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Numerical methods for nonlinear Dirac equation

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.