This paper introduces a general technique for the construction of multistep methods capable of integrating, without local truncation error, homogeneous linear ODEs with constant coefficients, including those, in particular, that result in oscillatory solutions. Moreover, these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies, even confluent ones that occur from nonhomogeneous terms in the differential equation. Our procedure allows the derivation of many of the existing codes with similar properties, as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency. The properties of the methods are studied within a general framework, and numerical examples are presented. These demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes, particularly when integrating the class of equations with perturbed oscillatory solutions. The methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known, as occurs in satellite orbit propagation. The underlying ideas have already been applied to the improvement of some Chebyshev methods that are not multistep.

Accurate Earth orientation parameter (EOP) predictions are needed for many applications, e.g., for the tracking and navigation of interplanetary spacecraft missions. One of the most difficult parameters to forecast is the length of day (LOD), which represents the variation in the Earth’s rotation rate since it is primarily affected by the torques associated with changes in atmospheric circulation. In this study, a new-generation time-series prediction algorithm is developed. The one-dimensional convolutional neural network (1D CNN), which is one of the deep learning methods, is introduced to model and predict the LOD using the IERS EOP 14 C04 and axial Z component of the atmospheric angular momentum (AAM), which was taken from the German Research Centre for Geosciences (GFZ) since it is strongly correlated with the LOD changes. The prediction procedure operates as follows: first, we detrend the LOD and Z-component series using the LS method, then, we obtain the residual series of each one to be used in the 1D CNN prediction algorithm. Finally, we analyze the results before and after introducing the AAM function. The results prove the potential of the proposed method as an optimal algorithm to successfully reconstruct and predict the LOD for up to 7 days.

The real-time estimation of polar motion (PM) is needed for the navigation of Earth satellite and interplanetary spacecraft. However, it is impossible to have real-time information due to the complexity of the measurement model and data processing. Various prediction methods have been developed. However, the accuracy of PM prediction is still not satisfactory even for a few days in the future. Therefore, new techniques or a combination of the existing methods need to be investigated for improving the accuracy of the predicted PM. There is a well-introduced method called Copula, and we want to combine it with singular spectrum analysis (SSA) method for PM prediction. In this study, first, we model the predominant trend of PM time series using SSA. Then, the difference between PM time series and its SSA estimation is modeled using Copula-based analysis. Multiple sets of PM predictions which range between 1 and 365 days have been performed based on an IERS 08 C04 time series to assess the capability of our hybrid model. Our results illustrate that the proposed method can efficiently predict PM. The improvement in PM prediction accuracy up to 365 days in the future is found to be around 40% on average and up to 65 and 46% in terms of success rate for the $${\hbox{PM}}_{x}$$ and $${\hbox{PM}}_{y}$$ , respectively.

In this paper we present a simple technique that allows us to limit the error growth in the long–term numerical integration of perturbed multi–dimensional oscillators, while using highly efficient and accurate special multi–step codes. After laying down the theoretical basis of their behaviour, we then illustrate them with some numerical examples, including a case with non–resonant frequencies, seldom found in the related literature.