Skip to main content
SpringerLink
Log in

A fully adaptive explicit stabilized integrator for advection–diffusion–reaction problems

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

A novel second order family of explicit stabilized Runge–Kutta–Chebyshev methods for advection–diffusion–reaction equations is introduced. The new methods outperform existing schemes for relatively high Peclet number due to their favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. An adaptive algorithm to implement the new scheme is proposed. This algorithm is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new algorithm are presented.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. Indeed, up to order two, the order conditions for nonlinear problems are the same as the order conditions for linear problems [14, Chap. III].

References

  1. Abdulle, A.: Chebyshev methods based on orthogonal polynomials. PhD Thesis, University of Geneva, Department of Mathematics. University of Geneva, (2001)

  2. Abdulle, A.: Fourth order Chebyshev methods with recurrence relation. SIAM J. Sci. Comput. 23(6), 2041–2054 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abdulle, A.: ROCK2 and ROCK4: software for stiff differential equations (discretized parabolic problems). Codes available under http://anmc.epfl.ch/, 2(002)

  4. Abdulle, A.: Explicit Stabilized Runge–Kutta Methods, pages 460–468. Encyclopedia of Applied and Computational Mathematics, Springer Berlin Heidelberg, (2015)

  5. Abdulle, A., Almuslimani, I., Vilmart, G.: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations. SIAM/ASA J. Uncertain. Quantif. 6(2), 937–964 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Abdulle, A., de Souza, G.R.: Explicit stabilized multirate method for stiff stochastic differential equations. arXiv:2010.15193, (2020)

  7. Abdulle, A., Li, T.: S-ROCK methods for stiff Ito SDEs. Commun. Math. Sci. 6(4), 845–868 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Abdulle, A., Medovikov, A.: Second order chebyshev methods based on orthogonal polynomials. Numer. Math. 90(1), 1–18 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Abdulle, A., Vilmart, G.: PIROCK: a swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise. J. Comput. Phys. 242, 869–888 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Abdulle, A., Vilmart, G., Zygalakis, K.C.: Weak second order explicit stabilized methods for stiff stochastic differential equations. SIAM J. Sci. Comput. 35(4), A1792–A1814 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Almuslimani, I.: Explicit Stabilized Methods for Stiff Stochastic Differential Equations and Stiff Optimal Control Problems. University of Geneva. PhD thesis, (2020)

  12. Almuslimani, I., Vilmart, G.: Explicit stabilized integrators for stiff optimal control problems. SIAM J. Sci. Comput. 43(2), A721–A743 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bakker, M., Analytical aspects of a minimax problem.: Technical Note TN 62 (in Dutch). Mathematical centre, Amsterdam (1971)

  14. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. Structure-preserving algorithms for ordinary differential equations

  15. Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems, volume 8. Springer Verlag Series in Comput. Math., Berlin, (1993)

  16. Hairer, E., Wanner, G.: Solving ordinary differential equations II. Stiff and differential-algebraic problems. Springer-Verlag, Berlin and Heidelberg (1996)

    MATH  Google Scholar 

  17. Hundsdorfer, W., Verwer, J.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Computational Mathematics, vol. 33. Springer-Verlag, Berlin (2003)

  18. Jeltsch, R., Torrilhon, M.: Flexible stability domains for explicit Runge-Kutta methods. In Some topics in industrial and applied mathematics, volume 8 of Ser. Contemp. Appl. Math. CAM, pages 152–180. Higher Ed. Press, Beijing, (2007)

  19. Ketcheson, D.I., Ahmadia, A.J.: Optimal stability polynomials for numerical integration of initial value problems. Commun. Appl. Math. Comput. Sci. 7(2), 247–271 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sommeijer, B., Shampine, L., Verwer, J.: RKC: an explicit solver for parabolic PDEs. J. Comput. Appl. Math. 88, 316–326 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sommeijer, B.P., Verwer, J.G.: A performance evaluation of a class of Runge-Kutta-Chebyshev methods for solving semidiscrete parabolic differential equations. Afdeling Numerieke Wiskunde [Department of Numerical Mathematics], 91. Mathematisch Centrum, Amsterdam, (1980)

  22. Tang, X., Xiao, A.: Improved runge-kutta-chebyshev methods. Math. Comput. Simul. 174, 59–75 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Torrilhon, M., Jeltsch, R.: Essentially optimal explicit Runge-Kutta methods with application to hyperbolic-parabolic equations. Numer. Math. 106(2), 303–334 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Van der Houwen, P., Sommeijer, B.: On the internal stage runge-kutta methods for largem-values. Z. Angew. Math. Mech. 60, 479–485 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. van der Houwen, P.J., Sommeijer, B.P.: On the internal stability of explicit, \(m\)-stage Runge-Kutta methods for large \(m\)-values. Z. Angew. Math. Mech. 60(10), 479–485 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Verwer, J., Hundsdorfer, W., Sommeijer, B.: Convergence properties of the runge-kutta-chebyshev method. Numer. Math. 57, 157–178 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  27. Verwer, J.G., Sommeijer, B.P.: An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. SIAM J. Sci. Comput. 25(5), 1824–1835 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Verwer, J.G., Sommeijer, B.P., Hundsdorfer, W.: RKC time-stepping for advection-diffusion-reaction problems. J. Comput. Phys. 201(1), 61–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zbinden, C.J.: Partitioned Runge-Kutta-Chebyshev methods for diffusion-advection-reaction problems. SIAM J. Sci. Comput. 33(4), 1707–1725 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to Gilles Vilmart for helpful discussions and comments.

Author information

Authors and Affiliations

  1. Univ Rennes, INRIA Rennes, IRMAR - UMR 6625, F-35000, Rennes, France

    Ibrahim Almuslimani

Authors
  1. Ibrahim Almuslimani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ibrahim Almuslimani.

Additional information

Communicated by Christian Lubich.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the Swiss National Science Foundation, project No: P2GEP2_195212.

Appendix

Appendix

1.1 Damping and number of stages for some choices of Peclet number

One can get more tables and increase the adaptivity of the algorithm with respect to damping. This will for sure increase the performance of the method. However, the method performs already very well with the tables we provide here.

Table 3 \(1/2<P_e'\le 1,\,1/4<\rho _A/\sqrt{\rho _D}\le 1/2\)
Full size table
Table 4 \(1<P_e'\le 3/2,\,1/2<\rho _A/\sqrt{\rho _D}\le 3/4\)
Full size table
Table 5 \(3/2<P_e'\le 2,\,3/4<\rho _A/\sqrt{\rho _D}\le 1\)
Full size table
Table 6 \(2<P_e'\le 2\sqrt{2},\,1<\rho _A/\sqrt{\rho _D}\le \sqrt{2}\)
Full size table
Table 7 \(P_e'>2\sqrt{2},\,\rho _A/\sqrt{\rho _D}>\sqrt{2}\)
Full size table

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Almuslimani, I. A fully adaptive explicit stabilized integrator for advection–diffusion–reaction problems. Bit Numer Math 63, 3 (2023). https://doi.org/10.1007/s10543-023-00945-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10543-023-00945-3

Keywords

Mathematics Subject Classification

Search

Navigation