An unstructured grid three-dimensional model based on the shallow water equations. (English) Zbl 0965.76061

Summary: A semi-implicit finite difference model based on the three-dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi-implicit algorithm that is robust, stable and very efficient. The resulting model conserves mass, can fit complicated boundaries, and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary.


76M20 Finite difference methods applied to problems in fluid mechanics
76D33 Waves for incompressible viscous fluids
86A05 Hydrology, hydrography, oceanography
86-08 Computational methods for problems pertaining to geophysics
Full Text: DOI


[1] Casulli, J. Comput. Phys. 86 pp 56– (1990) · Zbl 0681.76022
[2] Signell, J. Geophys. Res. 97 pp 15591– (1992)
[3] Cheng, Estuar. Coast. Shelf Sci. 36 pp 235– (1993)
[4] Casulli, Int. J. Numer. Methods Fluids 15 pp 629– (1992) · Zbl 0762.76068
[5] Casulli, Comput. Math. Appl. 27 pp 99– (1994) · Zbl 0796.76052
[6] and ?A description of a three dimensional coastal ocean circulation model?, in (ed.), Three Dimensional Coastal Ocean Circulation Models, Coastal and Estuarine Sciences 4, AGU, Washington, DC, 1987, pp. 1-16.
[7] and ?A non-orthogonal finite difference method for shallow water flow?, in (ed.), Tides in a Salt-Marsh, Febodruk BV, Enschhede, Netherlands, 1997.pp. 29-59.
[8] and Numerical Simulation of Reactive Flow, Elsevier, New York, 1987.
[9] Thacker, J. Phys. Oceanogr. 7 pp 284– (1977)
[10] Walters, Commun. Numer. Methods Eng. 14 pp 931– (1998) · Zbl 0915.76056
[11] (ed.), Finite Element Modeling of Environmental Problems, Wiley, New York, 1995.
[12] and ?A three-dimensional semi-implicit algorithm for environmental flows on unstructured grids?, Proc. Conf. on Numerical Methods for Fluid Dynamics, University of Oxford, 1998.
[13] Rebay, J. Comput. Phys. 106 pp 125– (1993) · Zbl 0777.65064
[14] ?The covolume approach to computing incompressible flows?, in and (eds.), Incompressible Computational Fluid Dynamics, Cambridge University Press, Cambridge, 1993, pp. 295-333. · Zbl 1189.76392
[15] Staniforth, Month. Weather Rev. 114 pp 2078– (1986)
[16] Gross, Int. J. Numer. Methods Fluids 28 pp 157– (1998) · Zbl 0912.76041
[17] and Matrix Computations, 3rd edn, J. Hopkins, London, 1996.
[18] Gravel, Month. Weather Rev. 20 pp 2633– (1992)
[19] Walters, Water Resour. Res. 16 pp 187– (1980)
[20] Henry, Commun. Numer. Methods Eng. 9 pp 555– (1993) · Zbl 0782.65139
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.