Local Discontinuous Galerkin Methods for the Boussinesq Coupled BBM System

Abstract

Nonlinear dispersive wave equations model a substantial number of physical systems that admit special solutions such as solitons and solitary waves. Due to the complex nature of the nonlinearity and dispersive effects, high order numerical methods are effective in capturing the physical system in computation. In this paper, we consider the Boussinesq coupled BBM system, and propose local discontinuous Galerkin (LDG) methods for solving the BBM system. For the proposed LDG methods, we provide two different choices of numerical fluxes, namely the upwind and alternating fluxes, as well as establish their stability analysis. The error estimate for the linearized BBM system is carried out for the LDG methods with the alternating flux. To present a time discretization that conserves the Hamiltonian numerically, the midpoint rule with a nontrivial nonlinear term in the discretization is proposed. Both Hamiltonian conserving and dissipating time discretizations are implemented, with multiple combinations of numerical flux and time discretization tested numerically. Numerical examples are provided to demonstrate the accuracy, long-time simulation, and Hamiltonian conservation properties of the proposed LDG methods for the coupled BBM system.