Skip to main content
SpringerLink
Log in

Criterion for the Stability of Difference Schemes for Nonlinear Differential Equations

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

Abstract

For abstract nonlinear difference schemes with operators acting in finite-dimensional Banach spaces, a stability criterion is stated and proved; namely, for a consistent finite-difference approximation to a well-posed differential problem, the solution of the difference scheme converges if and only if the scheme is unconditionally stable. In a sense, this criterion generalizes Lax’s equivalence theorem to nonlinear differential problems. The results obtained are used to study the stability of difference schemes that approximate quasilinear parabolic equations with nonlinearities of unbounded growth.

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

Similar content being viewed by others

REFERENCES

  1. Ryaben’kii, V.S. and Filippov, A.F., Ob ustoichivosti raznostnykh skhem (On the Stability of Difference Schemes), Moscow: GITTL, 1956.

    Google Scholar 

  2. Babenko, K.I., Osnovy chislennogo analiza (Fundamentals of Numerical Analysis), Moscow–Izhevsk: Regulyarnaya Khaoticheskaya Din., 2002.

    MATH  Google Scholar 

  3. Richtmyer, R.D. and Morton, K.W., Difference Methods for Initial-Value Problems, New York–London–Sydney: Interscience, 1967. Translated under the title: Raznostnye metody resheniya kraevykh zadach, Moscow: Mir, 1972.

    MATH  Google Scholar 

  4. Guo Ben-Yu (Kuo Pen-Yu), Generalized stability of discretization and its applications to numerical solutions of non-linear partial differential equations, Comput. Math. Math. Phys., 1992, vol. 32, no. 4, pp. 530–541.

    MathSciNet  MATH  Google Scholar 

  5. Yakut, L.I., Lax theorems for nonlinear evolution equations, Dokl. Akad. Nauk SSSR, 1964, vol. 156, no. 6, pp. 1304–1307.

    MathSciNet  Google Scholar 

  6. Yakut, L.I., To the issue of justifying the convergence of difference schemes, Dokl. Akad. Nauk SSSR, 1963, vol. 151, no. 1, pp. 76–79.

    MathSciNet  Google Scholar 

  7. Samarskii, A.A., Teoriya raznostnykh skhem (Theory of Difference Schemes), Moscow: Nauka, 1977.

    MATH  Google Scholar 

  8. Matus, P., Stability of difference schemes for non-linear time-dependent problems, Comput. Meth. Appl. Math., 2003, vol. 3, no. 2, pp. 313–329.

    Article  Google Scholar 

  9. Matus, P.P., Well-posedness of difference schemes for semilinear parabolic equations with weak solutions, Comput. Math. Math. Phys., 2010, vol. 50, no. 12, pp. 2155–2175.

    Article  MathSciNet  Google Scholar 

  10. Matus, P.P. and Lemeshevsky, S.V., Stability and monotonicity of difference schemes for nonlinear scalar conservation laws and multidimensional quasi-linear parabolic equations, Comput. Meth. Appl. Math., 2009, vol. 9, no. 3, pp. 253–280.

    Article  MathSciNet  Google Scholar 

  11. Matus, P.P., Stability with respect to the initial data and monotonicity of an implicit difference scheme for a homogeneous porous medium equation with a quadratic nonlinearity, Differ. Equations, 2010, vol. 46, no. 7, pp. 1019–1029.

    Article  MathSciNet  Google Scholar 

  12. Yakubuk, R.M., Stability with respect to the input data and monotonicity of an implicit finite-difference scheme for a quasilinear parabolic equation, Differ. Equations, 2012, vol. 48, no. 2, pp. 283–295.

    Article  MathSciNet  Google Scholar 

  13. Matus, P.P. and Chuiko, M.M., Investigation of the stability and convergence of difference schemes for a polytropic gas with subsonic flows, Differ. Equations, 2009, vol. 45, no. 7, pp. 1074–1085.

    Article  MathSciNet  Google Scholar 

  14. Martsinkevich, G.L., Matus, P.P., and Chuiko, M.M., Stability of difference schemes in terms of Riemann invariants for a polytropic gas, Comput. Math. Math. Phys., 2010, vol. 50, no. 6, pp. 1024–1037.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  16. Abrashin, V.N., Difference schemes for nonlinear hyperbolic equations. I, Differ. Uravn., 1973, vol. 9, no. 11, pp. 2029–2040.

    MathSciNet  MATH  Google Scholar 

  17. Lyashko, A.D. and Fedotov, E.M., Studying nonlinear two-level operator-difference schemes with weights, Differ. Uravn., 1985, vol. 21, no. 7, pp. 1217–1227.

    MATH  Google Scholar 

  18. Matus, P.P. and Stanishevskaya, L.V., On unconditional convergence of difference schemes for time-varying quasilinear equations of mathematical physics, Differ. Uravn., 1991, vol. 27, no. 7, pp. 1203–1219.

    MATH  Google Scholar 

  19. Matus, P., On convergence of difference schemes for IBVP for quasilinear parabolic equations with generalized solutions, Comput. Meth. Appl. Math., 2014, vol. 14, no. 3, pp. 361–371.

    Article  MathSciNet  Google Scholar 

  20. Godunov, S.K., A difference method for numerical calculation of discontinuous solutions of equations of hydrodynamics, Mat. Sb., 1959, vol. 47(89), no. 3, pp. 271–306.

    MathSciNet  MATH  Google Scholar 

  21. Karchevskii, M.M. and Pavlova, M.F., Uravneniya matematicheskoi fiziki. Dopolnitel’nye glavy (Equation of Mathematical Physics. Additional Chapters), Kazan: Izd. Kazan. Gos. Univ., 2008.

    Google Scholar 

  22. Matus, P., Hieu, L.M., and Vulkov, L.G., Analysis of second order difference schemes on non-uniform grids for quasilinear parabolic equations, J. Comput. Appl. Math., 2017, vol. 310, pp. 186–199.

    Article  MathSciNet  Google Scholar 

  23. Matus, P.P. and Utebaev, B.D., Compact and monotone difference schemes for parabolic equations, Mat. Model., 2021, vol. 33, no. 4, pp. 60–78.

    MathSciNet  MATH  Google Scholar 

Download references

ACKNOWLEDGMENTS

The author is grateful to Prof. B.S. Jovanovic for discussions of this work and helpful remarks.

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

Authors
  1. P. P. Matus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. P. Matus.

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. Criterion for the Stability of Difference Schemes for Nonlinear Differential Equations. Diff Equat 57, 805–813 (2021). https://doi.org/10.1134/S0012266121060082

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266121060082

Search

Navigation