Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime

Abstract

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\epsilon \ll 1$ 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 several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size $h$ and time step $\tau$ as well as the small parameter $\epsilon$. Based on the error bounds, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\epsilon \ll 1$, the FDTD methods share the same $\epsilon$-scalability on time step: $\tau =O({\epsilon }^3)$. Then we propose and analyze two numerical methods for the discretization of the Dirac equation 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 on time step is improved to $\tau =O({\epsilon }^2)$ when $0<\epsilon \ll 1$. Extensive numerical results are reported to support our error estimates.

34 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1511.01192

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• Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic regime without magnetic potential
• A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials
• Linear and nonlinear electromagnetic waves in modulated honeycomb media
• Study of instability of the Fourier split-step method for the massive Gross--Neveu model
• Gauge invariant canonical symplectic algorithms for real-time lattice strong-field quantum electrodynamics