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Adaptive Stokes Preconditioning for Steady Incompressible Flows

Part of: Hydrodynamic stability Numerical problems in dynamical systems Approximation methods and numerical treatment of dynamical systems Numerical linear algebra Incompressible viscous fluids Applications - Dynamical systems and ergodic theory Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  21 June 2017

Cédric Beaume
Cédric Beaume*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
*
*Corresponding author. Email address:c.m.l.beaume@leeds.ac.uk (C. Beaume)
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Abstract

This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier–Stokes flows based on Stokes preconditioning [42] which has been used successfully in studies of pattern formation in convection. The preconditioner takes the form of the Helmholtz operator I–ΔtL which maps the identity (no preconditioner) for Δt≪1 to Laplacian preconditioning for Δt≫1. It is built on a first order Euler time-discretization scheme and is part of the family of matrix-free methods. The preconditioner is tested on two fluid configurations: three-dimensional doubly diffusive convection and a two-dimensional projection of a shear flow. In the former case, it is found that Stokes preconditioning is more efficient for , away from the values used in the literature. In the latter case, the simple use of the preconditioner is not sufficient and it is necessary to split the system of equations into two subsystems which are solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications and the flexibility of the approach described, this preconditioner is expected to help in a wide range of applications.

Keywords

Fluid dynamicshydrodynamic stabilitynumerical continuationpreconditioningStokes preconditioningdoubly diffusive convectionshear flows

MSC classification

Secondary: 35Q30: Navier-Stokes equations 35Q35: PDEs in connection with fluid mechanics 37M20: Computational methods for bifurcation problems 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology 65F08: Preconditioners for iterative methods 65P30: Bifurcation problems 76D05: Navier-Stokes equations 76E05: Parallel shear flows 76E06: Convection
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 494 - 516
Copyright
Copyright © Global-Science Press 2017 

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