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Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations∗

Published online by Cambridge University Press:  01 April 2014

William Layton ,
Nathaniel Mays ,
Monika Neda  and
Catalin Trenchea
William Layton
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA. . wjl@pitt.edu
Nathaniel Mays
Affiliation:
Department of Mathematics, Wheeling Jesuit University, Wheeling, WV, 26003, USA.; nmays@wju.edu
Monika Neda
Affiliation:
Department of Mathematical Sciences, University of Nevada Las Vegas, USA.; Monika.Neda@unlv.edu
Catalin Trenchea
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA. ; trenchea@pitt.edu
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Abstract

We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing BDF2 time discretization in Step 1.1, and (ii) more general (linear or nonlinear) regularization operators in Step 1.1. We give a complete stability analysis, derive conditions on the Step 1.1 regularization operator for which the combination has good stabilization effects, characterize the numerical dissipation induced by Step 1.1, prove an asymptotic error estimate incorporating the numerical error of the method used in Step 1.1 and the regularizations consistency error in Step 1.1 and provide numerical tests.

Keywords

Modular regularizationBDF2 time discretizationNavier-Stokes equationsturbulencefinite element method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 3 , May 2014 , pp. 765 - 793
Copyright
© EDP Sciences, SMAI 2014

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