Numerical Methods for Fluids, Part 3: Volume 9
Published on 25. July 2003
Book
Hardback
1080 pages
978-0-444-51224-6 (ISBN)
Description
This book-size article is dedicated to the numerical simulation of unsteady incompressible viscous flow modelled by the Navier-Stokes equations, or by non-Newtonian variants of them. In order to achieve this goal a methodology has been developed based on four key tools. Time discretization by operator-splitting schemes such as Peaceman-Rachford's, Douglas Rachford's, Marchuk-Yanenko's, Strang's symmetrized, and the so-called "theta-scheme" introduced by the author in the mid-1980s. Projection methods (in L2 or H1) for the treatment of the incompressibility condition div u = 0. Treatment of the advection by: either a centered scheme leading to linear or nonlinear advection-diffusion problems solved by least squares/conjugate gradient algorithms, or to a linear wave-like equation well suited to finite element-based solution methods. Space approximation by finite element methods such as Hood-Taylor and Bercovier-Pironneau, which are relatively easy to implement.
In addition to the above topics the article contains detailed discussions of conjugate gradient algorithms, least-squares methods for boundary-value problems which are not equivalent to problems of the calculus of variations, Uzawa-type algorithms for the solution of saddle-point problems, embedding/fictitious domain methods for the solution of elliptic and parabolic problems. In fact many computational methods discussed in this article also apply to non-CFD problems although they were mostly designed for the solution of flow problems. Among the topics covered are: the direct numerical simulation of particulate flow; computational methods for flow control; splitting methods for viso-plastic flow a la Bingham; and more. It should also be mentioned that most methods discussed in this article are illustrated by the results of numerical experiments, including the simulation of three-dimensional flow. Due to their modularity the methods described in this article are relatively easy to implement - as is demonstrated by the fact that several practitioners in various institutions have been able to use them ab initio for the solution of complicated flow (and other) problems.
In addition to the above topics the article contains detailed discussions of conjugate gradient algorithms, least-squares methods for boundary-value problems which are not equivalent to problems of the calculus of variations, Uzawa-type algorithms for the solution of saddle-point problems, embedding/fictitious domain methods for the solution of elliptic and parabolic problems. In fact many computational methods discussed in this article also apply to non-CFD problems although they were mostly designed for the solution of flow problems. Among the topics covered are: the direct numerical simulation of particulate flow; computational methods for flow control; splitting methods for viso-plastic flow a la Bingham; and more. It should also be mentioned that most methods discussed in this article are illustrated by the results of numerical experiments, including the simulation of three-dimensional flow. Due to their modularity the methods described in this article are relatively easy to implement - as is demonstrated by the fact that several practitioners in various institutions have been able to use them ab initio for the solution of complicated flow (and other) problems.
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"The book is a remarkable sample of in-depth surveys, which include the most recent trends in mathematical finance.... Reflecting the state of the art in mathematical finance, this handbook is a useful guide in this area for both academics and practitioners."--reviewed in Mathematical ReviewsMore details
Series
Language
English
Place of publication
Oxford
United Kingdom
Publishing group
Elsevier Science & Technology
Target group
Professional and scholarly
Illustrations
Illustrations
ISBN-13
978-0-444-51224-6 (9780444512246)
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Schweitzer Classification
Other editions
Additional editions
Roland Glowinski
Handbook of Numerical Analysis IX
E-Book
07/2003
1st Edition
Elsevier
from
€175.00
Available for download
Content
Chapter I The Navier-Stokes equations for incompressible viscous fluids: derivation of the Navier-Stokes equations for viscous fluids; initial and boundary conditions; a stream function-vorticity formulation of the Navier-Stokes equations; a brief introduction to Sobolev spaces; variational formulations of the Navier-Stokes equations; a short review of mathematical results for the Navier-Stokes equations. Chapter II A family of operator splitting methods for initial value problems - application to the Navier-Stokes equations: a family of initial value problems; the Peaceman-Rachford method; the Douglas-Rachford method; A-scheme; application to the Navier-Stokes equations. Chapter III Iterative solution of the advection-diffusion subproblems: classical and variational formulations of the advection-diffusion subproblems associated with the operator splitting schemes; linear variational problems in Hilbert spaces; variational methods for the solution of the advection-diffusion problems (13.1) and (13.2); conjugate gradient methods for the solution of minimization problems in Hilbert spaces; least-squares solution of linear and nonlinear problems in Hilbert spaces; least-squares/conjugate gradient solution of problems (13.1) and (13.2). Chapter IV Iterative solution of the Stokes subproblems: mathematical properties of the generalized Stokes problem (GS)1; gradient methods for the Stokes problem; conjugate gradient methods for the Stokes problem (GS)1; iterative solution of the generalized Stokes problem (GS)2; on artificial compressibility methods and further comments. Chapter V Finite element approximation of the Navier-Stokes equations: solution of the Stokes problem with periodic boundary conditions; a Fourier analysis of the numerical instability mechanism; finite element implementation of the scheme (11.5)-(11.8); on the numerical solution of the discrete subproblems. Chapter VI Treatment of the advection by a wave-like equation method and by backward methods of characteristics: more on operator-splitting methods; a wave-like equation method for solving the Navier-Stokes equations; solution of the Navier-Stokes equations by backward methods of characteristics; on the treatment of the advection by upwinding. Chapter VII On L2-projection methods for the numerical treatment of the incompressibility: combining L2-projection methods with operator-splitting schemes la Peaceman-Rachford and Douglas-Rachford, and with the scheme; combining L2-projection methods with operator splitting schemes la Marchuk-Yanenko; numerical experiments. Chapter VIII Fictitious domain methods for incompressible viscous flow - application to particulate flow. Chapter X Complements - from stream function-vorticity to flow control.