Fourth-Order Uniformly Accurate Integrators With Long Time Near Conservations for the Nonlinear Dirac Equation in the Nonrelativistic Regime

Abstract. In this paper, we propose two novel fourth-order integrators that exhibit uniformly high accuracy and long-term near conservations for solving the nonlinear Dirac equation (NLDE) in the nonrelativistic regime. In this regime, the solution of the NLDE exhibits highly oscillatory behavior in time, characterized by a wavelength of [Formula: see text] with a small parameter [Formula: see text]. To ensure uniform temporal accuracy, we employ a two-scale approach in conjunction with exponential integrators, utilizing operator decomposition techniques for the NLDE. The proposed methods are rigorously proved to achieve fourth-order uniform accuracy in time for all [Formula: see text]. Furthermore, we successfully incorporate symmetry into the integrator, and the long-term near conservation properties are analyzed through the modulated Fourier expansion. The proposed schemes are readily extendable to the linear Dirac equation incorporating magnetic potential, the dynamics of traveling wave solutions, and the two-/three-dimensional Dirac equations. The validity of all theoretical findings and extensions is numerically substantiated through a series of numerical experiments.