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Muñoz Grajales, Juan Carlos

Error analysis of an implicit spectral scheme applied to the Schrödinger-Benjamin-Ono system. (English) Zbl 1353.65096

Int. J. Differ. Equ. 2016, Article ID 6930758, 12 p. (2016).
Summary: We develop error estimates of the semidiscrete and fully discrete formulations of a Fourier-Galerkin numerical scheme to approximate solutions of a coupled nonlinear Schrödinger-Benjamin-Ono system that describes the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The accuracy of the numerical solver is checked using some exact travelling wave solutions of the system.

Cited in 1 Document

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

error estimates; semidiscrete; fully discrete; Fourier-Galerkin numerical scheme; coupled nonlinear Schrödinger-Benjamin-Ono system; deep water; capillary-gravity waves; travelling wave solutions
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References:

[1] Funakoshi, M.; Oikawa, M., The resonant interaction between a long internal gravity wave and a surface gravity wave packet, Journal of the Physical Society of Japan, 52, 6, 1982-1995 (1983) · doi:10.1143/jpsj.52.1982
[2] Karpman, V. I., On the dynamics of sonic-langmuir solitons, Physica Scripta, 11, 5, 263-265 (1975) · doi:10.1088/0031-8949/11/5/003
[3] Djordjevic, V.; Redekoop, L., On two-dimensional packet of capillary-gravity waves, Journal of Fluid Mechanics, 79, 703-714 (1977) · Zbl 0351.76016
[4] Grimshaw, R. H., The modulation of an internal gravity-wave packet, and the resonance with the mean motion, Studies in Applied Mathematics, 56, 3, 241-266 (1977) · Zbl 0361.76029
[5] Benney, D. J., Significant interactions between small and large scale surface waves, Studies in Applied Mathematics, 55, 2, 93-106 (1976) · Zbl 0352.76017
[6] Benney, D. J., A general theory for interactions between short and long waves, Studies in Applied Mathematics, 56, 1, 81-94 (1977) · Zbl 0358.76011 · doi:10.1002/sapm197756181
[7] Angulo, J.; Montenegro, J. F., Existence and evenness of solitary-wave solutions for an equation of short and long dispersive waves, Nonlinearity, 13, 5, 1595-1611 (2000) · Zbl 0983.35005 · doi:10.1088/0951-7715/13/5/310
[8] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. Part 1, Annales De L Institut Henri Poincaré-Analyse Non Linéare, 1, 2, 109-145 (1984) · Zbl 0541.49009
[9] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. II, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 1, 4, 223-283 (1984) · Zbl 0704.49004
[10] Angulo, J., Stability of solitary wave solutions for equations of short and long dispersive waves, Electronic Journal of Differential Equations, 72, 1-8 (2006) · Zbl 1110.35071
[11] Bekiranov, D.; Ogawa, T.; Ponce, G., Interaction equations for short and long dispersive waves, Journal of Functional Analysis, 158, 2, 357-388 (1998) · Zbl 0909.35123 · doi:10.1006/jfan.1998.3257
[12] Pecher, H., Rough solutions of a Schrödinger-Benjamin-Ono system, Differential and Integral Equations. An International Journal for Theory & Applications, 19, 5, 517-535 (2006) · Zbl 1212.35456
[13] Angulo, J.; Matheus, C.; Pilod, D., Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system, Communications on Pure and Applied Analysis, 8, 3, 815-844 (2009) · Zbl 1168.35430 · doi:10.3934/cpaa.2009.8.815
[14] Muñoz Grajales, J. C., Error estimates for a Galerkin numerical scheme applied to a generalized BBM equation, Applicable Analysis, 94, 7, 1405-1419 (2015) · Zbl 1320.35278 · doi:10.1080/00036811.2014.932907
[15] Muñoz Grajales, J. C., Analysis of a fourier-galerkin numerical scheme for a 1D benney-luke-paumond equation, Revista Colombiana de Matemáticas, 49, 2, 213-234 (2015) · Zbl 1431.65190 · doi:10.15446/recolma.v49n2.60440
[16] Antonopoulos, D. C.; Dougalis, V. A., rror estimates for Galerkin approximations of the ‘classical’ Boussinesq system, Mathematics and Computers in Simulation, 82, 6, 984-1007 (2012) · Zbl 1320.76058
[17] Antonopoulos, D. C.; Dougalis, V. A.; Mitsotakis, D. E., Numerical solution of Boussinesq systems of the Bona-Smith family, Applied Numerical Mathematics, 60, 4, 314-336 (2010) · Zbl 1303.76082 · doi:10.1016/j.apnum.2009.03.002
[18] Pelloni, B.; Dougalis, V. A., Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations, Applied Numerical Mathematics, 37, 1-2, 95-107 (2001) · Zbl 0976.65086 · doi:10.1016/s0168-9274(00)00027-1
[19] Rashid, A.; Akram, S., Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction, Applications of Mathematics, 55, 4, 337-350 (2010) · Zbl 1224.65234 · doi:10.1007/s10492-010-0025-5
[20] Maday, Y.; Quarteroni, A., Error analysis for spectral approximation of the Korteweg-de Vries equation, RAIRO-Modélisation Mathématique et Analyse Numérique, 22, 3, 499-529 (1988) · Zbl 0647.65082
[21] Mercier, B., An Introduction to the Numerical Analysis of Spectral Methods. An Introduction to the Numerical Analysis of Spectral Methods, Lecture Notes in Physics, 318 (1983), New York, NY, USA: Springer, New York, NY, USA
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