Uniformly Accurate Nested Picard Iterative Integrators for the Dirac Equation in the Nonrelativistic Limit Regime

This paper is devoted to the construction and analysis of uniformly accurate nested Picard iterative integrators (NPI) for the Dirac equation in the nonrelativistic limit regime. In this regime, there is a dimensionless parameter $\varepsilon\in(0,1]$ inversely proportional to the speed of light and the equation admits propagating waves with $O(1)$ wavelength in space and $O(\varepsilon^2)$ wavelength in time. To overcome the difficulty induced by the temporal $\varepsilon$ dependent oscillation, we present the construction of several NPI methods which are uniformly first-, second-, and third-order convergent in time w.r.t. $\varepsilon$. The general idea is applying nested Picard iterations to the integral form of the Dirac equation and using exponential wave integrators to approximate the temporal integrals. Thanks to the nested Picard iterative idea, the NPI method can be extended to arbitrary higher-order in time with optimal and uniform accuracy. The implementation of the second-order in-time NPI method via Fourier pseudospectral discretization is clearly demonstrated, and the corresponding error bounds are rigorously established through the energy method as $h^{m_0}+\tau^2$, where $h$ is the mesh size, $\tau$ is the time step, and $m_0$ depends on the regularity of the solution. Numerical results are reported to confirm the error estimates for the second-order NPI method and show the uniform accurate properties (w.r.t. $\varepsilon$) for the first- and third-order NPI methods as well.