Advertisement

Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods

  • Leah IsherwoodEmail author
  • Zachary J. Grant
  • Sigal Gottlieb
Article
  • 16 Downloads

Abstract

Problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge–Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge–Kutta methods were established in Isherwood et al. (SIAM J Numer Anal 56(6):3276–3307, 2018), where it was shown that it is possible to define explicit integrating factor Runge–Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge–Kutta method that has non-decreasing abscissas. However, explicit SSP Runge–Kutta methods have an order barrier of \(p=4\), and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge–Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge–Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge–Kutta methods; furthermore, our explicit SSP two-step Runge–Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge–Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP integrating factor Runge–Kutta methods designed in Isherwood et al. (2018).

Keywords

Strong stability preserving Time stepping for hyperbolic PDEs Integrating factor methods Lawson type methods Multi-step Runge–Kutta methods 

Notes

Acknowledgements

This publication is based on work supported by AFOSR Grant FA9550-18-1-0383, and ONR-DURIP Grant N00014-18-1-2255. A part of this research is sponsored by the Office of Advanced Scientific Computing Research; US Department of Energy, and was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract no. De-AC05-00OR22725. This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

References

  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.
    Constantinescu, E.M., Sandu, A.: Optimal strong-stability-preserving general linear methods. SIAM J. Sci. Comput. 32(5), 3130–3150 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christlieb, A.J., Gottlieb, S., Grant, Z., Seal, D.C.: Explicit strong stability preserving multistage two-derivative time-stepping schemes. J. Sci. Comput. 68(3), 914–942 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Conde, S., Gottlieb, S., Grant, Z., Shadid, J.N.: Implicit and implicit-explicit strong stability preserving Runge–Kutta methods with high linear order. J. Sci. Comput. 73(2), 667–690 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cox, S., Matthews, P.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gaudreault, S., Rainwater, G., Tokman, M.: KIOPS: a fast adaptive Krylov subspace solver for exponential integrators. J. Comput. Phys. 372, 236–255 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gottlieb, S., Grant, Z., Isherwood, L.: Optimized strong stability preserving integrating factor two-step Runge–Kutta methods. https://github.com/SSPmethods/SSPIF-TSRK-methods
  9. 9.
    Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific Press, Singapore (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge–Kutta methods. Math. Comput. 67, 73–85 (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grant, Z., Gottlieb, S., Seal, D.C.: A strong stability preserving analysis for explicit multistage two-derivative time-stepping schemes based on Taylor series conditions. Commun. Appl. Math. Comput. 1(1), 21–59 (2019)CrossRefGoogle Scholar
  13. 13.
    Hesthaven, J.S.: Numerical Methods for Conservation Laws: From Analysis to Algorithms. SIAM Publishing, Philadelphia (2017)CrossRefGoogle Scholar
  14. 14.
    Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605–623 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Isherwood, L., Gottlieb, S., Grant, Z.: Strong stability preserving integrating factor Runge–Kutta methods. SIAM J. Numer. Anal. 56(6), 3276–3307 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Isherwood, L., Gottlieb, S., Grant, Z.: Downwinding for preserving strong stability in explicit integrating factor Runge–Kutta methods. Pure Appl. Math. Q. 14(1), 3–25 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126(1), 202–228 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ketcheson, D.I.: Highly efficient strong stability preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30, 2113–2136 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge–Kutta methods. Appl. Numer. Math. 52, 373–392 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ketcheson, D.I.: Computation of optimal monotonicity preserving general linear methods. Math. Comput. 78, 1497–1513 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ketcheson, D.I.: Step sizes for strong stability preservation with downwind-biased operators. SIAM J. Numer. Anal. 49(4), 1649–1660 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ketcheson, D.I., Gottlieb, S., Macdonald, C.B.: Strong stability preserving two-step Runge–Kutta methods. SIAM J. Numer. Anal. 49, 2618–2639 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bresten, C., Gottlieb, S., Grant, Z., Higgs, D., Ketcheson, D.I., Nemeth, A.: Explicit strong stability preserving multistep Runge–Kutta methods. Math. Comput. 86, 747–769 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kraaijevanger, J.F.B.M.: Contractivity of Runge–Kutta methods. BIT 31, 482–528 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lenferink, H.W.J.: Contractivity-preserving implicit linear multistep methods. Math. Comput. 56, 177–199 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. ETH Lectures in Mathematics Series. Birkhauser-Verlag, Basel (1990)CrossRefzbMATHGoogle Scholar
  27. 27.
    Niesen, J., Wright, W.M.: Algorithm 919: a Krylov subspace algorithm for evaluating the \(\phi \)-functions appearing in exponential integrators. ACM Trans. Math. Softw. 38(3), 22 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ruuth, S.J., Spiteri, R.J.: Two barriers on strong-stability-preserving time discretization methods. J. Sci. Comput. 17, 211–220 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shu, C.-W.: Total-variation diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sidje, R.B.: EXPOKIT: a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24(1), 130–156 (1998)CrossRefzbMATHGoogle Scholar
  32. 32.
    Spijker, M.N.: Contractivity in the numerical solution of initial value problems. Numer. Math. 42, 271–290 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Spijker, M.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45, 1226–1245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of Massachusetts DartmouthNorth DartmouthUSA
  2. 2.Department of Computational and Applied MathematicsOak Ridge National LaboratoryOak RidgeUSA

Personalised recommendations