Handbook of Numerical Analysis: Numerical methods for fluids (pt. 3)

Handbook of Numerical Analysis: Numerical methods for fluids (pt. 3)

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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. 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. 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.... time discretization of the following NavierStokes equations: 3u , 1 , h(u- V)u- vAu+-Vp = f ini2x(0, r), (29.6) dt p V-u=0 in. ... dx - I plv a–i yh dx = f f\-vhdx+l g\h- vhdrh, VviSVo/, , (29.15.1) JQt, J Hi, / V a€c u\qh dx = 0, Wqh e Ph, Section 29 All Finiteanbsp;...

Title:Handbook of Numerical Analysis: Numerical methods for fluids (pt. 3)
Author:Philippe G. Ciarlet, Jacques-Louis Lions
Publisher:Gulf Professional Publishing - 1990


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