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

In the nonrelativistic limit regime, nonlinear Dirac equations involve a small parameter $\epsilon >0$ which induces rapid temporal oscillations with frequency proportional to ${\epsilon }^{-2}$. Efficient time integrators are challenging to construct, since their accuracy has to be independent of $\epsilon$ or improve with smaller values of $\epsilon$. In [10], 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 cancellation effects in the global error to significantly simplify the NPI-2. We prove that for non-resonant step sizes $\tau \ge \frac{\pi }{4}{\epsilon }^2$, the NRNPI has the same accuracy as the 4NPI-2 and is thus more efficient. Moreover, we show that for arbitrary $\tau <\frac{\pi }{4}{\epsilon }^2$ the error decreases proportionally to ${\epsilon }^2\tau$ . We provide numerical experiments to illustrate the error behavior as well as the efficiency gain.