Communications in Mathematical Sciences

Volume 15 (2017)

Number 4

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

Pages: 1107 – 1128

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n4.a9

Authors

Mohammed Lemou (CNRS, Université de Rennes, IRMAR and INRIA, Rennes, France)

Florian Méhats (Université de Rennes, IRMAR and INRIA, Rennes, France)

Xiaofei Zhao (Université de Rennes, IRMAR, Rennes, France)

Abstract

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 $\varepsilon$ and $\varepsilon^2$ with $\varepsilon \to 0$ in the nonrelativistic limit regime. The small parameter causes high oscillations in time, which brings severe numerical burden for classical numerical methods. We transform our original problem as a two-scale formulation and present a general strategy to tackle a class of highly oscillatory problems involving the two small scales $\varepsilon$ and $\varepsilon^2$. Suitable initial data for the two-scale formulation is derived to bound the time derivatives of the augmented solution. Numerical schemes with uniform (with respect to $\varepsilon \in (0,1]$) spectral accuracy in space and uniform first order or second order accuracy in time are proposed. Numerical experiments are done to confirm the UA property.

Keywords

nonlinear Dirac equation, nonrelativistic limit, highly oscillatory equations, uniform accuracy, two-scale formulation

2010 Mathematics Subject Classification

35Q55, 65M12, 74Q10

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