@article {26623,
	title = {   Numerical methods for nonlinear Dirac equation},
	journal = {Journal of Computational Physics},
	volume = {245},
	year = {2013},
	pages = {131-149},
	abstract = {This paper presents a review of the current state-of-the-art of numerical methods for\&nbsp;nonlinear Dirac (NLD) equation.\&nbsp;Several methods are extendedly proposed for the (1+1)-dimensional NLD equation\&nbsp;with the scalar and vector self-interaction and analyzed in the way of the\&nbsp;accuracy and the time reversibility\&nbsp;as well as the conservation of the discrete charge, energy and linear momentum.\&nbsp;Those methods are the Crank-Nicolson (CN) schemes,\&nbsp;the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme,\&nbsp;a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes.\&nbsp;The nonlinear subproblems resulted from the OS schemes are analytically solved\&nbsp;by fully exploiting the local conservation laws of the NLD equation.\&nbsp;The effectiveness of the various numerical methods,\&nbsp;with special focus on the error growth and the computational cost,\&nbsp;is illustrated on two numerical experiments,\&nbsp;compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods.\&nbsp;Theoretical and numerical comparisons show that the high-order accurate OS schemes\&nbsp;may compete well with other numerical schemes discussed here in terms of the accuracy\&nbsp;and the efficiency.\&nbsp;A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics\&nbsp;of the NLD solitary waves under the scalar and vector self-interaction.\&nbsp;The results show that the interaction dynamics\&nbsp;of two NLD solitary waves \&nbsp; 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\&nbsp;under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.\&nbsp;},
	url = {http://dx.doi.org/10.1016/j.jcp.2013.03.031},
	author = {Jian Xu and Sihong Shao and Huazhong Tang}
}
