Implicit TVDLF Methods for Diffusion and Kinematic Flows
Publication: Journal of Hydraulic Engineering
Volume 139, Issue 9
Abstract
Diffusion-wave and kinematic-wave approximations of the St. Venant equations are commonly used in physically based, regional hydrologic models because they have high computational efficiency and use fewer equations. Increasingly, models based on these equations are being applied to cover larger areas of land with different surface and groundwater regimes and complicated topography. Existing numerical methods are not well suited for multiyear simulation of detailed flow behavior unless they can be run efficiently with large time steps and control numerical error. A numerical method also should be able to solve both diffusive and kinematic wave models. A total variation diminishing Lax-Friedrichs type method (TVDLF) that is stable and accurate with both diffusive- and kinematic-wave models and large time steps is presented as a means to address this problem. It uses a linearized conservative implicit formulation that makes it possible to avoid nonlinear iterations. The numerical method was tested successfully using steady flow profiles, analytical solutions for wave propagation, and observed data from a field experiment in a mountain stream of Sri Lanka. A grid convergence test and an error analysis are carried out to determine how the model errors of the numerical schemes behave with the discretization.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors wish to thank Joel VanArman, Zaki Moustafa, Randy Vanzee, Raul Novoa, Sharika Senarath, and Walter Wilcox at the South Florida Water Management District, West Palm Beach, Florida, for reviewing the manuscript and making valuable comments. The authors also wish to thank Mr. J. A. S. A. Jayasinghe, (Director River Basins), Mr. B. S. Layanagama (Deputy Director), Mr. Nilantha Dhanapala (Chief Engineer), and Mr. W. K. Jinadasa (IT Manager) of the Mahaweli Authority of Sri Lanka for supporting the field experiment at the Dambulu River in Sri Lanka.
References
Akan, A. O., and Yen, B. C. (1981). “Diffusion-wave flood routing in channel networks.” J. Hydr. Div., 107(6), 719–731.
Graham, D. N., and Butts, M. B. (2005). “Flexible integrated watershed modeling with MIKE SHE.” Watershed Models, V. P. Singh and D. K. Frevert, eds., CRC Press, Boca Raton, FL, 245–272.
Harten, A. (1983). “High resolution method for hyperbolic conservation laws.” J. Comput. Phys., 49, 357–393.
Henderson, F. M. (1966). Open channel flow, MacMillan, New York.
Hirsch, C. (2007). Numerical computation of internal and external flow, Wiley, New York.
Hromadka, T. V., II, McCuen, R. H., and Yen, C. C. (1987). Computational hydrology in flood control design and planning, Lighthouse.
Kadlec, R. H., and Wallace, S. T. (2009). Treatment wetlands, 2nd Ed., CRC Press, Boca Raton, FL.
Lal, A. M. W. (1998). “Weighted implicit finite-volume model for overland flow.” J. Hydraul. Eng., 124(4), 342–349.
Lal, A. M. W. (2000). “Numerical errors in groundwater and overland flow models.” Water Resour. Res., 36(5), 1237–1247.
Lal, A. M. W. (2008). “Development of a robust diffusion-kinematic flow algorithm for regional models operating with large time steps.” EWRI World Env. and Water Res. Conf., ASCE, Honolulu, HI.
Lal, A. M. W., Van Zee, R., and Belnap, M. (2005). “Case study: A model to simulate regional flow in south Florida.” J. Hydraul. Eng., 131(4), 247–258.
Ponce, V. M. (1991). “The Kinematic Wave Controversy.” J. Hydraul. Eng., 117(4), 511–525.
Ponce, V. M., Li, R. M., and Simons, D. B. (1978). “Applicability of Kinematic and Diffusion Models.” J. Hydr. Div., 104(3), 353–360.
Rodriguez, L. B., Cello, P. A., Vionette, C. A., and Goodrich, D. (2008). “Fully conservative coupling of HEC-RAS with MODFLOW to simulate stream-aquifer interactions in a Drainage Basin.” J. Hydrol., 353(1–2), 129–142.
Rosner, L. A., Shubinski, R. P., and Aldrich, J. A. (1984). “Stormwater management model user’s manual version III, addendum I EXTRAN.”, U.S. Environmental Protection Agency, Cincinnati, OH.
Singh, V. P. (1996). Kinematic wave modeling in water resources, Wiley-IEEE, New York.
Toth, G., Keppens, R., and Botchev, M. A. (1998). “Implicit and semi-implicit schemes in the versatile advection code: Numerical tests.” Astron. Astrophys., 332, 1159–1170.
Toth, G., and Odstrcil, D. (1996). “Comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magneto hydrodynamic problems.” J. Comput. Phys., 128, 82–100.
van Leer, B. (1979). “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method.” J. Comput. Phys., 32(1), 101–136.
Yee, H. C. (1987). “Construction of explicit and implicit symmetric TVD Schemes and their applications.” J. Comput. Phys., 68(1), 151–179.
Yee, H. C. (1989). “A class of high-resolution explicit and implicit shock-capturing methods.”, NASA Ames Research Center, Moffette Field, CA.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 139 • Issue 9 • September 2013
Pages: 974 - 983
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Feb 1, 2012
Accepted: Feb 19, 2013
Published online: Feb 21, 2013
Discussion open until: Jul 21, 2013
Published in print: Sep 1, 2013
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.