  This paper develops three high-order accurate discontinuous Galerkin (DG)
methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac
(NLD) equations with a general scalar self-interaction. They are the
Runge-Kutta DG (RKDG) method and the DG methods with the one-stage fourth-order
Lax-Wendroff type time discretizaiton (LWDG) and the two-stage fourth-order
accurate time discretization (TSDG). The RKDG method uses the spatial DG
approximation to discretize the NLD equations and then utilize the explicit
multistage high-order Runge-Kutta time discretization for the first-order time
derivatives, while the LWDG and TSDG methods, on the contrary, first give the
one-stage fourth-order Lax-Wendroff type and the two-stage fourth-order time
discretizations of the NLD equations, respectively, and then discretize the
first- and higher-order spatial derivatives by using the spatial DG
approximation. The $L^{2}$ stability of the 2D semi-discrete DG approximation
is proved in the RKDG methods for a general triangulation, and the
computational complexities of three 1D DG methods are estimated. Numerical
experiments are conducted to validate the accuracy and the conservative
properties of the proposed methods. The interactions of the solitary waves, the
standing and travelling waves are investigated numerically and the 2D breathing
pattern is observed.
