RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Model. Anal. Inform. Sist.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 529–538 (Mi mais519)  

This article is cited in 1 scientific paper (total in 1 paper)

Numerical methods of solving Cauchy problems with contrast structures

A. A. Belova, N. N. Kalitkinb

a Lomonosov Moscow State University, 1–2 Leninskie Gory, Moscow 119991, Russia
b Keldysh Institute of Applied Mathematics RAS, 4 Miusskaya sq., Moscow 125047, Russia

Abstract: Modern numerical methods allowing to solve contrast structure problems in the most efficient way are described. These methods include explicit-implicit Rosenbrock schemes with complex coefficients and fully implicit backward optimal Runge–Kutta schemes. As an integration argument, it is recommended to choose the length of the integral curve arc. This argument provides high reliability of the calculation and sufficiently decreases the complexity of computations for low-order systems. In order to increase the efficiency, we propose an automatic step selection algorithm based on curvature of the integral curve. This algorithm is as efficient as standard algorithms and has sufficiently larger reliability. We show that along with such an automatic step selection it is possible to calculate a posteriori asymptotically precise error estimation. Standard algorithms do not provide such estimations and their actual error quite often exceeds the user-defined tolerance by several orders. The applicability limitations of numerical methods are investigated. In solving superstiff problems, they sometimes do not provide satisfactory results. In such cases, it is recommended to imply approximate analytical methods. Consequently, numerical and analytical methods are complementary.

Keywords: stiff Cauchy problem, contrast structure, automatic step selection, curvature in multidimensional space, Richardson method estimations, singularity diagnostics, solution blow-up.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00161_а
16-31-00062_мол_а
This work was supported by RFBR grants, Projects №14-01-00161, 16-31-00062.


DOI: https://doi.org/10.18255/1818-1015-2016-5-529-538

Full text: PDF file (696 kB)
References: PDF file   HTML file

Bibliographic databases:

UDC: 519.6
Received: 20.06.2016

Citation: A. A. Belov, N. N. Kalitkin, “Numerical methods of solving Cauchy problems with contrast structures”, Model. Anal. Inform. Sist., 23:5 (2016), 529–538

Citation in format AMSBIB
\Bibitem{BelKal16}
\by A.~A.~Belov, N.~N.~Kalitkin
\paper Numerical methods of solving Cauchy problems with contrast structures
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 5
\pages 529--538
\mathnet{http://mi.mathnet.ru/mais519}
\crossref{https://doi.org/10.18255/1818-1015-2016-5-529-538}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=514844}
\elib{http://elibrary.ru/item.asp?id=27202302}


Linking options:
  • http://mi.mathnet.ru/eng/mais519
  • http://mi.mathnet.ru/eng/mais/v23/i5/p529

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. N. Kalitkin, A. A. Belov, P. E. Bulatov, “Raschet khimicheskoi kinetiki yavnymi skhemami s geometricheski-adaptivnym vyborom shaga”, Preprinty IPM im. M. V. Keldysha, 2018, 173, 32 pp.  mathnet  crossref
  • Моделирование и анализ информационных систем
    Number of views:
    This page:216
    Full text:81
    References:20

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020