Numerical modeling of solitary waves by 1-D Madsen and Sorensen extended Boussinesq equations

Abstract

This paper presents numerical solution of one-dimensional enhanced Boussinesq equations introduced by Madsen and Sorensen for modeling the solitary wave propagation in three different physical situations. Governing equations are spatially discretized using weighted residual Galerkin approach. To implement the linear finite element method, an auxiliary equation is introduced in order to simplify the discretization task of the third-order spatial derivatives present in momentum equation. Exact solitary wave solutions are used to specify the initial data for the incident solitary waves in the numerical model and for verification of the corresponding computed solutions. The proposed model has been used to simulate important wave phenomena including solitary wave propagation, shoaling, absorption and reflection. Validity of the developed code has been assessed by comparing the results against published analytical solutions and experimental measurements. In order to compare the current results against the numerical solution of the extended Boussinesq equation without the auxiliary form, propagation of solitary wave over a shelf is investigated. In all of the considered cases in the current study, the proposed model is shown capable of producing favorable agreements.

Keywords:

• Madsen–Sorensen extended Boussinesq equations
• solitary wave
• finite element method
• auxiliary equation

Disclosure statement

The authors declare that there is no conflict of interests regarding the publication of this article.