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

Krämer, Patrick; Schratz, Katharina; Zhao, Xiaofei

Abstract:
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, w ... mehr

 Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-876369 KITopen-ID: 1000087636 Verlag KIT, Karlsruhe Umfang 20 S. Serie CRC 1173 ; 2018/33 Schlagworte Dirac equation, time integration, splitting methods, error estimates, highly-oscillatory, nonrelativistic limit
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft KITopen Landing Page