Method to Solve 1D Unsteady Transport and Flow Equations
Publication: Journal of Hydraulic Engineering
Volume 121, Issue 5
Abstract
A method of solution for a fixed and nonstaggered grid is proposed. This method is obtained by a modification of the standard method of integration applied in the Galerkin procedure. The proposed approach can be used to the linear basis functions. For approximation of any function in an element, the weighted averages of weighting parameter ω are used. This approach yields a six-point implicit scheme that can be used either for the transport equation with dominated advection or for the full flow equations and their simplifications. It ensures a uniform approach to the solution of both problems. The particular cases of this method are the various well-known difference schemes and the standard finite-element method. The Fourier analysis carried out for the linear equations showed that the proposed approach has advantageous numerical properties. It has been confirmed by calculations carried out for the different kinds of transport and flow.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Adey, R. A., and Brebbia, C. A. (1974). “Finite-element solution for effluent dispersion.”Numerical methods in fluid dynamics, C. A. Brebbia and J. J. Connor, eds., Pentech Press, Ltd., London, England, 325–354.
2.
Bentley, L. R., and Pinder, G. F.(1992). “Eulerian-Langrangian solution of the vertically averaged groundwater transport equation.”Water Resour. Res., 28(11), 3011–3020.
3.
Cunge, J., Holly, F. M. Jr., and Verwey, A. (1980). Practical aspect of computational river hydraulics . Pitman Publishing, Ltd., London, England.
4.
Donea, J. (1984). “A Taylor-Galerkin method for convective transport problems.”Int. J. Numerical Methods in Engrg., Vol. 20, 101–119.
5.
Fletcher, C. A. J. (1991). Computational techniques for fluid dynamics, Vol. I . Springer-Verlag, New York, N.Y.
6.
Holly, F. Jr., and Preissmann, A.(1977). “Accurate calculation of transport in two dimensions.”J. Hydr. Engrg., ASCE, 103(11), 1259–1277.
7.
Katopodes, N. D.(1984). “A dissipative Galerkin scheme for open-channel flow.”J. Hydr. Engrg., ASCE, 110(4), 450–466.
8.
Kinnmark, I. P. E., and Gray, W. G. (1986). “A wave equations formulation of river flow.”Finite elements in water resources, A. Sa da Costa, A. Melo Baptista, W. G. Gray, C. A. Brebbia, and T. F. Pinder, eds., Springer-Verlag, Berlin, Germany, 599–605.
9.
Liggett, J. A., and Cunge, J. A. (1975). “Numerical methods of solution of the unsteady flow equations.”Unsteady flow in open channels, K. Mahmood and V. Yevjevich, eds., Water Res. Pub., Fort Collins, Colo., 89–182.
10.
Lynch, D. R., and Gray, W. G. (1979). “A wave equation model for finite element computations.”Computers and Fluids, Vol. 7, 207–228.
11.
Neuman, S. P. (1984). “Adaptive Eulerian-Lagrangian finite element method for advection-dispersion.”Int. J. Numerical Methods in Engrg., Vol. 20, 321–327.
12.
Szymkiewicz, R. (1991). “Finite-element method for the solution of the Saint Venant equations in an open channel network.”J. Hydro., Vol. 122, 275–287.
13.
Szymkiewicz, R.(1993). “Solution of the advection-diffusion equation using the spline function and finite elements.”Communications in numerical methods in engineering, 9(3), 197–206.
14.
Tan Weiyan (1992). Shalow water hydrodynamics . Elsevier, Amsterdam, The Netherlands.
15.
Yeh, G. T., Chang, J. R., and Short, T. E.(1992). “An exact peak capturing and oscillation—free scheme to solve advection-dispersion transport equations.”Water Resour. Res., 28(11), 2937–2951.
16.
Zienkiewicz, O. (1972). The finite element method . Arkady, Warsaw, Poland (Polish edition).
Information & Authors
Information
Published In
Copyright
Copyright © 1995 American Society of Civil Engineers.
History
Published online: May 1, 1995
Published in print: May 1995
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.