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

We propose a class of efficient and uniformly accurate nested Picard iterative integrators (NPI) for solving the nonlinear Dirac equation (NLDE) in the nonrelativistic regime, and apply it to study the convergence rates of the NLDE to its limiting models, the dynamics of traveling waves, and the two-dimensional dynamics. The NLDE involves a dimensionless parameter $\varepsilon\in (0, 1]$, and its solution is highly oscillatory in time with wavelength $O(\varepsilon^2)$ in the nonrelativistic regime. To gain uniform accuracies in time, the NPI method employs an operator decomposition technique for explicitly separating the highly oscillatory phases and utilizes exponential wave integrators for the time integrals. Moreover, with the help of nested Picard iterations, the NPI method could easily achieve uniform first- and second-order accuracies.