Advances in Numerical Analysis
Volume 2012 (2012), Article ID 162539, 32 pages
http://dx.doi.org/10.1155/2012/162539
Research Article

Convergence Analysis of a Fully Discrete Family of Iterated Deconvolution Methods for Turbulence Modeling with Time Relaxation

R. Ingram,1 C. C. Manica,2 N. Mays,3 and I. Stanculescu4

1Department of Mathematics, University of Pittsburgh, PA 15260, USA
2Departmento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre 91509-900, RS, Brazil
3Department of Mathematics, Wheeling Jesuit University, WV 26003, USA
4Farquhar College of Arts and Sciences, Nova Southeastern University, FL 33314, USA

Received 31 May 2012; Accepted 25 July 2012

Academic Editor: William J. Layton

Copyright © 2012 R. Ingram et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a general theory for regularization models of the Navier-Stokes equations based on the Leray deconvolution model with a general deconvolution operator designed to fit a few important key properties. We provide examples of this type of operator, such as the (modified) Tikhonov-Lavrentiev and (modified) Iterated Tikhonov-Lavrentiev operators, and study their mathematical properties. An existence theory is derived for the family of models and a rigorous convergence theory is derived for the resulting algorithms. Our theoretical results are supported by numerical testing with the Taylor-Green vortex problem, presented for the special operator cases mentioned above.