0
TECHNICAL PAPERS

Single Grid Error Estimation Using Error Transport Equation

[+] Author and Article Information
Ismail Celik, Gusheng Hu

Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26505-6106E-mail: Ismail.Celik@mail.wvu.edu

J. Fluids Eng 126(5), 778-790 (Dec 07, 2004) (13 pages) doi:10.1115/1.1792254 History: Received June 16, 2003; Revised March 02, 2004; Online December 07, 2004
Copyright © 2004 by ASME
Topics: Equations , Errors
Your Session has timed out. Please sign back in to continue.

References

Richardson,  L. F., 1910, “The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, With an Application to the Stresses in a Masonary Dam,” Philos. Trans. R. Soc. London, Ser. A, 210, pp. 307–357.
Richardson,  L. F., and Gaunt,  J. A., 1927, “The Deferred Approach to the Limit,” Philos. Trans. R. Soc. London, Ser. A, 226, pp. 299–361.
Celik, I., Chen, C. J., Roache, P. J., and Scheurer, G., eds., 1993, Quantification of Uncertainty in Computational Fluid Dynamics, ASME Publ. No. FED-Vol. 158, ASME Fluids Engineering Division Summer Meeting, Washington, DC, 20–24 June.
Roache, P. J., 1993, “A Method for Uniform Reporting of Grid Refinement Studies,” Proc. of Quantification of Uncertainty in Computation Fluid Dynamics, I. Celik et al., eds., ASME Fluids Engineering Division Spring Meeting, Washington, D.C., June 230–240, ASME Publ. No. FED-Vol. 158.
Roache, P. J., 1998, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque.
Celik, I., and Zhang, W.-M., 1993, “Application of Richardson Extrapolation to Some Simple Turbulent Flow Calculations,” Proceedings of the Symposium on Quantification of Uncertainty in Computational Fluid Dynamics, I. Celik et al., eds., ASME Fluids Engineering Division Spring Meeting, Washington, C.C., June 20–24, pp. 29–38.
Celik,  I., and Karatekin,  O., 1997, “Numerical Experiments on Application of Richardson Extrapolation With Nonuniform Grids,” ASME J. Fluids Eng., 119, pp. 584–590.
Stern,  F., Wilson,  R. V., Coleman,  H. W., and Paterson,  E. G., 2001, “Comprehensive Approach to Verification and Validation of CFD Simulations—Part 1: Methodology and Procedures,” ASME J. Fluids Eng., 123, pp. 793–802, December.
Cadafalch,  J., Perez-Segarra,  C. C., Consul,  R., and Oliva,  A., 2002, “Verification of Finite Volume Computations on Steady State Fluid Flow and Heat Transfer,” ASME J. Fluids Eng., 124, pp. 11–21.
Eca, L., and Hoekstra, M., 2002, “An Evaluation of Verification Procedures for CFD Applications,” 24th Symposium on Naval Hydrodynamics, Fukuoka, Japan, July 8–13.
Coleman,  H. W., Stern,  F., Di Mascio,  A., and Campana,  E., 2001, “The Problem With Oscillatory Behavior in Grid Convergence Studies,” ASME J. Fluids Eng., 123, pp. 438–439.
Van Straalen, B. P., Simpson, R. B., and Stubley, G. D., 1995, “A Posteriori Error Estimation for Finite Volume Simulations of Fluid Flow Transport,” Proceedings of the Third Annual Conference of the CFD Society of Canada, Vol. I, Banff, Alberta, 25–27 June, P. A. Thibault and D. M. Bergeron, eds.
Zhang, Z., Trepanier, J. Y., and Camarero, R., 1997, “A Posteriori Error Estimation Method Based on an Error Equation,” Proceedings of the AIAA 13th Computational Fluid, pp. 383–397.
Zhang, Z., Pelletier, D., Trepanier, J. Y., and Camarero, R., 2000, “Verification of Error Estimators for the Euler Equations,” 38th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2000-1001.
Wilson, R. V., and Stern, F., 2002, “Verification and Validation for RANS Simulation of a Naval Surface Combatant,” American Institute of Aeronautics & Astronautics (AIAA) 2000-0904, 40th Aerospace Sciences Meeting & Exhibit, January 14–17.
Celik, I., and Hu, G., 2002, “Discretization Error Estimation Using Error Transport Equation,” Proceedings of ASME FEDSM ’02, paper No. 31372, Montreal, Canada, July 14–18.
Celik, I., Hu, G., and Badeau, A., 2003, “Further Refinement and Benchmarking of a Single-Grid Error Estimation Technique,” Paper No. AIAA-2003-0628, 41st Aerospace Sciences Meeting & Exhibit, Reno, Nevada, January 6–9.
Qin, Y., and Shih, T. I.-P., 2003, “A Method for Estimating Grid-Induced Errors in Finite-Difference and Finite-Volume Methods,” Paper No. AIAA-2003-0845, 41st Aerospace Sciences Meeting & Exhibit, Reno, Nevada, January 6–9.
Hu, G., 2002, “Uncertainty Assessment for CFD Using Error Transport Equation,” Master Thesis, Mechanical & Aerospace Engineering Department, West Virginia University, May.
Ferziger, J. H., 1993, “Estimation and Reduction of Numerical Error,” ASME transaction on the Quantification of Uncertainty in Computational Fluid Dynamics, I. Celik et al., eds., FED-Vol. 158.
Hu, G., and Celik, I., 2004, “On Consistency in Evaluating the Residuals and Derivatives for Finite Difference/Volume Methods,” submitted to Communications in Numerical Methods in Engineering.

