We propose a new scheme for simulation of a high-order nonlinear Schrodinger equation with a trapped term by using the mid-point rule and Fourier pseudospectral method to approximate time and space derivatives, respectively. The method is proved to be both charge- and energy-conserved. Various numerical experiments for the equation in different cases are conducted. From the numerical evidence, we see the present method provides an accurate solution and conserves the discrete charge and energy invariants to machine accuracy which are consistent with the theoretical analysis.
We derive a new method for a coupled nonlinear Schrdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative.We prove the proposed method preserves the charge and energy conservation laws exactly.In numerical tests,we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions.Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws.These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.
Observational and bogus satellite data are directly assimilated into the Weather Research and Forecasting (WRF) model in simulations of Typhoon Kalmaegi (2008). The data assimilation is performed using the Radiative Transfer for TIROS-N Operational Vertical Sounder (RTTOV) model and the three-dimensional variational data assimilation (3DVAR) technique, with satellite observations taken from the National Oceanic and Atmospheric Administration-16 (NOAA-16) Advanced TIROS Vertical Sounder (ATOVS) system composed of the High-resolution Infrared Radiation Sounder (HIRS), the Advanced Microwave Sounding Unit-A (AMSU-A), and the Advanced Microwave Sounding Unit-B (AMSU-B). Data assimilation experiments are initialized at three different times. Improvements in the numerical simulation of the typhoon are discussed in the context of wind, temperature, pressure, and geopotential fields. The results indicate that assimilation of satellite data can improve both the representation of the initial conditions and the subsequent simulation of the typhoon. Different satellite data have different impacts on the typhoon track. In these simulations, data from AMSU-A play a greater role in improving the simulation of the typhoon than data from AMSU-B or HIRS. Assimilation of satellite data significantly affects the simulation of the subtropical high and the steering of the typhoon by the environmental flow. The subtropical high is enhanced and extends westward in the data assimilation experiments. The background flow therefore steers the typhoon more westward, improving the simulated typhoon track. Although direct assimilation of satellite brightness temperature improves the simulated environmental conditions, it does not significantly improve the simulated intensity of the typhoon. By contrast, initializing the typhoon simulation using bogus data in tandem with satellite data improves not only the environmental conditions but also the simulated inner-core structure of the typhoon. Assimilation of both types of data therefore improves
A local energy conservation law is proposed for the Klein–Gordon–Schr ¨odinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving(LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order O(τ2+ h2). The theoretical properties are verified by numerical experiments.
