Advertisement

Journal of Scientific Computing

, Volume 71, Issue 3, pp 944–958 | Cite as

Dense Output for Strong Stability Preserving Runge–Kutta Methods

  • David I. KetchesonEmail author
  • Lajos Lóczi
  • Aliya Jangabylova
  • Adil Kusmanov
Article

Abstract

We investigate dense output formulae (also known as continuous extensions) for strong stability preserving (SSP) Runge–Kutta methods. We require that the dense output formula also possess the SSP property, ideally under the same step-size restriction as the method itself. A general recipe for first-order SSP dense output formulae for SSP methods is given, and second-order dense output formulae for several optimal SSP methods are developed. It is shown that SSP dense output formulae of order three and higher do not exist, and that in any method possessing a second-order SSP dense output, the coefficient matrix A has a zero row.

Keywords

Runge-Kutta methods SSP methods Dense output Continuous extension 

Notes

Acknowledgements

We thank an anonymous referee for several suggestions that improved the presentation of this work.

References

  1. 1.
    Bellen, A.: Contractivity of continuous Runge–Kutta methods for delay differential equations. Appl. Numer. Math. 24(2), 219–232 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellen, A., Vermiglio, R.: Some applications of continuous Runge–Kutta methods. Appl. Numer. Math. 22(1), 63–80 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Enright, W.H., Jackson, K.R., Nørsett, S.P., Thomsen, P.G.: Interpolants for Runge–Kutta formulas. ACM Trans. Math. Softw. 12(3), 193–218 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gottlieb, Sigal, Ketcheson, David I., Shu, Chi-Wang: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251–289 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gottlieb, Sigal, Ketcheson, David I., Shu, Chi-Wang: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (1993)Google Scholar
  7. 7.
    Jeltsch, Rolf: Reducibility and contractivity of Runge–Kutta methods revisited. BIT Numer. Math. 46(3), 567–587 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Macdonald, Colin B., Gottlieb, Sigal, Ruuth, Steven J.: A numerical study of diagonally split Runge–Kutta methods for PDEs with discontinuities. J. Sci. Comput. 36(1), 89–112 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Shu, Chi-Wang, Osher, Stanley: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83(1), 32–78 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Spiteri, Raymond J., Ruuth, Steven J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Torelli, Lucio: A sufficient condition for GPN-stability for delay differential equations. Numer. Math. 59(1), 311–320 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Zennaro, Marino: Natural continuous extensions of Runge–Kutta methods. Math. Comput. 46(173), 119–133 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • David I. Ketcheson
    • 1
    Email author
  • Lajos Lóczi
    • 1
  • Aliya Jangabylova
    • 2
  • Adil Kusmanov
    • 2
  1. 1.King Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia
  2. 2.Nazarbayev UniversityAstanaKazakhstan

Personalised recommendations