Asymptotic Expansion of the Solutions to a Regularized Boussinesq System (Theory and Numerics)

• Published in: Acta applicandae mathematicae Vol. 191; no. 1; p. 12
• Main Authors: Safa, Ahmad, Le Meur, Hervé, Chehab, Jean-Paul, Talhouk, Raafat
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.06.2024; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Applications of Mathematics; Asymptotic series; Boussinesq equations; Calculus of Variations and Optimal Control; Optimization; Cauchy problems; Computational Mathematics and Numerical Analysis; Differential equations; Fourier transforms; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical analysis; Operators (mathematics); Partial Differential Equations; Probability Theory and Stochastic Processes; Surface water; Water waves; Wave propagation; Numerical Fourier transform; Energy estimate; 35Q35; 65T99; Boussinesq system; 35B30; 35B40; 35L56; Numerical stability; Cauchy problem; Numerical expansion; numerical Fourier transform; numerical expansion. Mathematics Subject Classification numbers: 35Q35; numerical stability
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-024-00660-3

We consider the propagation of surface water waves described by the Boussinesq system. Following (Molinet et al. in Nonlinearity 34:744–775, 2021 ), we introduce a regularized Boussinesq system obtained by adding a non-local pseudo-differential operator define by g λ [ ζ ] ˆ = | k | λ ζ ˆ k with λ ∈ ] 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 ϵ . Then, we compute numerically the function coefficients of the expansion (in ϵ ) and verify numerically the validity of this expansion up to order 2. We also check the numerical L 2 stability of the numerical algorithm.