Local discontinuous Galerkin methods for the abcd nonlinear Boussinesq system

TL;DR

This paper develops and analyzes local discontinuous Galerkin methods for the abcd Boussinesq system, providing error estimates and demonstrating effectiveness in simulating wave interactions and blow-up phenomena.

Contribution

It introduces LDG methods with specific flux choices for the abcd Boussinesq system and establishes rigorous error bounds across various parameters.

Findings

Numerical experiments confirm optimal convergence rates.

Methods accurately simulate wave collisions.

Effective in capturing finite-time blow-up behavior.

Abstract

Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, BBM-BBM system, Bona-Smith system etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.