Flexible exponential integration methods for large systems of differential equations

  • Dongping Li
  • Yuhao Cong
  • Kaifeng Xia
Original Research


In this paper, we describe a flexible variant of exponential integration methods for large systems of differential equations. This version possesses the flexibility and generality which allows to further exploit the special structure of the system. By using modified B-series and bi-coloured rooted trees, we can derive the general structure of the classical order conditions for these schemes. Some numerical schemes are constructed and the order conditions are derived. Numerical experiments with reaction-diffusion type problems are included.


Exponential integrators Rosenbrock methods Runge–Kutta methods Classical order conditions B-series 

Mathematics Subject Classification

65M12 65L06 65L20 



The authors gratefully acknowledge the supports of the National Science Foundation Grant (11471217) of China.


  1. 1.
    Ruuth, S.J.: Implicit-explicit methods for reaction-diffusion problems in pattern formation. J. Math. Biol. 34, 148–176 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit methods for time-dependent PDE’s. SIAM J. Numer. Anal. 32, 797–823 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kassam, A.K., Trefethen, L.N.: Fourth-order time stepping for stiff PDEs. SIAM J. Sci. Comput. 26, 1214–1233 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tokman, M.: Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods. J. Comput. Phys. 213, 748–776 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caliari, M., Ostermann, A.: Implementation of exponential Rosenbrocktype integrators. Appl. Numer. Math. 59, 568–581 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Loffeld, J., Tokman, M.: Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs. J. Comput. Appl. Math. 241, 45–67 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Minchev, B.V., Wright, W.M.: A review of exponential integrators for first order semi-linear problems, Technical report 2/05, Department of Mathematics, NTNU (2005)Google Scholar
  11. 11.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hochbruck, M., Ostermann, A.: Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 1069–1090 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hochbruck, M., Ostermann, A., Schweitzer, J.: Exponential rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786–803 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29, 209–228 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Butcher, J.C.: Trees, B-series and exponential integrators. IMA J. Numer. Anal. 30, 131–140 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rainwater, G., Tokman, M.: A new class of split exponential propagation iterative methods of RungeCKutta type (sEPIRK)forsemilinear systems of ODEs. J. Comput. Phys. 269, 40–60 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, Berlin (1993)zbMATHGoogle Scholar
  22. 22.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2005)zbMATHGoogle Scholar
  23. 23.
    Niesen, J., Wright, W.: Algorithm 919: A Krylov subspace algorithm for evaluating the phi- functions appearing in exponential integrators, ACM Trans. Math. Software, 38 (3), Article 22 (2012)Google Scholar
  24. 24.
    Gear, C.W.: The automatic integration of stiff ordinary differential equations. In: Proceedings of the IFIP Congress, pp. 81–85 (1968)Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2015

Authors and Affiliations

  1. 1.Mathematics and Science CollegeShanghai Normal UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsChangchun Normal UniversityChangchunPeople’s Republic of China

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