Stability and Accuracy of Weighted Four-Point Implicit Finite Difference Schemes for Open Channel Flow
Publication: Journal of Hydraulic Engineering
Volume 128, Issue 3
Abstract
The unsteady motion of streams, with a free surface, has been described by a system of equations that were proposed by Saint-Venant in 1871. These are universally known as “Saint-Venant’s equations.” Weighted four-point implicit finite difference schemes are largely used for their numerical solution in the case of one-dimensional flow. For these models, stability, dissipation, and dispersion are first investigated by looking at the truncation error and then by using Fourier’s classic linear analysis. Variations of the space and time weighting coefficients, the Courant number, the Froude number, and the frictional resistance term are examined in this paper. In particular, instabilities are analyzed that are due to the progressive accumulation of dispersion errors, which are the consequence of an increasing frictional resistance term. However, in this case the numerical scheme requires a dissipation mechanism. Computational suggestions are given for this mechanism.
Get full access to this article
View all available purchase options and get full access to this article.
References
Abbott, M. B., and Basco, D. R. (1989). Computational fluid dynamics: An introduction for engineers, Longman Scientific & Technical, London.
De Saint-Venant, B.(1871). “Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit.” Acad. Sci., Paris, C. R., 73, 147–154, 237–240 (in French).
Evans, E. P. (1977). “The behaviour of a mathematical model of open channel.” 17th Congress of the IAHR, Baden-Baden, Vol. 2, 173–180.
Fread, D. L. (1974). “Numerical properties of implicit four-point finite difference equations of unsteady flow.” NOAA Technical Memorandum NWS HYDRO-18.
Henderson, F. M. (1966). Open channel flow, Macmillan, New York.
Hicks, F. E., and Steffler, P. M. (1990). “Finite element modeling of open channel flow.” Water Resources Engineering, Rep. WRE 90-6, Dept. of Civil Engineering, Univ. of Alberta, Edmont, Alberta, Canada.
Hicks, F. E., and Steffler, P. M.(1994). “Comparison of finite element methods for the St. Venant equations.” Int. J. Numer. Methods Fluids, 20, 99–113.
Leendertse, J. J. (1967). “Aspects of a computational model for long-period water-wave propagation.” Memorandum RM-5294-PR, Rand Corporation, Santa Monica, Calif.
Liggett, J. A. (1975a). “Basic equations of unsteady flow.” Water Resources Publications, Ed. Mahmood, Yevjevich, Fort Collins, Colo., Vol. 1, 29–62.
Liggett, J. A. (1975b). “Stability.” Water Resources Publications, Ed. Mahmood, Yevjevich, Fort Collins, Colo., Vol. 1, 259–282.
Liggett, J. A., and Cunge, J. A. (1975). “Numerical methods of solution of the unsteady flow equations.” Water Resources Publications, Ed. Mahmood, Yevjevich, Fort Collins, Colo. Vol. 1, 89–182.
Lyn, D. A., and Goodwin, P.(1987). “Stability of a general Preissmann scheme.” J. Hydraul. Eng., 113(1), 16–28.
Meselhe, E. A., and Holly, F. M., Jr.(1997). “Invalidity of Preissmann scheme for transcritical flow.” J. Hydraul. Eng., 123(7), 652–655.
Preissmann, A. (1961). “Propagation des intumescences dans les canaux et rivières.” First Congress of the French Association for Computation, Grenoble, France, 433–442 (in French).
Richtmyer, R. D., and Morton, K. W. (1967). Difference methods for initial-value problems, Wiley, New York.
Samuels, P. G., and Skeels, C. P.(1990). “Stability limits for Preissmann’s scheme.” J. Hydraul. Eng., 116(8), 997–1012.
Skeels, C. P., and Samuels, P. G. (1989). “Stability and accuracy analysis of numerical schemes modelling open channel flow.” Proc., HYDROCOMP 89, Elsevier Applied Science, London, 148–157.
Information & Authors
Information
Published In
Copyright
Copyright © 2002 American Society of Civil Engineers.
History
Received: Apr 28, 2000
Accepted: Aug 29, 2001
Published online: Mar 1, 2002
Published in print: Mar 2002
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.