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

Krämer, Patrick; Schratz, Katharina; Zhao, Xiaofei


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.

Volltext §
DOI: 10.5445/IR/1000087636
Veröffentlicht am 19.11.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000087636
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 20 S.
Serie CRC 1173 ; 2018/33
Schlagwörter Dirac equation, time integration, splitting methods, error estimates, highly-oscillatory, nonrelativistic limit
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