Error estimates for Galerkin approximations of the “classical” Boussinesq system

Antonopoulos, D.; Dougalis, V.

doi:10.1090/S0025-5718-2012-02663-9

Error estimates for Galerkin approximations of the “classical” Boussinesq system
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by D. C. Antonopoulos and V. A. Dougalis PDF

Math. Comp. 82 (2013), 689-717 Request permission

Abstract:

We consider the “classical” Boussinesq system in one space dimension and its symmetric analog. These systems model two-way propagation of nonlinear, dispersive long waves of small amplitude on the surface of an ideal fluid in a uniform horizontal channel. We discretize an initial-boundary-value problem for these systems in space using Galerkin-finite element methods and prove error estimates for the resulting semidiscrete problems and also for their fully discrete analogs effected by explicit Runge-Kutta time-stepping procedures. The theoretical orders of convergence obtained are consistent with the results of numerical experiments that are also presented.

References

Karine Adamy, Existence of solutions for a Boussinesq system on the half line and on a finite interval, Discrete Contin. Dyn. Syst. 29 (2011), no. 1, 25–49. MR 2725279, DOI 10.3934/dcds.2011.29.25
Charles J. Amick, Regularity and uniqueness of solutions to the Boussinesq system of equations, J. Differential Equations 54 (1984), no. 2, 231–247. MR 757294, DOI 10.1016/0022-0396(84)90160-8
D.C. Antonopoulos, The Boussinesq system of equations: Theory and numerical analysis, Ph.D. Thesis, University of Athens, 2000 (In Greek).
D. C. Antonopoulos and V. A. Dougalis, Numerical solution of the ‘classical’ Boussinesq system, Math. Comput. Simulation 82 (2012), no. 6, 984–1007. MR 2903340, DOI 10.1016/j.matcom.2011.09.006
D. C. Antonopoulos and V. A. Dougalis, Numerical approximations of Boussinesq systems, Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000) SIAM, Philadelphia, PA, 2000, pp. 265–269. MR 1785905
D.C. Antonopoulos and V.A. Dougalis, Notes on error estimates for Galerkin approximations of the “classical” Boussinesq system and related hyperbolic problems, arXiv:1008.4248.
D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, Initial-boundary-value problems for the Bona-Smith family of Boussinesq systems, Adv. Differential Equations 14 (2009), no. 1-2, 27–53. MR 2478928
D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of Boussinesq systems of the Bona-Smith family, Appl. Numer. Math. 60 (2010), no. 4, 314–336. MR 2607793, DOI 10.1016/j.apnum.2009.03.002
D.C. Antonopoulos, V.A. Dougalis, and D.E. Mitsotakis, Galerkin approximations of periodic solutions of Boussinesq systems, Bull. Greek Math. Soc. 57(2010), 13–30.
Jerry L. Bona and Min Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Phys. D 116 (1998), no. 1-2, 191–224. MR 1621920, DOI 10.1016/S0167-2789(97)00249-2
J. L. Bona, M. Chen, and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci. 12 (2002), no. 4, 283–318. MR 1915939, DOI 10.1007/s00332-002-0466-4
J. L. Bona, M. Chen, and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory, Nonlinearity 17 (2004), no. 3, 925–952. MR 2057134, DOI 10.1088/0951-7715/17/3/010
Jerry L. Bona, Thierry Colin, and David Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 373–410. MR 2196497, DOI 10.1007/s00205-005-0378-1
J. L. Bona, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves, Math. Comput. Simulation 74 (2007), no. 2-3, 214–228. MR 2307859, DOI 10.1016/j.matcom.2006.10.004
J. L. Bona, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of Boussinesq systems of KdV-KdV type. II. Evolution of radiating solitary waves, Nonlinearity 21 (2008), no. 12, 2825–2848. MR 2461042, DOI 10.1088/0951-7715/21/12/006
V. A. Dougalis, A. Durán, M. A. López-Marcos, and D. E. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith family of Boussinesq systems, J. Nonlinear Sci. 17 (2007), no. 6, 569–607. MR 2371147, DOI 10.1007/s00332-007-9004-8
Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, Optimal $L_{\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475–483. MR 371077, DOI 10.1090/S0025-5718-1975-0371077-0
Todd Dupont, Galerkin methods for first order hyperbolics: an example, SIAM J. Numer. Anal. 10 (1973), 890–899. MR 349046, DOI 10.1137/0710074
A. S. Fokas and B. Pelloni, Boundary value problems for Boussinesq type systems, Math. Phys. Anal. Geom. 8 (2005), no. 1, 59–96. MR 2136653, DOI 10.1007/s11040-004-1650-6
Beatrice Pelloni, Spectral methods for the numerical solution of nonlinear dispersive wave equations, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–Yale University. MR 2695225
Beatrice Pelloni and Vassilios A. Dougalis, Numerical modelling of two-way propagation of non-linear dispersive waves, Math. Comput. Simulation 55 (2001), no. 4-6, 595–606. Nonlinear waves: computation and theory (Athens, GA, 1999). MR 1831398, DOI 10.1016/S0378-4754(00)00305-0
D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25(1966), 321–331.
D.H. Peregrine, Long waves on a beach, J. Fluid Mech. 27(1967), 815–827.
D.H. Peregrine, Equations for water waves and the approximations behind them, in Waves on Beaches and Resulting Sediment Transport, R.E. Meyer ed., Academic Press, New York 1972, pp. 95–121.
Maria Elena Schonbek, Existence of solutions for the Boussinesq system of equations, J. Differential Equations 42 (1981), no. 3, 325–352. MR 639225, DOI 10.1016/0022-0396(81)90108-X
Robert Schreiber, Finite element methods of high-order accuracy for singular two-point boundary value problems with nonsmooth solutions, SIAM J. Numer. Anal. 17 (1980), no. 4, 547–566. MR 584730, DOI 10.1137/0717047
Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170, DOI 10.1007/978-3-662-03359-3
Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835
G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954