  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.
