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

¹ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA.
wjl@pitt.edu
² Department of Mathematics, Wheeling Jesuit University, Wheeling, WV, 26003, USA.
nmays@wju.edu
³ Department of Mathematical Sciences, University of Nevada Las Vegas, USA.
Monika.Neda@unlv.edu
⁴ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA.
trenchea@pitt.edu

Received: 20 October 2011
Revised: 3 July 2013

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.