EPIRK-W and EPIRK-K Time Discretization Methods

  • Mahesh Narayanamurthi
  • Paul Tranquilli
  • Adrian Sandu
  • Mayya Tokman


Exponential integrators are special time discretization methods where the traditional linear system solves used by implicit schemes are replaced with computing the action of matrix exponential-like functions on a vector. A very general formulation of exponential integrators is offered by the Exponential Propagation Iterative methods of Runge–Kutta type (EPIRK) family of schemes. The use of Jacobian approximations is an important strategy to drastically reduce the overall computational costs of implicit schemes while maintaining the quality of their solutions. This paper extends the EPIRK class to allow the use of inexact Jacobians as arguments of the matrix exponential-like functions. Specifically, we develop two new families of methods: EPIRK-W integrators that can accommodate any approximation of the Jacobian, and EPIRK-K integrators that rely on a specific Krylov-subspace projection of the exact Jacobian. Classical order conditions theories are constructed for these families. Practical EPIRK-W methods of order three and EPIRK-K methods of order four are developed. Numerical experiments indicate that the methods proposed herein are computationally favorable when compared to a representative state-of-the-art exponential integrator, and a Rosenbrock–Krylov integrator.


Time integration Exponential integrator Krylov B-series Butcher trees 

Mathematics Subject Classification

65L05 65L04 65F60 65M22 



This work has been supported in part by NSF through Awards NSF DMS-1419003, NSF DMS-1419105, NSF CCF-1613905, by AFOSR through Award AFOSR FA9550-12-1-0293-DEF, and by the Computational Science Laboratory at Virginia Tech.


  1. 1.
    Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27(6), 1085–1095 (1979). CrossRefGoogle Scholar
  3. 3.
    Berland, H., Owren, B., Skaflestad, B.: B-series and order conditions for exponential integrators. SIAM J. Numer. Anal. 43(4), 1715–1727 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berland, H., Skaflestad, B., Wright, W.M.: Expint—a matlab package for exponential integrators. ACM Trans. Math. Softw. (TOMS) 33(1), 4 (2007)CrossRefGoogle Scholar
  5. 5.
    Butcher, J.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008). CrossRefzbMATHGoogle Scholar
  6. 6.
    Butcher, J.: Trees, B-series and exponential integrators. IMA J. Numer. Anal. 30, 131–140 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caliari, M., Ostermann, A.: Implementation of exponential rosenbrock-type integrators. Appl. Numer. Math. 59(3–4), 568–581 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Augustine, A.F.: MATLODE: A MATLAB ODE solver and sensitivity analysis toolbox. Masters Thesis, Virginia Tech (2018)Google Scholar
  9. 9.
    Hairer, E., Norsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993)zbMATHGoogle Scholar
  10. 10.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, 2nd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  11. 11.
    Heineken, W., Warnecke, G.: Partitioning methods for reaction–diffusion problems. Appl. Numer. Math. 56(7), 981–1000 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34(5), 1911–1925 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19(5), 1552–1574 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hochbruck, M., Ostermann, A.: Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43(3), 1069–1090 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Lorenz, E.N.: Predictability—a problem partly solved. In: Palmer, T., Hagedorn, R. (eds.) Predictability of Weather and Climate, pp. 40–58. Cambridge University Press (CUP), Cambridge (1996). Google Scholar
  18. 18.
    Lu, Y.Y.: Computing a matrix function for exponential integrators. J. Comput. Appl. Math. 161(1), 203–216 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Minchev, B.V., Wright, W.: A review of exponential integrators for first order semi-linear problems. Technical report (2005)Google Scholar
  20. 20.
    Niesen, J., Wright, W.M.: Algorithm 919: a krylov subspace algorithm for evaluating the \(\varphi \)-functions appearing in exponential integrators. ACM Trans. Math. Softw. 38(3), 22:1–22:19 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rainwater, G., Tokman, M.: A new approach to constructing efficient stiffly accurate Epirk methods. J. Comput. Phys. 323, 283–309 (2016). MathSciNetCrossRefGoogle Scholar
  22. 22.
    Roberts, S., Popov, A., Sandu, A.: ODE Test Problems. Accessed 16 Dec 2017
  23. 23.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  24. 24.
    Schulze, J.C., Schmid, P.J., Sesterhenn, J.L.: Exponential time integration using Krylov subspaces. Int. J. Numer. Meth. Fluids 60(6), 561–609 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sidje, R.B.: Expokit: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998). CrossRefzbMATHGoogle Scholar
  26. 26.
    Steihaug, T., Wolfbrandt, A.: An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comput. 33(146), 521–521 (1979). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tokman, M.: Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods. J. Comput. Phys. 213(2), 748–776 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tokman, M.: A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK). J. Comput. Phys. 230, 8762–8778 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tokman, M., Loffeld, J., Tranquilli, P.: New adaptive exponential propagation iterative methods of Runge–Kutta type. SIAM J. Sci. Comput. 34(5), A2650–A2669 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tranquilli, P., Glandon, S.R., Sarshar, A., Sandu, A.: Analytical Jacobian-vector products for the matrix-free time integration of partial differential equations. J. Comput. Appl. Math. (2016). zbMATHGoogle Scholar
  31. 31.
    Tranquilli, P., Sandu, A.: Exponential–Krylov methods for ordinary differential equations. J. Comput. Phys. 278, 31–46 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Tranquilli, P., Sandu, A.: Rosenbrock–Krylov methods for large systems of differential equations. SIAM J. Sci. Comput. 36(3), A1313–A1338 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Trefethen, L.N.: Evaluating matrix functions for exponential integrators via Carathéodory–Fejér approximation and contour integrals. Electron. Trans. Numer. Anal. 29, 1–18 (2007)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Verwer, J.G., Spee, E.J., Blom, J.G., Hundsdorfer, W.: A second-order rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20(4), 1456–1480 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    van der Vorst, H.A.: Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, Cambridge (2003). CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Computational Science Laboratory, Department of Computer ScienceVirginia TechBlacksburgUSA
  2. 2.School of Natural SciencesUniversity of CaliforniaMercedUSA

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