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A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations


Author: Ross Ingram
Journal: Math. Comp. 82 (2013), 1953-1973
MSC (2010): Primary 65M12, 65M60, 76D05, 76M10, 76M20; Secondary 76D15, 76M25, 65M22
DOI: https://doi.org/10.1090/S0025-5718-2013-02678-6
Published electronically: March 20, 2013
MathSciNet review: 3073187
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Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the stability of a fully-implicit, linearly extrapolated Crank-Nicolson (CNLE) time-stepping scheme for finite element spatial discretization of the Navier-Stokes equations. Although presented in 1976 by Baker and applied and analyzed in various contexts since then, all known convergence estimates of CNLE require a time-step restriction. We propose a new linear extrapolation of the convecting velocity for CNLE that ensures energetic stability without introducing an undesirable exponential Gronwall constant. Such a result is unknown for conventional CNLE for inhomogeneous boundary data (usual techniques fail!). Numerical illustrations are provided showing that our new extrapolation clearly improves upon stability and accuracy from conventional CNLE.


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Additional Information

Ross Ingram
Affiliation: University of Pittsburgh, 615 Thackeray Hall, Pittsburgh Pennsylvania 15260
Address at time of publication: 2259 Shady Avenue, Pittsburgh, Pennsylvania 15217
Email: rni1@psualum.com

DOI: https://doi.org/10.1090/S0025-5718-2013-02678-6
Keywords: Navier-Stokes, Crank-Nicolson, finite element, extrapolation, linearization, implicit, stability, analysis, inhomogeneous
Received by editor(s): June 14, 2011
Received by editor(s) in revised form: January 8, 2012
Published electronically: March 20, 2013
Additional Notes: The author was partially supported by NSF Grants DMS 0508260 and 080385
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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