Uniformly accurate numerical schemes for the nonlinear dirac equation in the nonrelativistic limit regime

Mohammed Lemou (1, 2) , Florian Méhats (2, 1) , Xiaofei Zhao (2)

1 IPSO - Invariant Preserving SOlvers
2 IRMAR - Institut de Recherche Mathématique de Rennes

We apply the two-scale formulation approach to propose uniformly accurate (UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic limit regime. The nonlinear Dirac equation involves two small scales ε and ε 2 with ε → 0 in the nonrelativistic limit regime. The small parameter causes high oscillations in time which bring severe numerical burden for classical numerical methods. We present a suitable two-scale formulation as a general strategy to tackle a class of highly oscillatory problems involving the two small scales ε and ε 2. A numerical scheme with uniform (with respect to ε ∈ (0,1]) second order accuracy in time and a spectral accuracy in space are proposed. Numerical experiments are done to confirm the UA property.

Mots clés: nonlinear Dirac equation; nonrelativistic limit; highly oscillatory equations; uniform accuracy; two-scale formulation

Domaines: Mathématiques [math]

Mohammed Lemou, Florian Méhats, Xiaofei Zhao. Uniformly accurate numerical schemes for the nonlinear dirac equation in the nonrelativistic limit regime. 2016. ⟨hal-01313976v1⟩