Cybernetics and Systems Analysis

, Volume 55, Issue 6, pp 988–998 | Cite as

A Splitting Scheme for Diffusion and Heat Conduction Problems

  • A. V. GladkyEmail author
  • Y. A. Gladka


The problem of mathematical modeling and optimization of nonstationary diffusion and heat conduction processes is considered. An approach that uses the idea of splitting and computation of the obtained difference schemes using explicit schemes of point to point computing is proposed for numerical solution of multidimensional diffusion and heat conduction initial–boundary-value problems. Construction of difference splitting schemes, approximation and stability on initial data are investigated. Differential properties of the quality functional are analyzed for the numerical solution of the optimal control problem for a parabolic equation. An iterative algorithm for finding the optimal control is proposed.


parabolic equation optimal control problem numerical method splitting methods difference scheme stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. I. Marchuk, Mathematical Modeling in the Environment Problem [in Russian], Nauka, Moscow (1982).Google Scholar
  2. 2.
    A. I. Egorov, Optimal Control of Thermal and Diffusion Processes [in Russian], Nauka, Moscow (1978).Google Scholar
  3. 3.
    A. I. Egorov, Fundamentals of Control Theory [in Russian], Fizmatgiz, Moscow (2004).Google Scholar
  4. 4.
    M. Z. Zgurovskii, V. V. Skopetskii, V. K. Khrushch, and N. N. Belyaev, Numerical Modeling of the Propagation of Contamination in the Environment [in Russian], Naukova Dumka, Kyiv (1997).Google Scholar
  5. 5.
    I. V. Sergienko, V. V. Skopetskii, and V. S. Deineka, Mathematical Modeling and Analysis of Processes in Inhomogeneous Media [in Russian], Naukova Dumka, Kyiv (1991).Google Scholar
  6. 6.
    A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer [in Russian], Editorial URSS, Moscow (2003).Google Scholar
  7. 7.
    V. K. Saul’ev, Integration of Parabolic Equations by the Grid Method [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
  8. 8.
    A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin (2007).CrossRefGoogle Scholar
  9. 9.
    P. N. Vabishchevich and V. I. Vasil’ev, “Computational algorithms for solving the coefficient inverse problem for parabolic equations,” Inverse Probl. Sci. Engin., Vol. 24, No. 1, 42–59 (2016).MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. N. Vabishchevich, V. I. Vasil’ev, and M. V. Vasil’eva, “Computational identification of the right-hand side of a parabolic equation,” Computational Math. and Math. Physics, Vol. 55, No. 6, 1015–1021 (2015).MathSciNetCrossRefGoogle Scholar
  11. 11.
    S. K. Godunov and V. S. Ryaben’kii, Difference Schemes [in Russian], Nauka, Moscow (1977).Google Scholar
  12. 12.
    V. I. Agoshkov, Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics [in Russian], IVM RAN, Moscow (2004).Google Scholar
  13. 13.
    V. S. Deineka and I. V. Sergienko, Models and Methods to Solve Problems in Inhomogeneous Media [in Russian], Naukova Dumka, Kyiv (2001).Google Scholar
  14. 14.
    V. P. Ilyin, Methods of Finite Differences and Finite Volumes for Elliptical Equations [in Russian], Izd. IM SO RAN, Novosibirsk (2001).Google Scholar
  15. 15.
    P. N. Vabishchevich and P. E. Zakharov, “Explicit-implicit splitting schemes for parabolic equations and systems,” in: Numerical Methods and Applications, Springer (2015), pp. 157–166.Google Scholar
  16. 16.
    G. I. Marchuk, Splitting Methods [in Russian], Nauka, Moscow (1988).Google Scholar
  17. 17.
    A. V. Gladky, “Stability of difference splitting schemes for convection diffusion equation,” Cybern. Syst. Analysis, Vol. 53, No. 2, 193–203 (2017).MathSciNetCrossRefGoogle Scholar
  18. 18.
    A. V. Gladky, “Analysis of splitting algorithms in convection–diffusion problems,” Cybern. Syst. Analysis, Vol. 50, No. 4, 548–559 (2014).CrossRefGoogle Scholar
  19. 19.
    P. N. Vabishchevich, “On a new class of additive (splitting) operator sub difference schemes,” Math. Comput., Vol. 81, No. 277, 267–276 (2012).CrossRefGoogle Scholar
  20. 20.
    P. N. Vabishchevich, “Flux-splitting schemes for parabolic problems,” Computational Math. and Math. Physics, Vol. 52, No. 8, 1128–1138 (2012).MathSciNetCrossRefGoogle Scholar
  21. 21.
    A. A. Samarskii and A. V. Gulin, Stability of Difference Schemes [in Russian], Nauka, Moscow (1973).Google Scholar
  22. 22.
    G. I. Marchuk and V. N. Agoshkov, An Introduction to Projection–Mesh Methods [in Russian], Nauka, Moscow (1981).Google Scholar
  23. 23.
    O. M. Alifanov, Extremum Methods to Solve Ill-Posed Problems [in Russian], Nauka, Moscow (1988).Google Scholar
  24. 24.
    P. F. Vasil’ev, Optimization Methods [in Russian], Faktorial Press, Moscow (2002).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine
  2. 2.Kyiv National Economic University Named after Vadym HetmanKyivUkraine

Personalised recommendations