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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

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

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


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 δi, measure the error \isini in each solution against the exact solution, the order is then obtained as the slope of the line that fits (log\isini,logδi). 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.

Keywords: numerical error estimates; finite volume method; impact of singularity

MSC 2010: 65N08; 65N15


  • [1]

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

  • [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

About the article

Received: 2017-01-19

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, DOI: https://doi.org/10.1515/ijnsns-2017-0016.

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