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Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation
Author(s) -
Xu Chuanju,
Lin Yumin
Publication year - 2000
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/(sici)1097-0363(20000330)32:6<619::aid-fld977>3.0.co;2-n
Subject(s) - mathematics , conjugate gradient method , euler's formula , inviscid flow , iterative method , convergence (economics) , euler equations , spectral element method , finite element method , mathematical analysis , mathematical optimization , mixed finite element method , classical mechanics , physics , economics , thermodynamics , economic growth
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.