Skip to main content
SpringerLink
Log in

A class of explicit multistep exponential integrators for semilinear problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

A class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed. It is shown that the k-step method achieves order k, for appropriate starting values, which can be computed by auxiliary routines or by one strategy proposed in the paper. Together with some implementation issues, numerical illustrations are also provided.

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. Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147, 362–387 (1998)

    Google Scholar 

  2. Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)

    Google Scholar 

  3. Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)

    Google Scholar 

  4. Friesner, R.A., Tuckerman, L.S., Dornblaser, B.C., Russo, T.V.: A method of exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comp. 4, 327–354 (1989)

    Google Scholar 

  5. González, C., Palencia, C.: Stability of Runge-Kutta methods for quasilinear parabolic problems. Math. Comput. 69, 609–628 (2000)

    Google Scholar 

  6. Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840, Springer, Berlin, 1981

  7. Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical Society, Providence, 1957

  8. Hochbruck, M., Lubich, Ch.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)

    Google Scholar 

  9. Hochbruck, M., Lubich, Ch.: Error analysis of Krylov methods in a nutshell. SIAM J. Sci. Comput. 19, 695–701 (1998)

    Google Scholar 

  10. Hochbruck, M., Lubich, Ch.: Exponential integrators for quantum-classical molecular dynamics. BIT 39, 620–645 (1999)

    Google Scholar 

  11. Hochbruck, M., Lubich, Ch., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. 19, 1552–1574 (1998)

    Google Scholar 

  12. Hochbruck, M., Ostermann, A.: Exponential Runge-Kutta methods for parabolic problems. Appl. Numer. Math. (to appear)

  13. Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. Preprint, 2004

  14. Kassam, A.-K., Trefethen, L.: Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. (to appear)

  15. Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)

    Google Scholar 

  16. Lawson, J.D.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372–380 (1967)

    Google Scholar 

  17. Le Roux, M.N.: Méthodes multipas pour des équations paraboliques non linéaires. Numer. Math. 35, 143–162 (1980)

    Google Scholar 

  18. Lubich, Ch., Ostermann, A.: Runge-Kutta approximation of quasilinear parabolic equations. Math. Comput. 64, 601–627 (1995)

    Google Scholar 

  19. Lubich, Ch., Ostermann, A.: Runge-Kutta time discretization of reaction-diffusion and Navier-Stokes equations: nonsmooth-data error estimates and applications to long-time behavior. Appl. Numer. Math. 22, 179–292 (1996)

    Google Scholar 

  20. McKee, S.: Generalised discrete Gronwall lemmas. Z. Angew. Math. Mech. 62, 429–434 (1982)

    Google Scholar 

  21. Minchev, B.V., Wright, W.M.: A review of exponential integrators. Preprint, 2004

  22. Nørsett, S.P.: An A-stable modification of the Adams-Bashforth methods. Springer Lect. Notes Math. 109, 214–219 (1969)

    Google Scholar 

  23. Ostermann, A., Thalhammer, M.: Non-smooth data error estimates for linearly implicit Runge-Kutta methods. IMA J. Numer. Anal. 20, 167–184 (2000)

    Google Scholar 

  24. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983

  25. Quarteroni, A., Saleri, F.: Scientific Computing with MATLAB. Springer, Berlin, 2003

  26. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978

  27. Van der Houwen, P.J., Verwer, J.G.: Generalized linear multistep methods I. Development of algorithms with zero-parasitic roots. Report NW 10/74, Mathematisch Centrum, Amsterdam, 1974

  28. Verwer, J.G.: Generalized linear multistep methods II. Numerical applications. Report NW 12/74, Mathematisch Centrum, Amsterdam, 1974

  29. Verwer, J.G.: On generalized linear multistep methods with zero-parasitic roots and an adaptive principal root. Numer. Math. 27, 143–155 (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Universidad de Valladolid, Spain

    M. P. Calvo & C. Palencia

Authors
  1. M. P. Calvo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Palencia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. P. Calvo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calvo, M., Palencia, C. A class of explicit multistep exponential integrators for semilinear problems. Numer. Math. 102, 367–381 (2006). https://doi.org/10.1007/s00211-005-0627-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0627-0

Mathematics Subject Classifications (2000)

Search

Navigation