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

摘要

Nonlinear Dirac equations describe the motion of relativistic spin-1/2 particles in presence of external electromagnetic fields, 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 infinity, 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 two-term splitting scheme cannot be straightforwardly applied. Thus, we propose and analyse 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 O (tau(2)c(4)) to O (tau(2)c(3)) if the magnetic potential in the system vanishes. Furthermore, we propose an efficient limit approximation scheme for solving nonlinear Dirac systems in the nonrelativistic limit regime c 1 which allows errors of order O (c(-1)) without any c-dependent time step restriction. (C) 2019 Elsevier B.V. All rights reserved.