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