  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.
