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.Comment: 34 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1511.0119

Using Open Source Data to Identify Transit Deserts in Four Major Chinese Cities

You are viewing an article from the International Journal of Geo-Information that was in the Good Systems Network Digest in February 2020.Office of the VP for Researc

Layout optimization for multi-bi-modulus materials system under multiple load cases

Financial support from the National Natural Science Foundation of China (Grant No. 51179164) and the Australian Research Council (Grant No. DP140103137) is acknowledged

A method for measuring rotation of a thermal carbon nanomotor using centrifugal effect

A thermal nanomotor is relatively easy to fabricate and regulate as it contains just a few or even no accessory devices. Since the double-wall carbon nanotube (CNT)-based rotary nanomotor was established in a thermostat, assessment of the rotation of the rotor (inner tube) in the stator (outer tube) of the nanomotor has been critical, but remains challenging due to two factors: the small size of the rotor (only a few nanometers) and the high rotational frequency (»1 GHz). To measure the rotation of the nanomotor, in the present study, a probe test method is proposed. Briefly, the rotor is connected to an end-tube (CNT) through a graphene (GN) nanoribbon. As the CNT-probe is on the trajectory of the end-tube which rotates with the rotor, it will collide with the end-tube. The sharp fluctuation indicating the probe tip deflection can be observed and recorded. As a curly GN by hydrogenation is adopted for connecting the rotor and the end-tube, collision between the end-tube and the probe tip occurs only when the centrifugal force is higher than a threshold which can be considered as the rotational frequency of the rotor being measured by the present method.The authors are grateful for financial support from the National Natural-Science-Foundation of China (Grant No. 11372100) and the Australian Research Council (Grant No. DP140103137)

A characterization of metacirculants

AbstractMetacirculants were introduced by Alspach and Parsons in 1982 and have been a rich source of various topics since then, including the Hamiltonian path problem in metacirculants. A metacirculant has a vertex-transitive metacyclic subgroup of automorphisms, and a long-standing interesting question in the area is if the converse statement is true, namely, whether a graph with a vertex-transitive metacyclic automorphism group is a metacirculant. We shall answer this question in the negative, and then present a classification of cubic metacirculants

Applications of mercury intrusion capillary pressure for pore structures: A review

The shape, size, and connectivity of porous structures control the overall storage capacity and flow in oil and gas reservoirs. The mercury intrusion capillary pressure (MICP) tech- nique is widely utilized to measure capillary pressure and calculate pore size distribution of core samples in the geo-energy industry. Combining the MICP capillary pressure data with parameters from other experimental methods (such as scanning electron microscopy, and nuclear magnetic resonance) or theoretical approaches (such as fractal theory) can more accurately describe the pore structure of reservoirs. In this paper, the latest advances on the application of primary drainage MICP curves from reservoir porous structures are reviewed in three main aspects: The measurement and calculation of MICP capillary pressure, estimation of pore size distributions making use of fractal characteristics, and determination of permeability. Experimental measurements and numerical simulation methods of MICP capillary pressure with its influencing factors are also discussed. MICP capillary pressure combined with other methods are argued to be one of the main directions for future research on reservoir pore structures.Cited as: Jiao, L., Andersen, P.Ø., Zhou, J., Cai, J. Applications of mercury intrusion capillary pressure for pore structures: A review. Capillarity, 2020, 3(4): 62-74, doi: 10.46690/capi.2020.04.0