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arXiv 2011.00934math.NAcs.NA
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一维和二维非线性Dirac方程的三种间断Galerkin方法

Three discontinuous Galerkin methods for one- and two-dimensional nonlinear Dirac equations with a scalar self-interaction

Shu-Cun Li, Huazhong Tang

摘要:

针对一维和二维具有一般标量自相互作用的非线性Dirac方程,发展了三种高精度间断Galerkin(DG)方法.它们是Runge-Kutta DG(RKDG)方法和具有单阶段四阶Lax-Wendroff型时间离散(LWDG)和两阶段四阶精确时间离散(TSDG)的DG方法。RKDG方法首先采用空间DG近似离散NLD方程,然后对一阶时间导数采用显式多级高阶Runge-Kutta时间离散,而LWDG和TSDG方法则相反,首先分别给出NLD方程的一级四阶Lax-Wendroff型和两级四阶时间离散,然后通过使用空间DG近似来离散一阶和高阶空间导数。证明了一般三角剖分下RKDG方法中二维半离散DG近似的$L^{2}$稳定性,并对3种一维DG方法的计算复杂性进行了估计.数值实验验证了所提方法的精度和保守性。数值研究了孤波、驻波和行波的相互作用,观察到了二维呼吸模式。

英文摘要:

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.

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