Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

In this paper, we are concerned with symmetric integrators for the nonlinear relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter $0<\varepsilon\ll 1$, which is inversely proportional to the speed of light. The highly oscillatory property in time of this model corresponds to the parameter $\varepsilon$ and the equation has strong nonlinearity when \eps is small. There two aspects bring significantly numerical burdens in designing numerical methods. We propose and analyze a novel class of symmetric integrators which is based on some formulation approaches to the problem, Fourier pseudo-spectral method and exponential integrators. Two practical integrators up to order four are constructed by using the proposed symmetric property and stiff order conditions of implicit exponential integrators. The convergence of the obtained integrators is rigorously studied, and it is shown that the accuracy in time is improved to be \mathcal{O}(\varepsilon^{3} \hh^2) and \mathcal{O}(\varepsilon^{4} \hh^4) for the time stepsize \hh. The near energy conservation over long times is established for the multi-stage integrators by using modulated Fourier expansions. These theoretical results are achievable even if large stepsizes are utilized in the schemes. Numerical results on a NRKG equation show that the proposed integrators have improved uniform error bounds, excellent long time energy conservation and competitive efficiency

Two-scale exponential integrators with uniform accuracy for three-dimensional charged-particle dynamics under strong magnetic field

The numerical simulation of three-dimensional charged-particle dynamics (CPD) under strong magnetic field is challenging. In this paper, we introduce a new methodology to design two-scale exponential integrators for three-dimensional CPD whose magnetic field's strength is inversely proportional to a dimensionless parameter $0<\varepsilon \ll 1$. By dealing with the transformed form of three-dimensional CPD, we linearize the magnetic field and put the rest part in a nonlinear function which can be shown to be small. Based on which and the proposed two-scale exponential integrators, a class of novel integrators is formulated. The corresponding uniform accuracy over $\mathcal{O}(1/\varepsilon^{\beta})$ time interval is $\mathcal{O}(\varepsilon^{r\beta} h^r)$ for the $r$-th order integrator with the time stepsize $h$, $r=1,2,3,4$ and $0<\beta<1$. A rigorous proof of this error bound is presented and a numerical test is performed to illustrate the error behaviour of the proposed integrators

Collaborative MR Workspace with Shared 3D Vision Based on Stereo Video Transmission

P.R.China Mixed reality (MR) research aims to develop technologies that inputting or mixing the rea

Dehydrogenation of formic acid over Pd/C catalysts: Insight into the cold plasma treatment

Safe and efficient generation of renewable hydrogen via dehydrogenation of cheap and sustainable formic acid using supported Pd catalysts has attracted significant interest. Non-thermal (cold) plasma is demonstrably a fast...</p

The Poverty Alleviation and Immigration Practice Model Effect Research in Liulin

Abstract By the way of field investigation and statistical analysis, I have carried on investigation to Liulin, migrant village and I have learnt something about migration forms, mainly interaction patterns of villages and towns aggregation model. This article mainly analyzes the work and the necessity of immigration for poverty alleviation and the actual effect of the two modes in Lilulin. Moreover, I put forward some suggestions to model improvement and hope that I can do some useful things to poverty reduction in Liulin. At the same time, it can provide favorable help for building a harmonious society

Development of a Flood Warning Simulation System:A Case Study of 2007 Tewkesbury Flood

Many flood warning systems were developed for 2D environments and limited on specific flood hazard. With the purpose of overcoming these disadvantages, it is necessary to propose new methodologies and techniques for 3D real time flood simulation. In this paper, a novel flood hazard warning system has been proposed. It describes and defines the relationship between the different parts of the simulation system in order to offer not only numeric data or figures, but also more meaningful and appealing 3D visual information. Consequently, the performance of this simulation system depends on the quality of the three sub systems: 3D real world modelling system with GIS data, 3D environment reconstruction system and 3D flood simulation system. A new flooding model has been developed which can handle dynamic flood behaviour and predict inundation areas in real time. In order to validate our flood warning system, the region of Tewkesbury in England has been simulated with a potential flood. The flood spreading process is shown during different time and the detailed inundation area is presented for further disaster evaluation. The study achieved two main objectives: implementing a useful flood simulation with real world model and reconstructed environment for flood hazard warning; producing a friendly simulation system interface for either a decision maker or experienced user