Absorbing-Generating Boundary Condition for Shallow Water Models
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 123, Issue 6
Abstract
An absorbing-generating boundary condition is derived for the two-dimensional-horizontal nonlinear shallow equations using the method of characteristics. It assumes local superposition of the incoming and outgoing long waves at the boundary, and uses a relationship between the flux and surface elevation of the waves. This boundary condition allows the outgoing waves to leave the computational domain through the boundaries with a minimum of reflection, while specifying incoming waves at the same boundaries. The boundary condition's absorbing properties are tested for both linear and nonlinear waves for a range of amplitudes and of angles of incidence. Its performance is compared to the classical Sommerfeld radiation condition for the linear case and is shown to cause significantly less reflection errors, especially for oblique angles. Also, a case of simultaneous absorption and generation of waves at the same boundary is analyzed where it is shown that the errors are of the same order as for the case of absorption only. Finally, the boundary condition is extended to include known currents.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Abbott, M. B. (1979). Computational hydraulics. Elements of the theory of free surface flow. Pitman Publishing, Ltd., London, U.K.
2.
Atkinson, K. E. (1978). An introduction to numerical analysis. John Wiley & Sons, Inc., New York, N.Y.
3.
Broeze, J., and Van Daalen, E. F. G.(1992). “Radiation boundary conditions for the two-dimensional wave equation from a variational principle.”Math. Comp., 58(197), 73–82.
4.
Engquist, B., and Majda, A.(1977). “Absorbing boundary conditions for the numerical simulation of waves.”Math. Comp., 31(139), 629–651.
5.
Givoli, D.(1991). “Non-reflective boundary conditions.”J. Comp. Phys., 94, 1–29.
6.
Hibberd, S. (1977). “Surf and run-up,” PhD thesis, School of Mathematics, Univ. of Bristol, Bristol, U.K.
7.
Higdon, R. L.(1986). “Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation.”Math. Comp., 47(176), 437–459.
8.
Higdon, R. L.(1987). “Numerical absorbing boundary conditions for the wave equation.”Math. Comp., 49(179), 65–90.
9.
Keys, R. G.(1985). “Absorbing boundary conditions for acoustic media.”Geophysics, 50(6), 892–902.
10.
Kobayashi, N., Otta, A. K., and Roy, I.(1987). “Wave reflection and run-up on rough slopes.”J. Wtrwy., Port, Coast., and Oc. Engrg., ASCE, 113(3), 282–298.
11.
Longuet-Higgins, F. R. S., and Cokelet, E. D.(1976). “The deformation of steep surface waves on water. I. A numerical method of computation.”Proc., Royal Soc. Lond. A., London, England, 350, 1–26.
12.
Sommerfeld, A. (1964). Lectures on theoretical physics. Academic Press, Inc., San Diego.
13.
Strikwerda, J. C. (1989). Finite difference schemes and partial differential equations. Wadsworth & Brooks/Cole Mathematical Series, Wadsworth, Inc., Belmont, Calif.
14.
Svendsen, I. A. (1974). “Cnoidal waves over a gently sloping bottom.”Series paper #6, Inst. of Hydrodynamics and Hydr. Engrg., Tech. Univ. of Denmark, Lyngby, Denmark.
15.
Svendsen, I. A., and Justesen, P. (1984). “Forces on slender cylinders from very high waves and spilling breakers.”Symp., on description and modelling of directional seas, Copenhagen, Denmark, Chap. D-7.
16.
Svendsen, I. A., and Putrevu, U. (1996). “Surf zone hydrodynamics.”Advances in coastal and ocean engineering, 2, P. L.-F. Liu, ed., World Scientific Publishing, Co., Singapore, 1–78.
17.
Van Dongeren, A. R., Sancho, F. E., Svendsen, I. A., and Putrevu, U. (1994). “SHORECIRC: A quasi 3-D nearshore model.”Proc., 24th ICCE, American Society of Civil Engineers, New York, N.Y., 2741–2754.
18.
Van Dongeren, A. R., Svendsen, I. A., and Sancho, F. E. (1995). “Application of the Q-3D SHORECIRC model to surfbeat.”Proc., Coastal Dynamics, American Society of Civil Engineers, New York, N.Y., 233–244.
19.
Verboom, G. K., and Slob, A.(1984). “Weakly-reflective boundary conditions for two-dimensional shallow water flow problems.”Adv. Water Resour., 7, 192–197.
20.
Verboom, G. K., Stelling, G. S., and Officier, M. J. (1981). “Boundary conditions for the shallow water equations.”Engineering applications of computational hydraulics, I, M. B. Abbott and J. A. Cung, eds., Pitman Publishing, Ltd., London, U.K., 230–262.
Information & Authors
Information
Published In
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Nov 1, 1997
Published in print: Nov 1997
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.