On the Numerical Solution of the One-dimensional Shallow Water Equations in Constant-depth Environment

This paper presents a parametric finite-difference scheme concerning the numerical solution of theone-dimensional Boussinesq-type set of equations, as they were introduced byPeregrine, in the case of wavesrelatively long with small amplitudes in water of constant depth. The method which is used can be considered asa generalization of the Crank-Nickolson method and it has been applied successfully to a problem used by Bejiand Battjes. REFERENCES[1] Abbott, M.B, Petersen, H.M. and Skovgaard, O. (1978), “On the numerical modeling of short waves inshallow water”, J. Hydraul. Res., Vol. 16 (3), pp. 173-203.[2] Abbott, M.B., McCowan, A.D. and Warren, I.R. (1984), “Accuracy of short wave numerical models”, J.Hydraul. Eng., Vol. 110(10), pp. 1287-1301.[3] Beji, S. and Battjes, J.A. (1994), “Numerical simulation of nonlinear wave propagation over a bar”, CoastalEng., Vol. 23, pp. 1-16.[4] Beji, S. and Nadaoka, K. (1996), “Formal derivation and numerical modelling of the improved Boussinesqequations for varying depth”, Ocean Eng., Vol. 23, pp. 691-704.[5] Berkhoff, J.C.W. (1972), “Computation of combined refraction-diffraction”, Proceedings of 13 CoastalEngineering Conference, ASCE, Vancouver, Vol. 1, pp. 471-490.[6] Bratsos, A.G. (1998), “The solution of the Boussinesq equation using the method of lines”, Comput.Methods Appl. Mech. Engrg., Vol. 157, pp. 33-44.[7] Bratsos, A.G. (2001), “A parametric scheme for the numerical solution of the Boussinesq equation”, KoreanJ. Comput. Appl. Math., Vol. 8, No. 1, pp. 45-57.[8] Bratsos, A.G., Prospathopoulos, A.M. and Famelis, I.Th. (2006), “On the numerical solution of the one-dimensional shallow sea waves”, Proceedings of 5th MATHMOD (IMACS International Symposium onMathematical Modelling), Vienna University of Technology, Vienna, Austria, 8-10 February. [9] Courant, R., Friedrichs, H. and Lewy (1928), “Uber die partiellen Differenzen-Greichungen dermathematischen Physik”, Mathematische Annalen, Vol. 100, pp. 32-74.[10] Hamdi, S., Enright, W.H., Ouellet, Y. and Schiesser, W.E. (2004), “Exact solutions of extended Boussinesqequations”, Numer. Algorithms, Vol. 37, pp. 165-175.[11] Kirby, J.T. (2003), Boussinesq models and applications to nearshore wave propagation, surf zone processesand wave-induced currents, Advances in Coastal Engineering (Ed. Lakhan, C.), Elsevier, Amsterdam.[12] Madsen, P.A., Murray, R. and Sørensen, O.R. (1991), “A New Form of the Boussinesq Equations withImproved Linear Dispersion Characteristics (Part 1)”, Coastal Eng., Vol. 15, No. 4, pp. 371-388.[13] Madsen, P.A. and Sørensen, O.R. (1992), “A New Form of the Boussinesq Equations with Improved Linear Famelis, I.Th., Prospathopoulos, A.M., Sarantopoulos, S. and Bratsos, A.G. Dispersion Characteristics, Part 2: A Slowly-Varying Bathymetry”, Coastal Eng., Vol. 18, No. 1, pp. 183-204. [14] Madsen, P.A., Sørensen, O.R. and Sch ̈affer, H.A. (1997), “Surf zone dynamics simulated by a Boussinesqtype model. Part I. Model description and cross-shore motion of regular waves”, Coastal Eng., Vol. 32, pp.255-287. [15] Madsen, P.A. and Sch ̈affer, H.A. (1999), A review of Boussinesq-type equations for surface gravitywaves, Advances in Coastal and Ocean Engineering (Ed. Liu, P.L.-F.), Vol. 5, World Scientific Publ., pp.1-95.[16] Peregrine, D.M. (1967), “Long waves on a beach”, J. Fluid Mech., Vol. 27, Issue 4, pp. 815-827. [17] Schäffer, H.A., Madsen, P.A. and Deigaard, R. (1993), “A Boussinesq model for waves breaking in shallowwater”, Coastal Eng., Vol. 20, pp. 185-202. [18] Twizell, E.H. (1984), Computational Methods for Partial Differential Equations, Ellis Horwood Limited,England.[19] Wei, G. and Kirby, T. (1995), “Time-dependent numerical code for extended Boussinesq equations”, J.Waterw. Port Coast. Ocean Eng., ASCE, Vol. 121 (5), pp. 251-261.