Fast evaluation of multivariate monomials for speeding up numerical integration in space dynamics

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Abstract

Many differential equations of Dynamics (i.e. Celestial Mechanics, Molecular Dynamics, and so on) one can reduce to polynomial form, i.e. to system of differential equations with polynomial (in unknowns) right-hand sides. It implies that at every step of numerical integration of these equations, one needs to evaluate many different multivariate monomials for many values of variables, and that is why minimizing the evaluation cost of a system of monomials in the right-hand sides is an important problem. In Alesova, Babadzanjanz, et al., “Schemes of Fast Evaluation of Multivariate Monomials for Speeding up Numerical Integration of Equations in Dynamics” (AIP, volume 1978, issue 1, 2018) we considered a scheme of successive multiplications minimizing the total cost of evaluation of multivariate monomials of a system of monomials and the algorithm, which for a given system of third order monomials reduces the original problem to the linear programming problem, and computes such a scheme. Then we proposed the algorithm and the corresponding Mathematica program that, given an arbitrary system of multivariate cubic monomials constructs the linear programming problem mentioned. We have also presented the results of corresponding numerical experiments and have shown that the total evaluation cost of systems of monomials has reduced substantially. An important requirement for the process of solving differential equations in Dynamics is high accuracy at large time intervals. One of effective tools for obtaining such solutions is the Taylor series method. In Alesova, Babadzanjanz, et al., “High-Precision Numerical Integration of Equations in Dynamics” (AIP, volume 1959, issue 1, 2018) we considered the equations of the N-body problem in various polynomial forms (with and without additional third order polynomial perturbations). This allowed us to obtain effective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. Moreover, we considered a number of corresponding numerical experiments, which showed the effectiveness of the Taylor series method implementation presented. In present work, we generalize the results mentioned above on the case of systems of multivariate fifth order monomials (using in corresponding numerical experiments differential equations of the N-body problem with additional fifth order polynomial perturbations).

Original languageEnglish
Title of host publication19th International Multidisciplinary Scientific Geoconference, SGEM 2019; Albena; Bulgaria; 30 June 2019 до 6 July 2019
Pages647-654
ISBN (Electronic)9786197408898
DOIs
StatePublished - 2019
Event19th International Multidisciplinary Scientific GeoConference SGEM 2019 - Албена, Bulgaria
Duration: 30 Jun 20196 Jul 2019

Publication series

NameInternational Multidisciplinary Scientific GeoConference-SGEM
Number6.2
Volume19
ISSN (Print)1314-2704

Conference

Conference19th International Multidisciplinary Scientific GeoConference SGEM 2019
CountryBulgaria
CityАлбена
Period30/06/196/07/19

Scopus subject areas

  • Geotechnical Engineering and Engineering Geology
  • Geology

Keywords

  • Dynamics
  • Multivariate monomials
  • Numerical integration
  • Taylor series method

Fingerprint Dive into the research topics of 'Fast evaluation of multivariate monomials for speeding up numerical integration in space dynamics'. Together they form a unique fingerprint.

Cite this