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

TL;DR

This paper introduces uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit, effectively handling high oscillations caused by small parameters and ensuring spectral accuracy in space and high order in time.

Contribution

The authors develop a two-scale formulation approach that provides the first uniformly accurate schemes for the nonlinear Dirac equation across the nonrelativistic limit regime.

Findings

Schemes achieve spectral accuracy in space uniformly in 

First and second order accuracy in time uniformly in 

Numerical experiments confirm the uniform accuracy property

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 $\varepsilon\to0$ 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.