method described in chapter 6 is one of the best ways to solve the vorticity-stream function equations as it is most straightforward, especially when associated with collocation technique. Chapter 7 presents Navier-Stokes equations in velocity-pressure formulation for three-dimensional fully periodic flows, for two-dimensional flows with one periodic direction, and for two-dimensional flows without periodicity. Usually it is difficult to solve Navier-Stokes equations in primitive variables as one has to calculate the pressure field ensuring the solenoidal character of velocity field, except in fully periodic cases. Thus two types of approaches are given according to the nature of elliptic problem to be solved at each time cycle of integration process.

The third part of the book discusses in chapter 8 the solution of stiff and singular problems, and the domain decomposition method in chapter 9. The formulas related to Chebyshev polynomials and to their general properties are given in appendix A, the solution of quasi-tridiagonal system is discussed in appendix B, and the last appendix C presents theorems on zeros of a polynomial.

This book gives a good understanding of spectral methods which deal with the flows of incompressible viscous fluid governed by Navier-Stokes equations. Other books dealing with the entire field of spectral methods are also given for easy reference. On the whole, this book is an excellent text for graduate students or researchers interested in learning spectral methods and in their application to incompressible fluid flows, and is a good addition to any library.

The first chapter introduces the general principles on which the spectral methods are based. Fourier method and its applications to differential equations with constant and variable coefficients are discussed in chapter 2. The pseudospectral technique and the associated problem of aliasing are given in detail. In chapter 3, the author presents in detail the basic properties of Chebyshev polynomials and series, and Galerkin-tau method and collocation method for obtaining approximate solutions of boundary value problems associated with second-order elliptic equations. Chapter 4, which ends the first part, describes the time discretization of unsteady equations. One-step and multistep methods are considered in various situations in explicit, semi-implicit and fully implicit schemes. The stability of time discretization is investigated for Fourier and Chebyshev methods.

Efficient methods for computation of complex flows governed by Navier-Stokes equations are the subject of the entire second part. Two classical formulations, namely velocity-pressure and vorticity-stream function formulations for studying the flows of incompressible viscous fluids are dealt with in chapter 5. The Boussinesq approximation which is commonly used in convection problems is also presented here. The influence matrix method described in chapter 6 is one of the best ways to solve the vorticity-stream function equations as it is most straightforward, especially when associated with collocation technique. Chapter 7 presents Navier-Stokes equations in velocity-pressure formulation for three-dimensional fully periodic flows, for two-dimensional flows with one periodic direction, and for two-dimensional flows without periodicity. Usually it is difficult to solve Navier-Stokes equations in primitive variables as one has to calculate the pressure field ensuring the solenoidal character of velocity field, except in fully periodic cases. Thus two types of approaches are given according to the nature of elliptic problem to be solved at each time cycle of integration process.

The third part of the book discusses in chapter 8 the solution of stiff and singular problems, and the domain decomposition method in chapter 9. The formulas related to Chebyshev polynomials and to their general properties are given in appendix A, the solution of quasi-tridiagonal system is discussed in appendix B, and the last appendix C presents theorems on zeros of a polynomial.

This book gives a good understanding of spectral methods which deal with the flows of incompressible viscous fluid governed by Navier-Stokes equations. Other books dealing with the entire field of spectral methods are also given for easy reference. On the whole, this book is an excellent text for graduate students or researchers interested in learning spectral methods and in their application to incompressible fluid flows, and is a good addition to any library.

Reviewer: Adabala Ramachandra Rao (Bangalore)

### MSC:

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

76M22 | Spectral methods applied to problems in fluid mechanics |

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