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On a Comparison of Several Numerical Integration Methods for Ordinary Systems of Differential Equations

  • Anatoliy G. Korotchenko
  • Valentina M. SmoryakovaEmail author
Conference paper
  • 11 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11974)

Abstract

The paper considers the numerical integration methods for ordinary systems of differential equations in which the end of the integration interval is a priori undefined but is defined during the integration process instead. Moreover, the calculation of right hand sides of such systems is an expensive procedure. The paper describes a new integration strategy based on an implicit fourth order method. The proposed strategy employs the behavior of obtained solution to control the integration process. In addition, the number of integration nodes selected by the mentioned method is minimal at every fixed interval under the limitations defined by the local error which results from the approximation of system derivatives.

Keywords

Finite difference formulas Integration strategies Optimal strategy 

Notes

Acknowledgments

The work is financially supported by the Federal Targeted Program for Research and Development in Priority Areas of Development of the Russian Scientific and Technological Complex for 2014–2020 under the contract No. 14.578.21.0246 (unique identifier RFMEFI57817X0246).

References

  1. 1.
    Effati, S., Roohparvar, H.: Iterative dynamic programming for solving linear and nonlinear differential equations. Appl. Math. Comput. 175, 247–257 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Korotchenko, A.G., Lapin, A.V.: On an numerical integration algorithm with optimal choice of step. Vestn. Lobachevsky State Univ. Nizhni Novgorod 24(2), 270–278 (2001)zbMATHGoogle Scholar
  3. 3.
    Korotchenko, A.G., Lapin, A.V.: About construction of the approximately optimal algorithm of numerical integration. Vestn. Lobachevsky State Univ. Nizhni Novgorod 1(26), 189–195 (2003)Google Scholar
  4. 4.
    Korotchenko, A., Smoryakova, V.: On a method of construction of numerical integration formulas. In: AIP Conference on Proceedings, Numerical Computations: Theory and Algorithms (NUMTA-2016) 1776, 090012 (2016)Google Scholar
  5. 5.
    Sergeyev, Y.D.: Solving ordinary differential equations on the infinity computer by working with infinitesimals numerically. Appl. Math. Comput. 219, 10668–10681 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Shampine, L.: The Matlab ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.N.I. Lobachevsky State UniversityNizhni NovgorodRussia

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