Skip to main content
SpringerLink
Log in

Exponential Euler and Backward Euler Methods for Nonlinear Heat Conduction Problems

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).

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

REFERENCES

  1. W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations (Springer, Berlin, 2003).

    Book  MATH  Google Scholar 

  2. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Courier, New York, 2013).

    Google Scholar 

  3. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge Univ. Press, Cambridge, 1991).

    Book  MATH  Google Scholar 

  4. M. Hochbruck and A. Ostermann, Acta Numer. 19, 209 (2010).

    Article  MathSciNet  Google Scholar 

  5. K. Dekker and J. G. Verwer, Stability of Runge-Kutta Methods for Stiff Non-Linear Differential Equations (North-Holland Elsevier Science, Amsterdam, 1984).

    MATH  Google Scholar 

  6. F. R. Gantmacher, The Theory of Matrices (AMS Chelsea, Providence, RI, 1998), Vol. 1.

    MATH  Google Scholar 

  7. N. J. Higham, Functions of Matrices: Theory and Computation (Soc. Ind. Appl. Math., Philadelphia, PA, USA, 2008).

    Book  MATH  Google Scholar 

  8. M. A. Botchev, L. Knizhnerman, and E. E. Tyrtyshnikov, SIAM J. Sci. Comput. 43, A3733 (2021). https://doi.org/10.1137/20M1375383

    Article  Google Scholar 

  9. V. T. Zhukov, Math. Models Comput. Simul. 3, 311 (2011). https://doi.org/10.1134/S2070048211030136

    Article  MathSciNet  Google Scholar 

Download references

Funding

The work of the first author is supported by the Russian Science Foundation, grant no. 19-11-00338. The first authors thanks Leonid Knizhherman for a useful suggestion.

Author information

Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, 125047, Moscow, Russia

    M. A. Botchev & V. T. Zhukov

  2. Marchuk Institute of Numerical Mathematics of Russian Academy of Sciences, 119333, Moscow, Russia

    M. A. Botchev

Authors
  1. M. A. Botchev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. T. Zhukov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to M. A. Botchev or V. T. Zhukov.

Additional information

(Submitted by A. B. Muravnik)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Botchev, M.A., Zhukov, V.T. Exponential Euler and Backward Euler Methods for Nonlinear Heat Conduction Problems. Lobachevskii J Math 44, 10–19 (2023). https://doi.org/10.1134/S1995080223010067

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080223010067

Keywords:

Search

Navigation