Employing nonresonant step sizes for time integration of highly oscillatory nonlinear Dirac equations

Abstract

In the nonrelativistic limit regime, nonlinear Dirac equations involve a small parameter $\varepsilon>0$ which induces rapid temporal oscillations with frequency proportional to $\varepsilon ^{-2}$. Efficient time integrators are challenging to construct, since their accuracy has to be independent of $\varepsilon $ or improve with smaller values of $\varepsilon $. Yongyong Cai and Yan Wang have presented a nested Picard iterative integrator (NPI-2), which is a uniformly accurate second-order scheme. We propose a novel method called the nonresonant nested Picard iterative integrator (NRNPI), which takes advantage of cancelation effects in the global error to significantly simplify the NPI-2. We prove that for nonresonant step sizes $\tau \geq \frac{\pi }{4} \varepsilon ^{2}$, the NRNPI has the same accuracy as the NPI-2 and is thus more efficient. Moreover, we show that for arbitrary $\tau < \frac{\pi }{4} \varepsilon ^{2}$, the error decreases proportionally to $\varepsilon ^{2} \tau $. We provide numerical experiments to illustrate the error behavior as well as the efficiency gain.