Numerical methods for nonlinear Dirac equation
Numerical methods for nonlinear Dirac equation
复制标题
非线性狄拉克方程的数值方法
DOI：
10.1016/j.jcp.2013.03.031
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发表时间：
2012-12
影响因子：
4.1
通讯作者：
Tang, Huazhong
中科院分区：
文献类型：
--
作者：
Xu, Jian;Shao, Sihong;Tang, Huazhong
本文综述了非线性狄拉克方程数值方法的研究现状。推广了几种具有标量和矢量自相互作用的（1+1）维NLD方程的求解方法，并从精度和时间可逆性以及离散电荷、能量和线性动量守恒的角度进行了分析。这些方法是Crank-Nicolson （CN）格式，线性化CN格式，奇偶跳棋格式，跳跃格式，半隐式有限差分格式和指数算子分裂（OS）格式。通过充分利用NLD方程的局部守恒律，解析求解了由OS格式引起的非线性子问题。通过两个数值实验，对比了两种高阶精确龙格-库塔不连续伽辽金方法，说明了各种数值方法的有效性，特别关注误差增长和计算成本。理论和数值比较表明，高阶精确的OS格式在精度和效率方面可以与本文讨论的其他数值格式相媲美。进一步应用四阶精确OS格式研究了标量和矢量自相互作用下NLD孤立波的相互作用动力学。结果表明：两个NLD孤立波的相互作用动力学取决于NLD方程中自相互作用的指数幂；在三次矢量自相互作用下，两个相等的单峰NLD孤立波在碰撞后会发生坍缩，而在相应的二次情形下则没有坍缩散射。
This paper presents a review of the current state-of-the-art of numerical methods for nonlinear Dirac (NLD) equation. Several methods are extendedly proposed for the (1+1)-dimensional NLD equation with the scalar and vector self-interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge, energy and linear momentum. Those methods are the Crank–Nicolson (CN) schemes, the linearized CN schemes, the odd–even hopscotch scheme, the leapfrog scheme, a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes. The nonlinear subproblems resulted from the OS schemes are analytically solved by fully exploiting the local conservation laws of the NLD equation. The effectiveness of the various numerical methods, with special focus on the error growth and the computational cost, is illustrated on two numerical experiments, compared to two high-order accurate Runge–Kutta discontinuous Galerkin methods. Theoretical and numerical comparisons show that the high-order accurate OS schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency. A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics of the NLD solitary waves under the scalar and vector self-interaction. The results show that the interaction dynamics of two NLD solitary waves depend on the exponent power of the self-interaction in the NLD equation; collapse happens after collision of two equal one-humped NLD solitary waves under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.
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影响因子：
5
作者：
P. Mathieu;T. Morris
通讯作者：
P. Mathieu;T. Morris
影响因子：
1.3
作者：
J. Stubbe
通讯作者：
J. Stubbe
影响因子：
2.9
作者：
Andrew T. Sornborger;E. Stewart
通讯作者：
Andrew T. Sornborger;E. Stewart
DOI：
10.1103/physrevd.31.2701
发表时间：
1985-05
期刊：
Physical review. D, Particles and fields
影响因子：
--
作者：
Álvarez
通讯作者：
Álvarez
影响因子：
1.2
作者：
Sihong Shao;Huazhong Tang
通讯作者：
Sihong Shao;Huazhong Tang