Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction (Communications in Computational Physics, 2021)
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-Wendrofftype time discretization (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretizethe NLD equations and then utilize the explicit multistage high-order Runge-Kuttatime discretization for the first-order time derivatives, while the LWDG and TSDGmethods, on the contrary, first give the one-stage fourth-order Lax-Wendroff type andthe two-stage fourth-order time discretizations of the NLD equations, respectively, andthen discretize the first- and higher-order spatial derivatives by using the spatial DGapproximation. The $L^2$ stability of the 2D semi-discrete DG approximation is provedin the RKDG methods for a general triangulation, and the computational complexities of three 1D DG methods are estimated. Numerical experiments are conducted tovalidate the accuracy and the conservation properties of the proposed methods. Theinteractions of the solitary waves, the standing and travelling waves are investigatednumerically and the 2D breathing pattern is observed.