Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation (NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter 0 < epsilon a parts per thousand currency sign 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(epsilon (2)) and O(1) in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step tau as well as the small parameter 0 < epsilon a parts per thousand currency sign 1. Based on the error bound, in order to obtain 'correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 < epsilon a parts per thousand currency sign 1, the CNFD method requests the epsilon-scalability: tau = O(epsilon (3)) and . Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their epsilon-scalability is improved to tau = O(epsilon (2)) and h = O(1) when 0 < epsilon a parts per thousand currency sign 1. Extensive numerical results are reported to confirm our error estimates.