<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Jian Xu</style></author><author><style face="normal" font="default" size="100%">Sihong Shao</style></author><author><style face="normal" font="default" size="100%">Huazhong Tang</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">   Numerical methods for nonlinear Dirac equation</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Computational Physics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2013</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://dx.doi.org/10.1016/j.jcp.2013.03.031</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">245</style></volume><pages><style face="normal" font="default" size="100%">131-149</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">This paper presents a review of the current state-of-the-art of numerical methods for&amp;nbsp;nonlinear Dirac (NLD) equation.&amp;nbsp;Several methods are extendedly proposed for the (1+1)-dimensional NLD equation&amp;nbsp;with the scalar and vector self-interaction and analyzed in the way of the&amp;nbsp;accuracy and the time reversibility&amp;nbsp;as well as the conservation of the discrete charge, energy and linear momentum.&amp;nbsp;Those methods are the Crank-Nicolson (CN) schemes,&amp;nbsp;the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme,&amp;nbsp;a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes.&amp;nbsp;The nonlinear subproblems resulted from the OS schemes are analytically solved&amp;nbsp;by fully exploiting the local conservation laws of the NLD equation.&amp;nbsp;The effectiveness of the various numerical methods,&amp;nbsp;with special focus on the error growth and the computational cost,&amp;nbsp;is illustrated on two numerical experiments,&amp;nbsp;compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods.&amp;nbsp;Theoretical and numerical comparisons show that the high-order accurate OS schemes&amp;nbsp;may compete well with other numerical schemes discussed here in terms of the accuracy&amp;nbsp;and the efficiency.&amp;nbsp;A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics&amp;nbsp;of the NLD solitary waves under the scalar and vector self-interaction.&amp;nbsp;The results show that the interaction dynamics&amp;nbsp;of two NLD solitary waves &amp;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&amp;nbsp;under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.&amp;nbsp;</style></abstract></record></records></xml>