A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime^★

Beijing Computational Science Research Center, Beijing 100193, P.R. China

^* Corresponding author: e-mail: matwyan@csrc.ac.cn

Received: 3 July 2017
Accepted: 22 February 2018

Abstract

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε²) and O (1) in time and space, respectively. In the nonrelativistic regime, i.e., 0 < ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds in ε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as h^m₀+τ²/ε² and h^m₀ + τ² + ε², where h is the mesh size, τ is the time step and m₀ depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ²) in the regimes when either ε = O (1) or 0 < ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.