Structure-preserving properties of Strmer-Verlet scheme for mathematical pendulum

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conservation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Strmer-Verlet scheme is first constructed in a Hamiltonian framework. The conservation law of the Strmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Strmer-Verlet scheme associated with the conservation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the Strmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the Strmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Strmer-Verlet scheme.

Precise orbit determination for BDS satellites

Since the frst pair of BeiDou satellites was deployed in 2000,China has made continuous eforts to establish its own independent BeiDou Navigation Satellite System(BDS)to provide the regional radio determination satellite service as well as regional and global radio navigation satellite services,which rely on the high quality of orbit and clock products.This article summarizes the achievements in the precise orbit determination(POD)of BDS satellites in the past decade with the focus on observation and orbit dynamic models.First,the disclosed metadata of BDS satellites is presented and the contribution to BDS POD is addressed.The complete optical properties of the satellite bus as well as solar panels are derived based on the absorbed parameters as well the material properties.Secondly,the status and tracking capabilities of the L-band data from accessible ground networks are presented,while some low earth orbiter satellites with onboard BDS tracking capability are listed.The topological structure and measurement scheme of BDS Inter-Satellite-Link(ISL)data are described.After highlighting the progress on observation models as well as orbit perturbations for BDS,e.g.,phase center corrections,satellite attitude,and solar radiation pressure,diferent POD strategies used for BDS are summarized.In addition,the urgent requirement for error modeling of the ISL data is emphasized based on the analysis of the observation noises,and the incompatible characteristics of orbit and clock derived with L-band and ISL data are illuminated and discussed.The further researches on the improvement of phase center calibration and orbit dynamic models,the refnement of ISL observation models,and the potential contribution of BDS to the estimation of geodetic parameters based on L-band or ISL data are identifed.With this,it is promising that BDS can achieve better performance and provides vital contributions to the geodesy and navigation.