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:
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