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Unconditional long-time stability of a velocity–vorticity method for the 2D Navier–Stokes equations

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Abstract

We prove unconditional long-time stability for a particular velocity–vorticity discretization of the 2D Navier–Stokes equations. The scheme begins with a formulation that uses the Lamb vector to couple the usual velocity–pressure system to the vorticity dynamics equation, and then discretizes with the finite element method in space and implicit–explicit BDF2 in time, with the vorticity equation decoupling at each time step. We prove the method’s vorticity and velocity are both long-time stable in the \(L^2\) and \(H^1\) norms, without any timestep restriction. Moreover, our analysis avoids the use of Gronwall-type estimates, which leads us to stability bounds with only polynomial (instead of exponential) dependence on the Reynolds number. Numerical experiments are given that demonstrate the effectiveness of the method.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA

    Timo Heister & Leo G. Rebholz

  2. Department of Mathematics, University of Houston, Houston, TX, 77004, USA

    Maxim A. Olshanskii

Authors
  1. Timo Heister
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  2. Maxim A. Olshanskii
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  3. Leo G. Rebholz
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Corresponding author

Correspondence to Leo G. Rebholz.

Additional information

T. Heister was partially supported by the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation under Award No. EAR-0949446 and The University of California—Davis and NSF Grant DMS1522191. M. A. Olshanskii was partially supported by Army Research Office Grant 65294-MA and NSF Grant DMS1522192. L. G. Rebholzz was partially supported by Army Research Office Grant 65294-MA and NSF Grant DMS1522191.

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Heister, T., Olshanskii, M.A. & Rebholz, L.G. Unconditional long-time stability of a velocity–vorticity method for the 2D Navier–Stokes equations. Numer. Math. 135, 143–167 (2017). https://doi.org/10.1007/s00211-016-0794-1

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  • DOI: https://doi.org/10.1007/s00211-016-0794-1

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