TY - JOUR
T1 - Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction
AU - Li , Shu-Cun
AU - Tang , Huazhong
JO - Communications in Computational Physics
VL - 4
SP - 1150
EP - 1184
PY - 2021
DA - 2021/08
SN - 30
DO - http://doi.org/10.4208/cicp.OA-2020-0226
UR - https://global-sci.org/intro/article_detail/cicp/19397.html
KW - Nonlinear Dirac equation, discontinuous Galerkin method, Lax-Wendroff type time discretization, two-stage fourth-order accurate time discretization, Runge-Kutta method, solitary wave interaction.
AB - <p style="text-align: justify;">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 discretization (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 conservation 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.</p>