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On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds

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Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

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Abstract

The problem of the existence of explicit and at the same time conservative finite difference schemes that approximate a system of ordinary differential equations is investigated. An autonomous system of nonlinear ordinary differential equations on an algebraic manifold V is considered. A difference scheme for solving this system is called conservative, if the calculations of this scheme do not go beyond V, i.e., preserve it exactly. An explicit scheme is understood as such a difference scheme in which a system of linear equations is required to proceed to the next layer. We formulate the problem of constructing an explicit conservative scheme approximating a given autonomous system on a given manifold. For the case of 1-manifold, a solution to this problem is given and geometric obstacles to the existence of such difference schemes are indicated. Namely, it is proved that the scheme exists only if the genus of the integral curve is 1 or 0.

The publication has been prepared with the support of the “RUDN University Program 5-100”  and funded by RFBR according to the research projects Nos. 18-07-00567, 18-51-18005, and 19-01-00645 A.

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Author information

Authors and Affiliations

  1. Joint Institute for Nuclear Research (Dubna), Joliot-Curie, 6, Dubna, Moscow Region, 141980, Russia

    E. A. Ayryan & L. A. Sevastianov

  2. Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia

    M. D. Malykh, L. A. Sevastianov & Yu Ying

  3. Department of Algebra and Geometry, Kaili University, 3 Kaiyuan Road, Kaili, 556011, China

    Yu Ying

Authors
  1. E. A. Ayryan
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  2. M. D. Malykh
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  3. L. A. Sevastianov
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  4. Yu Ying
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Corresponding author

Correspondence to M. D. Malykh .

Editor information

Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Ayryan, E.A., Malykh, M.D., Sevastianov, L.A., Ying, Y. (2019). On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_23

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  • DOI: https://doi.org/10.1007/978-3-030-26831-2_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26830-5

  • Online ISBN: 978-3-030-26831-2

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