On a new method of local error estimation in Gauss – Everhart methods

This study is devoted to the advancement of the Gauss–Everhart collocation method. The main focus of the article is on the estimation of the local integration error and step correction of the numerical method. The method originally proposed by E. Everhart is analyzed, and new (author’s) method for estimating the local error is introduced. The author’s method is based on constructing an auxiliary solution whose approximation order is not lower than the stage order of the original method reduced by one. The advantages of the authors’ method were confirmed through numerical experiments on orbital dynamics problems. It was demonstrated that the authors’ method provides a more accurate estimate of the local error. The numerical results showed that applying the authors’ method allows for a more precise adjustment of the integration step that satisfies a given local tolerance for constructing the solution. In this case, the step size increase reaches up to 180% compared to the step found by the Everhart method. The highest efficiency was achieved with the scheme that excludes the second collocation point. The obtained results indicate that the author’s method is promising for prediction the motion of near-Earth space objects.