Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems....

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Bibliographic Details
Published in:International journal for numerical methods in fluids 2000-03, Vol.32 (6), p.619-646
Main Authors: Xu, Chuanju, Lin, Yumin
Format: Article
Language:English
Subjects:
Algorithms
Approximation theory
Computational complexity
Computational methods in fluid dynamics
Convergence of numerical methods
Exact sciences and technology
Fluid dynamics
Fundamental areas of phenomenology (including applications)
global Uzawa algorithm
Incompressible flow
Iterative methods
Mathematical models
Navier Stokes equations
Navier-Stokes/Euler coupled equations
Physics
pre-conditioned conjugate gradient method
spectral element approximation
Variational techniques
Viscous flow
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Summary:Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.
ISSN:0271-2091
1097-0363
DOI:10.1002/(SICI)1097-0363(20000330)32:6<619::AID-FLD977>3.0.CO;2-N