Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, p. 765-793

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.

DOI : https://doi.org/10.1051/m2an/2013120
Classification:  35Q30,  76F65
Keywords: modular regularization, BDF2 time discretization, Navier-Stokes equations, turbulence, finite element method
@article{M2AN_2014__48_3_765_0,
author = {Layton, William and Mays, Nathaniel and Neda, Monika and Trenchea, Catalin},
title = {Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {3},
year = {2014},
pages = {765-793},
doi = {10.1051/m2an/2013120},
zbl = {1293.35210},
mrnumber = {3264334},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_3_765_0}
}

Layton, William; Mays, Nathaniel; Neda, Monika; Trenchea, Catalin. Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 3, pp. 765-793. doi : 10.1051/m2an/2013120. http://www.numdam.org/item/M2AN_2014__48_3_765_0/

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