Analytical and Rothe time-discretization method for a Boussinesq-type system over an uneven bottom

• Published in: Communications in nonlinear science & numerical simulation 2021-11, Vol.102, p.105951, Article 105951
• Main Authors: Mejía, Luis Fernando, Grajales, Juan Carlos Muñoz
• Format: Article
• Language: English
• Subjects: Boussinesq equations; Boussinesq-type system; Discretization; FEniCS project; Finite difference method; Finite element analysis; Finite element method; Incident waves; Interpolation; Mathematical models; Moving bathymetry; Propagation; Solitary waves; Surface waves; Topography; Two dimensional analysis; Two dimensional models; Water waves; Wave propagation; Wave reflection
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.105951

•This paper studies analytically and numerically a Galerkin Finite element method for approximating the solutions of a Boussinesq-type system to model water wave propagation over a time dependent variable topography.•Numerical methods are valuable tools for studying dynamics of complicated evolution systems and for exploring open questions on water wave phenomena described by dispersivetype equations. We study analytically and numerically a 2D version of a Boussinesq-type model considered by M. Chen (2003) to describe water wave propagation on the surface of a channel with an irregular moving topography. Following a semidiscrete horizontal line method (Rothe’s method) implemented with FEniCS, we first discretize the temporal variable by using a finite-difference second-order Crank-Nicholson-type scheme, and then, at each time step, the spatial variables are discretized with an efficient Galerkin/Finite Element Method (FEM) using triangular-finite elements based on 2D piecewise-linear Lagrange interpolation. The numerical experiments presented are in accordance with the previous theoretical and experimental studies and show that the so-called Bragg resonant reflection emerges when surface waves modelled by the Boussinesq formulation studied interact with periodically varying bottoms. We also present some experiments to examine the interaction of incident waves with variable topographies such as the shoaling of a solitary wave on a slope and the generation of surface waves by moving topography.