Skip to main content
SpringerLink
Log in

A Hybrid Difference Scheme under Generalized Approximation Condition in the Space of Undetermined Coefficients

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Construction of difference schemes of high approximation orders for hyperbolic problems is still an important problem. For the construction of grid-characteristic methods, difference schemes were earlier analyzed in the space of undetermined coefficients, where the coefficients of high order derivatives in the first differential approximation of the difference scheme were used as the objective function to be minimized. Other reasonable functionals in the space of undetermined coefficients that are linear in the coefficients of the scheme may be used. By solving a linear programming problem, difference schemes meeting various conditions can be chosen. An example of the linear functional related to the approximation properties of the problem is discussed. It is proposed to call it the generalized approximation condition. Based on this condition, a difference scheme of a novel class is built that has no analogs in the literature. The presentation uses the transport equation with a constant coefficient as an example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.

Similar content being viewed by others

REFERENCES

  1. A. S. Kholodov, “On the construction of difference scheme with positive approximation for hyperbolic equations,” Zh. Vychisl. Mat. Mat. Fiz. 18, 1476–1492 (1978).

    MathSciNet  MATH  Google Scholar 

  2. M.-K. M. Magomedov and A. S. Kholodov, Grid–Characteristic Numerical Methods (Nauka, Moscow, 1988) [in Russian].

    MATH  Google Scholar 

  3. A. S. Kholodov, in Encyclopedia of Low-Temperature Plasma, Ser. B, Vol. VII-1, Part 2: Mathematical Modeling in Low-Temperature Plasma, Vol. VII-1 (Yanus-K, Moscow 2009), p. 141.

  4. V. M. Goloviznin and A. A. Samarskii, “Properties of the Iserles difference scheme,” Mat. Model. 10 (1), 101–116 (1998).

    MathSciNet  MATH  Google Scholar 

  5. B. V. Rogov and M. N. Mikhailovskaya, “Monotonic bicompact schemes for linear transport equations,” Math. Models Comput. Simul. 4, 92–100 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. V. Rogov and M. N. Mikhailovskaya, “Bicompact monotonic schemes for a multidimensional linear transport equation,” Math. Models Comput. Simul. 4, 355–362 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. I. Lobanov, “Difference schemes in the undetermined coefficient space and dual problems of linear programming,” Comput. Math. Math. Phys. 58 (6) (2018).

  8. S. K. Godunov and V. S. Ryaben’kii, “Finite difference schemes for the transport equation satisfying a generalized approximation condition,” Komp’ut. issled. modelir. 10 (2) (2018).

  9. S. K. Godunov and V. S. Ryaben’kii, Finite Difference Schemes. Introduction to the Theory (Nauka, Moscow, 1977) [in Russian].

    Google Scholar 

  10. S. K. Godunov, “A difference scheme for numerical calculation of discontinuous solution of hydrodynamic equations,” Mat. Sb. 47 (3), 271–306 (1959).

    MathSciNet  MATH  Google Scholar 

  11. R. P. Fedorenko, “Application of high-order difference schemes for the numerical solution of hyperbolic equations,” Zh. Vychisl. Mat. Mat. Fiz. 2, 1122–1128 (1962).

    Google Scholar 

  12. R. P. Fedorenko, Introduction to Computational Physics (Intellekt, Dolgoprudnyi, 2009) [in Russian].

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology, 141700, Dolgoprudnyi, Moscow oblast, Russia

    A. I. Lobanov & F. Kh. Mirov

Authors
  1. A. I. Lobanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Kh. Mirov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Lobanov.

Additional information

Translated by A. Klimontovich

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lobanov, A.I., Mirov, F.K. A Hybrid Difference Scheme under Generalized Approximation Condition in the Space of Undetermined Coefficients. Comput. Math. and Math. Phys. 58, 1270–1279 (2018). https://doi.org/10.1134/S0965542518080134

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542518080134

Keywords:

Search

Navigation