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BIT Numerical Mathematics

, Volume 53, Issue 3, pp 567–594 | Cite as

Exponential almost Runge-Kutta methods for semilinear problems

  • John Carroll
  • Eoin O’Callaghan
Article

Abstract

We present a new class of one-step, multi-value Exponential Integrator (EI) methods referred to as Exponential Almost Runge-Kutta (EARK) methods which involve the derivatives of a nonlinear function of the solution. In order to approximate such derivatives to a sufficient accuracy, the EARK methods will be implemented within the broader framework of Exponential Almost General Linear Methods (EAGLMs) to accommodate past values of this nonlinear function and becoming multistep in nature as a consequence. Established EI methods, such as Exponential Time Differencing (ETD) methods, Exponential Runge-Kutta (ERK) methods and Exponential General Linear Methods (EGLMs) become special cases of EAGLMs. We present order conditions which facilitate the construction of two- and three-stage EARK methods and, when cast in an EAGLM format, we perform a stability analysis to enable a comparison with existing EI methods. We conclude with some numerical experiments which confirm the convergence order and also demonstrate the computational efficiency of these new methods.

Keywords

Exponential integrators Exponential almost Runge-Kutta EARK Exponential almost general linear methods EAGLM Semilinear Stiff PDEs 

Mathematics Subject Classification (2010)

35K58 65L04 65L05 

References

  1. 1.
    Al-Mohy, A., Higham, N.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bergamaschi, L., Caliari, M., Martínez, A., Vianello, M.: Comparing Leja and Krylov approximations of large scale matrix exponentials. In: ICCS 2006, Reading (UK), Springer LNCS, vol. 3994, pp. 685–692 (2006) CrossRefGoogle Scholar
  3. 3.
    Bergamaschi, L., Vianello, M.: Efficient computation of the exponential operator for large, sparse, symmetric matrices. Numer. Linear Algebra Appl. 7(1), 27–45 (2000) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berland, H., Skaflestad, B., Wright, W.: Expint—a Matlab package for exponential integrators. ACM Trans. Math. Softw. 33(1), 1–17 (2007) CrossRefGoogle Scholar
  5. 5.
    Beylkin, G., Keiser, J., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147(2), 362–387 (1998) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Butcher, J.: On the convergence of numerical solutions to ordinary differential equations. Math. Comput. 20, 1–10 (1966) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Butcher, J.: General linear methods. Comput. Math. Appl. 31(4–5), 105–112 (1996) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Butcher, J.: An introduction to “Almost Runge-Kutta” methods. Appl. Numer. Math. 24(2–3), 331–342 (1997) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Caliari, M., Ostermann, A.: Implementation of exponential Rosenbrock-type integrators. Appl. Numer. Math. 59(3–4), 568–581 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Calvo, M., Portillo, A.: Variable step implementation of ETD methods for semilinear problems. Appl. Math. Comput. 196(2), 627–637 (2008) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Cox, S., Matthews, P.: Exponential time differencing for stiff systems. J. Comput. Phys. 176(2), 430–455 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Friesner, R., Tuckerman, L., Dornblaser, B., Russo, T.: A method for exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comput. 4(4), 327–354 (1989) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13(5), 1236–1264 (1992) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, vol. 2. Springer, Berlin (2002). Second revised edition with 137 figures Google Scholar
  15. 15.
    Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203(1), 72–88 (2005) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lu, Y.: Exponentials of symmetric matrices through tridiagonal reductions. Linear Algebra Appl. 279, 317–324 (1998) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Lu, Y.: Computing a matrix function for exponential integrators. J. Comput. Appl. Math. 161(1), 203–216 (2003) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Maset, S., Zennaro, M.: Unconditional stability of explicit exponential Runge-Kutta methods for semi-linear ordinary differential equations. Math. Comput. 78, 957–967 (2009) MathSciNetMATHGoogle Scholar
  21. 21.
    Minchev, B.: Exponential integrators for semilinear problems. Ph.D. thesis, Department of Informatics, University of Bergen (2004) Google Scholar
  22. 22.
    Minchev, B.: Integrating factor methods as exponential integrators. In: Proceedings of the 5th International Conference on Large-Scale Scientific Computing, LSSC’05, pp. 380–386. Springer, Berlin (2006) CrossRefGoogle Scholar
  23. 23.
    Minchev, B.: Wright., W.: A review of exponential integrators for first order semi-linear problems. Tech. rep. 2/05, NTNU (2005) Google Scholar
  24. 24.
    Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Niesen, J., Wright, W.M.: A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators. ACM TOMS 38(3), 22 (2012) MathSciNetCrossRefGoogle Scholar
  26. 26.
    O’Callaghan, E.: The analysis and implementation of exponential almost Runge-Kutta methods for semilinear problems. Ph.D. thesis, School of Mathematical Sciences, Dublin City University (2011) Google Scholar
  27. 27.
    Ostermann, A., Thalhammer, M., Wright, W.: A class of explicit exponential general linear methods. BIT Numer. Math. 46(2), 409–431 (2006) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Schmelzer, T., Trefethen, L.: Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximations and contour integrals. Electron. Trans. Numer. Anal. 29, 1–18 (2007) MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesDublin City UniversityDublinIreland

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