Numerical solution of Boussinesq systems of the Bona-Smith family

Applied Numerical Mathematics

Volumn 60, Issue 4, 2010, Pages 314-336

  Antonopoulos, D C ^a   Dougalis, V A ^a,b   Mitsotakis, D E ^c  

^a UNIVERSITY OF ATHENS   (Greece)

^b INSTITUTE OF COMPUTER SCIENCE   (Greece)

^c UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Bona Smith systems; Boussinesq approximation; Error estimates; Fully discrete Galerkin finite element methods; Initial boundary value problems; Numerical methods; Solitary waves; Water waves

Indexed keywords

BOUSSINESQ APPROXIMATIONS; ERROR ESTIMATES; FULLY DISCRETE; GALERKIN; INITIAL-BOUNDARY VALUE PROBLEMS; SOLITARY WAVE; SOLITARY WAVES;

COASTAL ENGINEERING; FINITE ELEMENT METHOD; FLUID DYNAMICS; GALERKIN METHODS; HYDRODYNAMICS; INITIAL VALUE PROBLEMS; NUMBER THEORY; RUNGE KUTTA METHODS; SOLITONS; WATER WAVES;

NUMERICAL METHODS;

EID: 77950298355     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2009.03.002     Document Type: Article

References (39)

1. D.C. Antonopoulos, The Boussinesq system of equations: Theory and numerical analysis, Ph.D. Thesis, University of Athens, 2000 (in Greek)
2. D.C. Antonopoulos, V.A. Dougalis, Galerkin methods for the Bona-Smith version of the Boussinesq equations, in: P.S. Theocaris (Ed.), Proceedings of the Fifth National Congress on Mechanics, Ioannina, Greece, 1998, pp. 1001-1008
3. Antonopoulos D.C., and Dougalis V.A. Numerical approximations of Boussinesq systems. In: Bermudez A., Gomez D., Hazard C., Joly P., and Roberts J. (Eds). Proceedings of the 5th International Conference on Mathematical and Numerical Aspects of Wave Propagation (2000), SIAM, Philadelphia 265-269
4. Antonopoulos D.C., Dougalis V.A., and Mitsotakis D.E. Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems. Adv. Differential Equations 14 (2009) 27-53
5. Baker G., Dougalis V.A., and Karakashian O.A. Convergence of Galerkin approximation for the Korteweg-deVries equation. Math. Comp. 40 (1983) 419-433
6. Bona J.L., and Chen M. A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D 116 (1998) 191-224
7. Bona J.L., Chen M., and Saut J.-C. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283-318
8. Bona J.L., Chen M., and Saut J.-C. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory. Nonlinearity 17 (2004) 925-952
9. Bona J.L., Dougalis V.A., Karakashian O.A., and McKinney W.R. Conservative high-order numerical schemes for the generalized Korteweg-deVries equation. Phil. Trans. R. Soc. London A 351 (1995) 107-164
10. Bona J.L., Dougalis V.A., and Mitsotakis D.E. Numerical solution of KdV-KdV systems of Boussinesq equations I. The numerical scheme and generalized solitary waves. Math. Comp. Simul. 74 (2007) 214-228
11. Bona J.L., Pritchard W.G., and Scott L.R. Solitary-wave interaction. Phys. Fluids 29 (1980) 438-441
12. Bona J.L., and Smith R. A model for the two-way propagation of water waves in a channel. Math. Proc. Cambridge Philos. Soc. 79 (1976) 167-182
13. Chatzipantelidis P. Explicit multistep methods for nonstiff partial differential equations. Appl. Numer. Math. 27 (1998) 13-31
14. Chen M. Exact traveling-wave solutions to bi-directional wave equations. Int. J. Theor. Phys. 37 (1998) 1547-1567
15. Chen M. Solitary-wave and multipulsed traveling-wave solutions of Boussinesq systems. Appl. Anal. 75 (2000) 213-240
16. Craig W., Guyenne P., Hammack J., Henderson D., and Sulem C. Solitary water wave interactions. Phys. Fluids 18 (2006) 57-106
