Numerical integration of ordinary differential equations on the basis of local polynomial approximations

S. K. Tatevyan^a, N. A. Sorokin^a, S. F. Zaletkin<sup>b

a</sup> Institute of Astronomy, Russian Academy of Sciences
^b Lomonosov Moscow State University, Research Computing Center

Abstract: The theory of numerical integration of first and second order ordinary differential equations on the basis of approximation of the solution by algebraic polynomials is considered. Polynomial approximations are constructed on segments whose lengths are equal to the integration step chosen in such a way that a prescribed accuracy is achieved. In order to construct an interpolating polynomial on each segment for the right-hand side of a differential equation, the corresponding segment is subdivided into subsegments by nodes of Markov's quadratures. By this is meant that the subdivision of the integration step is performed with the aid of nodes of quadratures with the highest algebraic order of accuracy. The computation of the solution and its derivatives at a required set of points (this set is often determined from experiments) is reduced to the evaluation of polynomials. This approach is especially convenient and useful for problems of astrodynamics and satellite geodesy.

Keywords: Cauchy problem, ordinary differential equations, polynomial approximations, Markov's quadratures, quadratures of highest algebraic degree of accuracy.

UDC: 519.622