Adaptive interpolation algorithm for solving problems of celestial mechanics with interval uncertainties


DOI: 10.34759/trd-2022-123-24

Аuthors

Morozov A. Y.

Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, 44-2, Vavilova str., Moscow, 119333, Russia

e-mail: morozov@infway.ru

Abstract

The presented work performs the application of the previously developed adaptive interpolation algorithm to the problem of XF11 asteroid motion simulation with regard for the interval uncertainties in the asteroid position and velocity. The XF11asteroid motion around the Sun is being considered without accounting for the effect of the other planets. The original problem is formulated as a Cauchy problem for a system of ordinary differential equations (ODE) with interval initial conditions.

The essence of the adaptive interpolation algorithm consists in plotting for each time point piecewise polynomial function that interpolates the dependence of the solution to the problem on the point values of the interval parameters. An adaptive grid is being created over the region of the parameter uncertainty. Each node of the grid corresponds to the original problem solution with the parameters values determined by the node position in space. The grid adaptation is being performed depending on the interpolation error. In the places where the error is large, new nodes are being added, and in the places where the error is small, the grid is being rarefied.

The article presents the description of various existing methods and corresponding software libraries, such as AWA, COZY-VI, RiOT, verifyode, for solving this class of problems. Employing the adaptive interpolation algorithm, the obtained interval system of ODEs is numerically integrated and compared with known results from the standpoint of the interval estimates accuracy and computational costs. Application of the fundamentally different approach to solving interval problems, allowed the adaptive interpolation algorithm obtaining solution boundaries with controlled accuracy. The algorithm is not subjected to the wrapping effect, and runs orders of magnitude faster than its analogs.

Keywords:

asteroid, interval system of ordinary differential equations, adaptive interpolation algorithm, software libraries

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