Abstract
For the time dependent flow simulation, despite the existence of extensive literature dealing with either the convection or the unsteady terms, their interaction has not been adequately investigated. In order to shed more light on this important issue, three time stepping methods, including the first-order backward Euler scheme, the second-order Crank-Nicolson scheme, and the third-order Adams-Bashforth/Adams-Moulton predictor-corrector scheme are studied along with four convection schemes, including the first-order upwind, the second-order upwind, the second-order central differencing, and QUICK schemes. The Burgers equation of both linear and nonlinear forms is used as the test problem, aided by the von Neumann stability analysis and the FFT spectral analysis. The results indicate that a second-or higher-order accuracy for both time and space discretizations can produce satisfactory results for smooth solution profiles. Overall, among the schemes tested, either a combination of first-order upwind for convection and Crank-Nicolson for time, or a combination of second-order upwind for convection and backward Euler for time performs better. It appears that by selectively utilizing the dispersive and diffusive characteristics of the time stepping and convection schemes in complementary manners, overall accuracy can be improved.
Similar content being viewed by others
References
Anderson, D. A.; Tannehill, T. C.; Pletcher, R. H. (1984): Computational fluid mechanics and heat transfer. Washington D.C.: Hemisphere
Benton, E. R.; Platzman, G. W. (1972): A table of the solutions of the one-dimensional Burgers equation. Quart. Appl. Math. 30, 195–212
Correa, S. M.; ShyyW. (1987): Computational models and methods for continuous gaseous turbulent combustion. Prog. in Energy Combust. Sci. 13, 249–292
Fletcher, C. A. J. (1988): Computational techniques for fluid dynamics. Vols I and II. Berlin, Heidelberg, New York: Springer
Hirsch, C. (1990): Numerical computation of internal and external flows. Vol. 2, New York: Wiley
Leonard, B. P. (1979): A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Eng. 19, 59–98
Lohar, B. L.; Jain, P. C. (1981): Variable mesh cubic spline technique for n-wave solution of Burgers' equation. J. Comput. Phys. 39, 433–442
Morton, K. W. (1971): Stability and convergence in fluid flow problems. Proc. Roy. Soc. Lond. A. 323, 237–253
Roe, P. L. (1986): Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mech. 18, 337–365
Shyy, W. (1985): A study of finite difference approximations to steady-state, convection-dominated flow problems. J. Comput. Phys. 57(3), 415–438
Shyy, W. (1984): A note on assessing finite difference procedure for large Peclet/Reynolds number flow calculation. In MillerJ. J. H. (ed.): Boundary and interior layers: computational and asymptotic methods, Vol. 3, pp. 303–308. Dublin: Boole Press
Warming, R.; Hyett, B. J. (1974): The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comput. Phys. 14, 159–179
Warming, R. F.; Kutler, P.; Lomax, H. (1973): Second-and third-order noncentral difference schemes for nonlinear hyperbolic equations. AIAA J. 11, 189–196
Author information
Authors and Affiliations
Additional information
Communicated by S. N. Atluri, September 17, 1991
Rights and permissions
About this article
Cite this article
Shyy, W., Liang, SJ. Interaction of time stepping and convection schemes for unsteady flow simulation. Computational Mechanics 9, 285–304 (1992). https://doi.org/10.1007/BF00370036
Issue Date:
DOI: https://doi.org/10.1007/BF00370036