Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System

doi:10.1007/s42967-021-00119-4

Abstract

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, the BBM-BBM system, the 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.

Key words: Local discontinuous Galerkin methods, Boussinesq equations, Coupled BBM equations, Error estimate, Numerical fluxes, Head-on collision

CLC Number:

Jiawei Sun, Shusen Xie, Yulong Xing. Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 381-416.

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