Application of the Fast Expansion Method in Space–Related Problems

Mikhail Ivanovich Popov,Alexey Viktorovich Chernyshov,Roman Alexandrovich Druzhinin,Sergey Yurievich Sablin,Vladimir Valentinovich Nikitin,Vyacheslav Gennadievich Kozlov,Aleksey Vasilyevich Skrypnikov,Alexander Danilovich Chernyshov

doi:10.13189/ms.2022.100206

Abstract

In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

Highlights

The calculation of the spacecraft trajectory in the atmosphere of the earth or other planets is of great importance in the preparation of flights of satellites and space vehicles [1]
The study of the obtained solution has revealed that, in order to achieve high accuracy of the solution in the fast expansion method, it is necessary to increase the order of the boundary function, and leave the number of terms in the Fourier series minimal
Because of limited computer resources, it is impossible to achieve an accuracy of the order of 10-38, obtained by the fast expansion method, using finite differences

Summary

Introduction

The calculation of the spacecraft trajectory in the atmosphere of the earth or other planets is of great importance in the preparation of flights of satellites and space vehicles [1]. It is important to obtain initial information on flight trajectory parameters for calculating onboard flight mission coefficients. Such a calculation is needed to implement optimal control to minimize costs. Mathematics and Statistics 10(2): 320-328, 2022 of outer space remote from the Earth. The cost of such the acceleration of gravity, β is the aerodynamic drag flights is very high, so the task of calculating the trajectory coefficient, Px , Py , Pz are the components of the reactive becomes more relevant [2]

Methods

Results

Conclusion