Numerische Mathematik

, Volume 102, Issue 3, pp 367–381 | Cite as

A class of explicit multistep exponential integrators for semilinear problems



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.

Mathematics Subject Classifications (2000)

65J15 65M12 65L05 65M20 


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  1. 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. 2.
    Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)Google Scholar
  3. 3.
    Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)Google Scholar
  4. 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. 5.
    González, C., Palencia, C.: Stability of Runge-Kutta methods for quasilinear parabolic problems. Math. Comput. 69, 609–628 (2000)Google Scholar
  6. 6.
    Henry, D.: Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840, Springer, Berlin, 1981Google Scholar
  7. 7.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-groups. American Mathematical Society, Providence, 1957Google Scholar
  8. 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. 9.
    Hochbruck, M., Lubich, Ch.: Error analysis of Krylov methods in a nutshell. SIAM J. Sci. Comput. 19, 695–701 (1998)Google Scholar
  10. 10.
    Hochbruck, M., Lubich, Ch.: Exponential integrators for quantum-classical molecular dynamics. BIT 39, 620–645 (1999)Google Scholar
  11. 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. 12.
    Hochbruck, M., Ostermann, A.: Exponential Runge-Kutta methods for parabolic problems. Appl. Numer. Math. (to appear)Google Scholar
  13. 13.
    Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. Preprint, 2004Google Scholar
  14. 14.
    Kassam, A.-K., Trefethen, L.: Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. (to appear)Google Scholar
  15. 15.
    Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)Google Scholar
  16. 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. 17.
    Le Roux, M.N.: Méthodes multipas pour des équations paraboliques non linéaires. Numer. Math. 35, 143–162 (1980)Google Scholar
  18. 18.
    Lubich, Ch., Ostermann, A.: Runge-Kutta approximation of quasilinear parabolic equations. Math. Comput. 64, 601–627 (1995)Google Scholar
  19. 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. 20.
    McKee, S.: Generalised discrete Gronwall lemmas. Z. Angew. Math. Mech. 62, 429–434 (1982)Google Scholar
  21. 21.
    Minchev, B.V., Wright, W.M.: A review of exponential integrators. Preprint, 2004Google Scholar
  22. 22.
    Nørsett, S.P.: An A-stable modification of the Adams-Bashforth methods. Springer Lect. Notes Math. 109, 214–219 (1969)Google Scholar
  23. 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. 24.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin, 1983Google Scholar
  25. 25.
    Quarteroni, A., Saleri, F.: Scientific Computing with MATLAB. Springer, Berlin, 2003Google Scholar
  26. 26.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam, 1978Google Scholar
  27. 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, 1974Google Scholar
  28. 28.
    Verwer, J.G.: Generalized linear multistep methods II. Numerical applications. Report NW 12/74, Mathematisch Centrum, Amsterdam, 1974Google Scholar
  29. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Departamento de Matemática AplicadaUniversidad de ValladolidSpain

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