Application of weight Sobolev spaces

in dynamics of satellite librations in elliptic orbit

Ivan I. Kossenko

Moscow State University of Service

e-mail: cosenco@mail.ru

 

Differential equation using to describe planar librations of a satellite is considered. Satellite center of mass performs its motion in an elliptic orbit. Solution is computed over the time segment corresponding to one orbital revolution. Right hand side of ODE has a singularity for an eccentricity value equals to unity. Simultaneously solutions can't be prolonged over the limit interval of an independent variable range.

Metric to make the computational method to be depended regularly upon the orbital eccentricity e as e → 1 is chosen in space of solutions. When using an integral metrics in weight spaces such regularization of solution approximation is performed automatically in a vicinity of the limit case. The results obtained make possible to construct a computational algorithms of a prolongation the solutions with respect to the eccentricity up to its limit value e = 1. In order to resolve this problem one can use Newton's method in weight spaces with integral metrics such that the same metric "covers" both regular, and limit cases.

If e → 1, then solution has an oscillations of increasing frequency tending to infinity when approaching to singular points. For this reason usual numeric technique can't be applied without performing preliminary regularizing procedure. This latter is reduced to the change of an independent variable. However in this case to build the solution when integrating over increasing segment of an independent variable one need the computational resources also growing. Computational algorithms applied usually to approximate solutions of the problem under consideration are depended on the eccentricity singularly. Indeed, time of computation need tends to infinity as e → 1.

In current article the algorithm having the computational complexity independent upon the eccentricity has been considered. Such a complexity one can evaluate using the estimation for the convergence rate of Newton's iterative process. Computational experiments performed have shown the accuracy high enough, near 10^-4 relative to norm of the weight space was achieved in three or four iterations. Moreover numerical explorations show number of iterations need to compute the solution is independent on the value of the eccentricity. In other words the technique has been considered here is not sensitive to closeness of eccentricity to unit.

In the paper the method of regularization for an algorithm to compute solutions of singular equations in problems of mechanics is presented. Metrics of weight spaces play role of a measure for computational complexity of similar problems. The technique developed to approximate solutions can be used to investigate the families of solutions for differential equations having nonanalytical and even discontinuous right hand sides. The technique regarded makes possible to prolong the solutions numerically via Newton's method both on parameters of the problem, and on initial data in the phase space.