Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Numerical methods and comparison for the Dirac equation in the
  nonrelativistic limit regime   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 $0<\varepsilon\ll 1$ 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 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 $\varepsilon$. Based on the error bounds, in order
to obtain `correct' numerical solutions in the nonrelativistic limit regime,
i.e. $0<\varepsilon\ll 1$, the FDTD methods share the same
$\varepsilon$-scalability on time step: $\tau=O(\varepsilon^3)$. 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 $\varepsilon$-scalability on time step is improved to
$\tau=O(\varepsilon^2)$ when $0<\varepsilon\ll 1$. Extensive numerical results
are reported to support our error estimates.

23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc