Journal of Scientific Computing

, Volume 75, Issue 2, pp 1040–1056 | Cite as

\(L_2\) Stability of Explicit Runge–Kutta Schemes



Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations are often conducted only in the semidiscrete setting. Here, strong-stability of semidiscretisations for linear transport equations, resulting in ODEs with semibounded operators, are investigated. For the first time, it is proved that the fourth-order, ten-stage SSP method of Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) is strongly stable for general semibounded operators. Additionally, insights into fourth-order methods with fewer stages are presented.


Runge–Kutta methods Strong stability Strong-stability preserving 

Mathematics Subject Classication

65L20 65L06 65M12 



The authors would like to thank the anonymous reviewers for their helpful comments.


  1. 1.
    Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Conway, J.B.: A Course in Functional Analysis. Springer Science New York Inc, New York (1997)Google Scholar
  4. 4.
    Cooper, G.: Stability of Runge–Kutta methods for trajectory problems. IMA J. Numer. Anal. 7(1), 1–13 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)CrossRefMATHGoogle Scholar
  7. 7.
    Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer Science & Business Media, New York (2006)MATHGoogle Scholar
  9. 9.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, Berlin (2000)MATHGoogle Scholar
  10. 10.
    Higueras, I.: Monotonicity for Runge–Kutta methods: inner product norms. J. Sci. Comput. 24(1), 97–117 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar
  12. 12.
    Ketcheson, D.I.: Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30(4), 2113–2136 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer Science & Business Media, New York (2009)CrossRefMATHGoogle Scholar
  14. 14.
    Kopriva, D.A., Gassner, G.J.: An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems. SIAM J. Sci. Comput. 36(4), A2076–A2099 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kopriva, D.A., Jimenez, E.: An assessment of the efficiency of nodal discontinuous Galerkin spectral element methods. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, pp. 223–235. Springer, Berlin (2013)CrossRefGoogle Scholar
  16. 16.
    Levy, D., Tadmor, E.: From semidiscrete to fully discrete: stability of Runge–Kutta schemes by the energy method. SIAM Rev. 40(1), 40–73 (1998)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rackauckas, C., Nie, Q.: Differentialequations.jl—a performant and feature-rich ecosystem for solving differential equations in Julia. J. Open Res. Softw. 5(1), 15 (2017)Google Scholar
  18. 18.
    Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Riesz, F., Nagy, B.S.: Functional Analysis. Dover Publications, New York (1990)MATHGoogle Scholar
  20. 20.
    Roman, S.: Advanced Linear Algebra. Springer Science & Business Media, LLC, New York (2008)CrossRefMATHGoogle Scholar
  21. 21.
    Sun, Z., Shu, C.W.: Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations (2016),, to appear in Annals of Mathematical Sciences and Applications
  22. 22.
    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Tadmor, E.: From semidiscrete to fully discrete: stability of Runge–Kutta schemes by the energy method II. In: Estep, D.J., Tavener, S. (eds.) Collected Lectures on the Preservation of Stability Under Discretization, vol. 109, pp. 25–49. SIAM, Philadelphia (2002)Google Scholar
  24. 24.
    Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47(1), 50–72 (2011)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Vincent, P.E., Farrington, A.M., Witherden, F.D., Jameson, A.: An extended range of stable-symmetric-conservative flux reconstruction correction functions. Comput. Methods Appl. Mech. Eng. 296, 248–272 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.TU BraunschweigBraunschweigGermany

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