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

Abstract

We present several numerical methods and establish their error estimates for\nthe discretization of the nonlinear Dirac equation in the nonrelativistic limit\nregime, involving a small dimensionless parameter $0<\\varepsilon\\ll 1$ which is\ninversely proportional to the speed of light. In this limit regime, the\nsolution is highly oscillatory in time, i.e. there are propagating waves with\nwavelength $O(\\varepsilon^2)$ and $O(1)$ in time and space, respectively. We\nbegin with the conservative Crank-Nicolson finite difference (CNFD) method and\nestablish rigorously its error estimate which depends explicitly on the mesh\nsize $h$ and time step $\\tau$ as well as the small parameter $0<\\varepsilon\\le\n1$. Based on the error bound, in order to obtain `correct' numerical solutions\nin the nonrelativistic limit regime, i.e. $0<\\varepsilon\\ll 1$, the CNFD method\nrequests the $\\varepsilon$-scalability: $\\tau=O(\\varepsilon^3)$ and\n$h=O(\\sqrt{\\varepsilon})$. Then we propose and analyze two numerical methods\nfor the discretization of the nonlinear Dirac equation by using the Fourier\nspectral discretization for spatial derivatives combined with the exponential\nwave integrator and time-splitting technique for temporal derivatives,\nrespectively. Rigorous error bounds for the two numerical methods show that\ntheir $\\varepsilon$-scalability is improved to $\\tau=O(\\varepsilon^2)$ and\n$h=O(1)$ when $0<\\varepsilon\\ll 1$ compared with the CNFD method. Extensive\nnumerical results are reported to confirm our error estimates.