This paper presents a review of the current state-of-the-art of numerical methods for nonlinear Dirac (NLD) equation. Several methods are extendedly proposed for the (1+1)-dimensional NLD equation with the scalar and vector self-interaction and analyzed in the way of the accuracy and the time reversibility as well as the conservation of the discrete charge, energy and linear momentum. Those methods are the Crank-Nicolson (CN) schemes, the linearized CN schemes, the odd-even hopscotch scheme, the leapfrog scheme, a semi-implicit finite difference scheme, and the exponential operator splitting (OS) schemes. The nonlinear subproblems resulted from the OS schemes are analytically solved by fully exploiting the local conservation laws of the NLD equation. The effectiveness of the various numerical methods, with special focus on the error growth and the computational cost, is illustrated on two numerical experiments, compared to two high-order accurate Runge-Kutta discontinuous Galerkin methods. Theoretical and numerical comparisons show that the high-order accurate OS schemes may compete well with other numerical schemes discussed here in terms of the accuracy and the efficiency. A fourth-order accurate OS scheme is further applied to investigating the interaction dynamics of the NLD solitary waves under the scalar and vector self-interaction. The results show that the interaction dynamics of two NLD solitary waves depend on the exponent power of the self-interaction in the NLD equation; collapse happens after collision of two equal one-humped NLD solitary waves under the cubic vector self-interaction in contrast to no collapse scattering for corresponding quadric case.
A 2-D lateral heterogeneous model was constructed to simulate basin-edge effects using PSM/FDM method. Effects of basin-edge geometry and source depth were simulated. PGV of different models are given to illustrate the effects, and it suggests that the basin geometry and the depth of soft sediment play crucial roles in seismic ground motion study for sedimentary basin.
Ma S, Wang S, Gao W. Overview of IEEE 1857 video coding standard, in IEEE International Conference on Image Processing, ICIP 2013, Melbourne, Australia, September 15-18, 2013.; 2013:1500–1504. 访问链接
This study examines the hygroscopicity of newly formed particles (diameters range 25-45 nm) during two atmospheric new particle formation (NPF) events in the German mid-level mountains during the Hill Cap Cloud Thuringia 2010 (HCCT-2010) field experiment. At the end of the NPF event involving clear particle growth, we measured an unusually high soluble particle fraction of 58.5% at 45 nm particle size. The particle growth rate contributed through sulfuric acid condensation only accounts for around 6.5% of the observed growth rate. Estimations showed that sulfuric acid condensation explained, however, only around 10% of that soluble particle fraction. Therefore, the formation of additional water-soluble matter appears imperative to explain the missing soluble fraction. Although direct evidence is missing, we consider water-soluble organics as candidates for this mechanism. For the case with clear growth process, the particle growth rate was determined by two alternative methods based on tracking the mode diameter of the nucleation mode. The mean particle growth rate obtained from the inter-site data comparison using Lagrangian consideration is 3.8 (+/- 2.6) nm h(-1). During the same period, the growth rate calculated based on one site data is 5.0 nm h(-1) using log-normal distribution function method. In light of the fact that considerable uncertainties could be involved in both methods, we consider both estimated growth rates consistent.