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

Abstract

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.