Skip to main content
SpringerLink
Log in

Compact Difference Schemes on a Three-Point Stencil for Hyperbolic-Parabolic Equations with Constant Coefficients

  • NUMERICAL METHODS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

Stable compact weighted difference schemes of \(4+2 \) and \(4+4 \) approximation orders are studied for a hyperbolic-parabolic equation with constant coefficients. The results obtained are generalized to the cases of an equation with a variable coefficient, a quasilinear equation, and a multidimensional equation. A priori estimates for the stability and convergence of the difference solution in strong norms are obtained. It is shown that the test numerical calculations presented in the paper are consistent with the theoretical conclusions.

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.

Similar content being viewed by others

REFERENCES

  1. Tikhonov, A.N. and Samarskii, A.A., Equations of Mathematical Physics, New York: Dover, 1990.

    Google Scholar 

  2. Huang, Ya. and Yin, Zh., The compact finite difference method of two–dimensional Cattaneo model, J. Func. Spaces, 2020, vol. 1, pp. 1–12.

    MathSciNet  MATH  Google Scholar 

  3. Samarskii, A.A., Vabishchevich, P.N., and Matus, P.P., Raznostnye skhemy s operatornymi mnozhitelyami (Difference Schemes with Operator Multipliers), Minsk: ZAO TsOTZh, 1998.

    Google Scholar 

  4. Zolina, L.A., On a boundary value problem for a model equation of hyperbolic-parabolic type, Comput. Math. Math. Phys., 1966, vol. 6, no. 6, pp. 63–78.

    Article  MathSciNet  MATH  Google Scholar 

  5. Samarskii, A.A., Vabishchevich, P.N., Lemeshevskii, S.V., and Matus, P.P., Difference schemes for the problem of fusing hyperbolic and parabolic equations, Sib. Math. J., 1998, vol. 39, no. 4, pp. 825–833.

    Article  Google Scholar 

  6. Korzyuk, V.I., Lemeshevskii, S.V., and Matus, P.P., Conjugation problem about jointly separate flow of viscoelastic and viscous fluids in the plane duct, Math. Model. Anal., 1999, vol. 4, no. 1, pp. 114–123.

    Article  MathSciNet  MATH  Google Scholar 

  7. Korzyuk, V.I., Lemeshevskii, S.V., and Matus, P.P., Conjugation problem of jointly separate flow of viscoelastic and viscous fluids in the plane pipe, Dokl. Nats. Akad. Nauk Belarusi, 2000, vol. 44, no. 2, pp. 5–8.

    MathSciNet  MATH  Google Scholar 

  8. Chetverushkin, B.N., Morozov, D.N., Trapeznikova, M.A., Churbanova, N.G., and Shil’nikov, E.V., An explicit scheme for the solution of the filtration problems, Math. Models Comput. Simul., 2010, vol. 2, no. 6, pp. 669–677.

    Article  MathSciNet  Google Scholar 

  9. Vragov, V.N., On a mixed problem for a class of hyperbolic-parabolic equations, Dokl. Akad. Nauk SSSR, 1975, vol. 224, no. 2, pp. 273–276.

    MathSciNet  Google Scholar 

  10. Vong, S.W., Pang, H.K., and Jin, X.Q., A high-order difference scheme for the generalized Cattaneo equation, East Asian J. Appl. Math., 2012, vol. 2, no. 2, pp. 170–184.

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhao, X. and Sun, Z.Z., Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium, J. Sci. Comput., 2015, vol. 62, no. 3, pp. 747–771.

    Article  MathSciNet  MATH  Google Scholar 

  12. Samarskii, A.A., Schemes of high-order accuracy for the multi-dimensional heat conduction equation, Comput. Math. Math. Phys., 1963, vol. 3, no. 5, pp. 1107–1146.

    Article  MATH  Google Scholar 

  13. Valiullin, A.N. and Paasonen, V.I., Economical difference schemes of higher order of accuracy for the multidimensional equation of oscillations, Chisl. Metody Mekh. Sploshnoi Sredy, 1970, vol. 1, no. 1, pp. 17–30.

    Google Scholar 

  14. Matus, P.P. and Utebaev, B.D., Compact and monotone difference schemes for parabolic equations, Math. Models Comput. Simul., 2021, vol. 13, no. 6, pp. 1038–1048.

    Article  MATH  Google Scholar 

  15. Matus, P.P. and Hoang Thi Kieu Anh, Compact difference schemes on a three-point stencil for second-order hyperbolic equations, Differ. Equations, 2021, vol. 57, no. 7, pp. 934–946.

    Article  MathSciNet  MATH  Google Scholar 

  16. Matus, P.P. and Hoang Thi Kieu Anh, Compact difference schemes for the multidimensional Klein–Gordon equation, Differ. Equations, 2022, vol. 58, no. 1, pp. 120–138.

    Article  MathSciNet  MATH  Google Scholar 

  17. Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Difference Schemes), Moscow: Nauka, 1989.

    Google Scholar 

  18. Lapinska-Chrzczonowicz, M. and Matus, P., Exact difference scheme and difference scheme of higher order of approximation for a convection–diffusion equation. I, Ann. UMCS. Inf. AI , 2013, vol. 13, no. 1, pp. 37–51.

    MathSciNet  MATH  Google Scholar 

  19. Matus, P.P., Churbanova, N.G., and Shchadinskii, D.A., On the role of conservation laws and input data in the generation of peaking modes in quasilinear multidimensional parabolic equations with nonlinear source and in their approximations, Differ. Equations, 2016, vol. 52, no. 7, pp. 942–950.

    Article  MathSciNet  MATH  Google Scholar 

  20. Jovanović, B.S. and Matus, P.P., Coefficient stability of second-order operator-differential equations, Differ. Equations, 2002, vol. 38, no. 10, pp. 1460–1466.

  21. Wloka, J., Partial Differential Equations, Cambridge: Cambridge Univ. Press, 1987.

    Book  MATH  Google Scholar 

  22. Jovanović, B., Lemeshevsky, S., and Matus, P., On the stability of differential-operator equations and operator-difference schemes as \(t\to \infty \), Comput. Meth. Appl. Math., 2002, vol. 2, no. 2, pp. 153–170.

  23. Valiullin, A.N., Skhemy povyshennoi tochnosti dlya zadach matematicheskoi fiziki (Schemes of Increased Accuracy for Problems of Mathematical Physics) Novosibirsk: Novosib. Gos. Univ., 1973.

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, 220072, Belarus

    P. P. Matus

  2. The John Paul Catholic University of Lublin, Lublin, 20-950, Poland

    P. P. Matus & D. Pylak

  3. Belarusian State University, Minsk, 220030, Belarus

    Hoang Thi Kieu Anh

Authors
  1. P. P. Matus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hoang Thi Kieu Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Pylak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to P. P. Matus, Hoang Thi Kieu Anh or D. Pylak.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Matus, P.P., Hoang Thi Kieu Anh & Pylak, D. Compact Difference Schemes on a Three-Point Stencil for Hyperbolic-Parabolic Equations with Constant Coefficients. Diff Equat 58, 1277–1286 (2022). https://doi.org/10.1134/S0012266122090129

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266122090129

Search

Navigation