Commun. Comput. Phys., 16 (2014), pp. 817-840.


Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

E. Ferrer 1*, D. Moxey 2, R. H. J. Willden 3, S. J. Sherwin 2

1 ETSIA -- School of Aeronautics, Universidad Politecnica de Madrid, Pza. Cardenal Cisneros 3, Madrid, E-28040, Spain.
2 Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK.
3 Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK.

Received 29 January 2014; Accepted (in revised version) 17 April 2014
Available online 17 July 2014
doi:10.4208/cicp.290114.170414a

Abstract

This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used. Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant-Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

AMS subject classifications: 76D05, 35Q30, 65N30

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Key words: Incompressible Navier-Stokes equations, projection methods, velocity-correction, continuous Galerkin, discontinuous Galerkin, inf-sup LBB condition.

*Corresponding author.
Email: esteban.ferrer@upm.es (E. Ferrer), d.moxey@imperial.ac.uk (D. Moxey), richard.willden@eng.ox.ac.uk (R. H. J. Willden), s.sherwin@imperial.ac.uk (S. J. Sherwin)
 

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