Splitting methods for nonlinear Dirac equations with thirring type interaction in the nonrelativistic limit regime

Nonlinear Dirac equations describe the motion of relativistic spin-$\frac{1}{2}$ particles in presence of external electromagnetic felds, modelled by an electric and magnetic potential, and taking into account a nonlinear particle self-interaction. In recent years, the construction of numerical splitting schemes for the solution of these systems in the nonrelativistic limit regime, i.e., the speed of light c formally tending to infnity, has gained a lot of attention. In this paper, we consider a nonlinear Dirac equation with Thirring type interaction, where in contrast to the case of the Soler type nonlinearity a classical twoterm splitting scheme cannot be straightforwardly applied. Thus, we propose and analyze a three-term Strang splitting scheme which relies on splitting the full problem into the free Dirac subproblem, a potential subproblem, and a nonlinear subproblem, where each subproblem can be solved exactly in time. Moreover, our analysis shows that the error of our scheme improves from $\mathcal{O}$ ($r$${}^2$$c$${}^4$) to $\mathcal{O}$ ($r$${}^2$$c$${}^3$) if the magnetic potential in the system vanishes. Furthermore, we propose an effcient limit approximation scheme for solving nonlinear Dirac systems in the nonrelativistic limit regime $c$ $\gg$ 1 which allows errors of $\mathcal{O}$ ($c$${}^{-1}$) without any $c$-dependent time step restriction.