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We present several numerical methods and establish their error estimates forthe discretization of the nonlinear Dirac equation in the nonrelativistic limitregime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which isinversely proportional to the speed of light. In this limit regime, thesolution is highly oscillatory in time, i.e. there are propagating waves withwavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Webegin with the conservative Crank-Nicolson finite difference (CNFD) method andestablish rigorously its error estimate which depends explicitly on the meshsize $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le1$. Based on the error bound, in order to obtain `correct' numerical solutionsin the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD methodrequests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and$h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methodsfor the discretization of the nonlinear Dirac equation by using the Fourierspectral discretization for spatial derivatives combined with the exponentialwave integrator and time-splitting technique for temporal derivatives,respectively. Rigorous error bounds for the two numerical methods show thattheir $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and$h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensivenumerical results are reported to confirm our error estimates.

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