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 in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ 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 $O(\varepsilon^2)$ 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 $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear 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 $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288

Prospects for Detecting Neutrino Signals from Annihilating/Decaying Dark Matter to Account for the PAMELA and ATIC results

Recent PAMELA data show that positron fraction has an excess above several GeV while anti-proton one is not. Moreover ATIC data indicates that electron/positron flux have a bump from 300 GeV to 800 GeV. Both annihilating dark matter (DM) with large boost factor and decaying DM with the life around $10^{26} s$ can account for the PAMELA and ATIC observations if their main final products are charged leptons ($e$, $\mu$ and $\tau$). In this work, we calculated the neutrino flux arising from $\mu$ and $\tau$ which originate from annihilating/decaying DM, and estimated the final muon rate in the neutrino telescopes, namely Antares and IceCube. Given the excellent angular resolution, Antares and IceCube are promising to discover the neutrino signals from Galactic center and/or large DM subhalo in annihilating DM scenario, but very challenging in decaying DM scenario.Comment: 21 pages, 7 figures, 2 tables. V2: references added. V3: the number density of massive subhalo has been discussed in the appendix; accepted by PR

A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method ($S_\text{4c}$). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in the case of the Dirac equation, it turns out that only those steps involving potentials need to be amended, and the scheme remains efficient, accurate, as well as easy to implement. Numerical examples in 1D and 2D are given to validate the scheme.Comment: 24pages, 8 figure