Show Summary Details
More options …

# International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

8 Issues per year

IMPACT FACTOR 2016: 0.890

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2016: 0.07

Online
ISSN
2191-0294
See all formats and pricing
More options …
Volume 19, Issue 3-4

# Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities

Chenguang Zhang
• Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
• Other articles by this author:
/ Krishnaswamy Nandakumar
• Corresponding author
• Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA
• Email
• Other articles by this author:
Published Online: 2018-06-05 | DOI: https://doi.org/10.1515/ijnsns-2017-0016

## Abstract

Evaluating the order of accuracy (order) is an integral part of the development and application of numerical algorithms. Apart from theoretical functional analysis to place bounds on error estimates, numerical experiments are often essential for nonlinear problems to validate the estimates in a reliable answer. The common workflow is to apply the algorithm using successively finer temporal/spatial grid resolutions ${\mathrm{\delta }}_{i}$, measure the error ${\text{\isin}}_{i}$ in each solution against the exact solution, the order is then obtained as the slope of the line that fits $\left(log{\text{\isin}}_{i},log{\mathrm{\delta }}_{i}\right)$. We show that if the problem has singularities like divergence to infinity or discontinuous jump, this common workflow underestimates the order if solution at regions around the singularity is used. Several numerical examples with different levels of complexity are explored. A simple one-dimensional theoretical model shows it is impossible to numerically evaluate the order close to singularity on uniform grids.

MSC 2010: 65N08; 65N15

## References

• [1]

P. Shankar and M. Deshpande, Fluid mechanics in the driven cavity. Annual review of fluid mechanics. 32 (2000), 93–136.

• [2]

J. Shen, Hopf bifurcation of the unsteady regularized driven cavity flow. Journal of computational physics. 95 (1991), 228–245.Google Scholar

• [3]

The OpenFOAM Foundation, 2018. OpenFOAM: open source field operation and manipulation library. URL http://www.openfoam.org/

• [4]

M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Courier corporation. 1964.Google Scholar

• [5]

G.K. Batchelor, An introduction to fluid dynamics. Cambridge university press. 631, 1967.Google Scholar

• [6]

X. Nie, M.O. Robbins and S. Chen, Resolving singular forces in cavity flow: multiscale modeling from atomic to millimeter scales. Phys. Rev. Lett. 96 (2006), 2.Google Scholar

Accepted: 2018-02-02

Published Online: 2018-06-05

Published in Print: 2018-06-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 3-4, Pages 281–287, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.