Asymptotic expansion of the solutions to a regularized Boussinesq system (theory and numerics). (English) Zbl 1542.35326
Summary: We consider the propagation of surface water waves described by the Boussinesq system. Following L. Molinet et al. [Nonlinearity 34, No. 2, 744–775 (2021; Zbl 1460.35295)], we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator define by \(\widehat{g_{\lambda}[\zeta]}=|k|^{\lambda}\hat{\zeta}_k\) with \(\lambda \in ]0,2]\). In this paper, we display a twofold approach: first, we study theoretically the existence of an asymptotic expansion for the solution to the Cauchy problem associated to this regularized Boussinesq system with respect to the regularizing parameter \(\epsilon\). Then, we compute numerically the function coefficients of the expansion (in \(\epsilon)\) and verify numerically the validity of this expansion up to order 2. We also check the numerical \(L^2\) stability of the numerical algorithm.
MSC:
35Q35 | PDEs in connection with fluid mechanics |
76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
35L56 | Initial value problems for higher-order hyperbolic systems |
35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |
35C20 | Asymptotic expansions of solutions to PDEs |
35B65 | Smoothness and regularity of solutions to PDEs |
35A35 | Theoretical approximation in context of PDEs |
35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
65T50 | Numerical methods for discrete and fast Fourier transforms |
Keywords:
Boussinesq system; energy estimate; Cauchy problem; numerical stability; numerical Fourier transform; numerical expansionCitations:
Zbl 1460.35295References:
[1] | Amick, C. J., Regularity and uniqueness of solutions to the Boussinesq system of equations, J. Differ. Equ., 54, 231-247, 1984 · Zbl 0557.35074 · doi:10.1016/0022-0396(84)90160-8 |
[2] | Bona, J. L.; Chen, M.; 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, 283-318, 2002 · Zbl 1022.35044 · doi:10.1007/s00332-002-0466-4 |
[3] | Bona, J. L.; Chen, M.; Saut, J.-C., Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory, Nonlinearity, 17, 925-952, 2004 · Zbl 1059.35103 · doi:10.1088/0951-7715/17/3/010 |
[4] | Boussinesq, J., Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17, 55-108, 1872 · JFM 04.0493.04 |
[5] | Brezis, H., Analyse Fonctionnelle: théorie et applications - \(2^e\) tirage, 1987 |
[6] | Droniou, J.; Gallouet, T.; Vovelle, J., Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3, 499-521, 2002 · Zbl 1036.35123 · doi:10.1007/s00028-003-0503-1 |
[7] | Guillopé, C.; Mneimné, A.; Talhouk, R., Asymptotic behaviour, with respect to the isothermal compressibility coefficient, for steady flows of weakly compressible viscoelastic fluids, Asymptot. Anal., 35, 2, 127-150, 2003 · Zbl 1054.76007 |
[8] | Israwi, S., Large time existence for 1D Green-Naghdi equations, Nonlinear Anal., 74, 81-93, 2011 · Zbl 1381.86012 · doi:10.1016/j.na.2010.08.019 |
[9] | Lannes, D., The Water Waves Problem: Mathematical Analysis and Asymptotics, 2013 · Zbl 1410.35003 |
[10] | Mammeri, Y.; Zhang, Y., Comparison of solutions of Boussinesq systems, Advances in Pure and Applied Mathematics, 5, 101-115, 2014 · Zbl 1293.35064 · doi:10.1515/apam-2014-0013 |
[11] | Molinet, L.; Talhouk, R.; Zaiter, I., The Boussinesq system revisited, Nonlinearity, 34, 744-775, 2021 · Zbl 1460.35295 · doi:10.1088/1361-6544/abcea6 |
[12] | Schonbek, M. E., Existence of solutions for the Boussinesq system of equations, J. Differ. Equ., 42, 325-352, 1981 · Zbl 0476.35067 · doi:10.1016/0022-0396(81)90108-X |
[13] | Taylor, M. E., Partial Differential Equations III, Nonlinear Equations, 2011, Berlin: Springer, Berlin · Zbl 1206.35004 · doi:10.1007/978-1-4419-7049-7 |
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