Figures

Grahic Jump Location
Illustration of three-point stencil for implementation of boundary condition on a typical boundary, here denoted as the south boundary. (a) First grid node at the boundary, (b) first grid node outside the boundary.
Grahic Jump Location
Typical computational cell arrangement used in finite volume discretization
Grahic Jump Location
Influence circle with radius r in unstructured grid. Nodes filled with black color fall into the influence domain of the node 16.
Grahic Jump Location
Exact versus calculated error for steady 1D convection diffusion equation, first order upwind scheme, 41 nodes, velocity varies to obtain different Peclet numbers, (a) Pe=10, PeΔ=0.25, (b) Pe=20, PeΔ=0.5, (c) Pe=100, PeΔ=2.5, (d) Pe=200, PeΔ=5.0
Grahic Jump Location
Exact versus calculated error for steady 1D convection diffusion equation, central difference scheme, 41 nodes, velocity varies to obtain different Peclet numbers, (a) Pe=10, PeΔ=0.25, (b) Pe=20, PeΔ=0.5
Grahic Jump Location
Exact versus calculated error for 2D Poisson equation, central difference scheme, 21* 21 grid; (a) exact error, (b) calculated error
Grahic Jump Location
Exact versus calculated error for 2D Poisson equation along the line of y=0.5, central difference scheme, (a) 21* 21 grid, (b) 41* 41 grid
Grahic Jump Location
Exact versus calculated error for 2D convection diffusion equation, 1st order upwind scheme, 41* 41 grid; (a) exact error, (b) calculated error
Grahic Jump Location
Exact versus calculated error for 2D convection diffusion equation along the diagonal of the square domain, 1st order upwind scheme, x′ =distance from the origin along the diagonal line, (a) 41* 41 grid, (b) 81* 81 grid
Grahic Jump Location
Exact versus calculated error for 2D convection diffusion equation, central difference scheme, 41* 41 grid; (a) exact error, (b) calculated error
Grahic Jump Location
Exact versus calculated error for 2D convection diffusion equation, central difference scheme, (a) 41* 41 grid, (b) 81* 81 grid
Grahic Jump Location
Exact versus calculated error for steady 1D Burger’s equation, first order upwind scheme, 41 nodes, inlet velocity varies to obtain different Reynolds numbers, (a) Re=10, ReΔ=0.25, (b) Re=200, ReΔ=5
Grahic Jump Location
Exact versus calculated error for steady 1D Burger’s equation, central difference scheme, 41 nodes, inlet velocity varies to obtain different Peclet numbers, (a) Re=10, ReΔ=0.25, (b) Re=60, ReΔ=1.5

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In