论文标题
具有标量自相互作用的一维非线性狄拉克方程的三种不连续的galerkin方法
Three discontinuous Galerkin methods for one- and two-dimensional nonlinear Dirac equations with a scalar self-interaction
论文作者
论文摘要
本文为具有一般标量自我相互作用的一维（1D）和二维（2D）非线性DIRAC（NLD）方程开发了三个高阶准确不连续的盖尔金（DG）方法。它们是带有一阶段四阶Lax-Wendroff Type Time Divetizaiton（LWDG）的Runge-Kutta DG（RKDG）方法和DG方法和两阶段的四阶准确时间离散化（TSDG）。 RKDG方法使用空间DG近似来离散NLD方程，然后利用一阶时间导数的明确的多阶段高级runge-kutta时间离散化，而LWDG和TSDG方法相反，首先给出了单阶段的四阶段式的四分之二的时间，并给出了两阶段的四分之一阶段的时间订单。然后分别通过使用空间DG近似来离散一阶和高阶空间衍生物。在RKDG方法中证明了$ l^{2} $稳定性用于一般三角剖分的RKDG方法，并估计了三种1D DG方法的计算复杂性。进行数值实验以验证所提出方法的准确性和保守特性。孤立波，站立和行进波的相互作用是数值研究的，并观察到了2D呼吸模式。
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.