17. DeBoor C., and Fix G.J. Spline approximation by quasiinterpolants. J. Approx. Theory 8 (1973) 19-45
18. Dougalis V.A., Duran A., Lopez-Marcos M.A., and Mitsotakis D.E. A numerical study of the stability of solitary waves of the Bona-Smith system. J. Nonlinear Sci. 17 (2007) 569-607
19. Dougalis V.A., and Karakashian O.A. On some high-order accurate fully discrete Galerkin methods for the Korteweg-deVries equation. Math. Comp. 45 (1985) 329-345
20. Dougalis V.A., and Mitsotakis D.E. Solitary waves of the Bona-Smith system. In: Fotiadis D., and Massalas C. (Eds). Advances in Scattering Theory and Biomedical Engineering (2004), World Scientific, Singapore 286-294
21. Dougalis V.A., and Mitsotakis D.E. Theory and numerical analysis of Boussinesq systems: A review. In: Kampanis N.A., Dougalis V.A., and Ekaterinaris J.A. (Eds). Effective Computational Methods in Wave Propagation (2008), Chapman and Hall/CRC, Boca Raton 63-110
22. Dupont T. Galerkin methods for first order hyperbolics: An example. SIAM J. Numer. Anal. 10 (1973) 890-899
23. Fenton J.D. A ninth-order solution for the solitary wave. J. Fluid Mech. 53 (1972) 257-271
24. Lax P.D. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21 (1968) 467-490
25. D.E. Mitsotakis, Theory and numerical analysis of nonlinear dispersive wave equations: Boussinesq systems in one and two dimensions, Ph.D. Thesis, University of Athens, 2007 (in Greek)
26. Pego R.L., and Weinstein M.I. Convective linear stability of solitary waves for the Boussinesq equations. Stud. Appl. Math. 99 (1997) 311-375
27. B. Pelloni, Spectral methods for the numerical solution of nonlinear solution dispersive wave equations, Ph.D. Thesis, Yale University, 1996
28. Pelloni B., and Dougalis V.A. Numerical modelling of two-way propagation of nonlinear dispersive waves. Math. Comput. Simul. 55 (2001) 595-606
29. Renouard D.P., Santos F.J.S., and Temperville A.M. Experimental study of the generation, damping, and reflexion of a solitary wave. Dynamics of Atmospheres and Oceans 9 (1985) 341-358
30. Russell J.S. Report on waves. 14th Meeting of the British Association for the Advancement of Science (1844), John Murray, London 311-390
31. Schreiber R. Finite element methods of high-order accuracy for singular two-point boundary value problems with nonsmooth solutions. SIAM J. Numer. Anal. 17 (1980) 546-566
32. Su C.-H., and Mirie R.M. On head-on collisions between two solitary waves. J. Fluid Mech. 98 (1980) 509-525
33. Su C.-H., and Zou Q.-S. Waves generated by collisions of solitary waves. Phys. Rev. A 35 (1987) 4738-4742
34. Thomée V., and Wahlbin L.B. Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable. Numer. Math. 41 (1983) 345-371
35. Thomée V., and Wendroff B. Convergence estimates for Galerkin methods for variable coefficient initial value problems. SIAM J. Numer. Anal. 11 (1974) 1059-1068
36. Toland J.F. Solitary wave solutions for a model of the two-way propagation of water waves in a channel. Math. Proc. Cambridge Philos. Soc. 90 (1981) 343-360
37. Toland J.F. Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waver. Commun. Math. Phys. 94 (1984) 239-254
38. Toland J.F. Existence of symmetric homoclinic orbits for systems of Euler-Lagrange equations. Proceedings of Symposia in Pure Mathematics vol. 45 (1986), American Mathematical Society, Providence, RI 447-459
39. Winther R. A finite element method for a version of the Boussinesq equation. SIAM J. Numer. Anal. 19 (1982) 561-570