Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65–98 (2017)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2008)
Conway, J.B.: A Course in Functional Analysis. Springer Science New York Inc, New York (1997)
Cooper, G.: Stability of Runge–Kutta methods for trajectory problems. IMA J. Numer. Anal. 7(1), 1–13 (1987)
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)
Gottlieb, S., Ketcheson, D.I., Shu, C.W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific, Singapore (2011)
Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67(221), 73–85 (1998)
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)
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, Berlin (2000)
Higueras, I.: Monotonicity for Runge–Kutta methods: inner product norms. J. Sci. Comput. 24(1), 97–117 (2005)
Iserles, A.: A First Course in the Numerical Analysis of Differential Equations. Cambridge University Press, Cambridge (2009)
Ketcheson, D.I.: Highly efficient strong stability-preserving Runge–Kutta methods with low-storage implementations. SIAM J. Sci. Comput. 30(4), 2113–2136 (2008)
Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer Science & Business Media, New York (2009)
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)
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)
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)
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)
Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016)
Riesz, F., Nagy, B.S.: Functional Analysis. Dover Publications, New York (1990)
Roman, S.: Advanced Linear Algebra. Springer Science & Business Media, LLC, New York (2008)
Sun, Z., Shu, C.W.: Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations (2016), https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/Stability%20of%20the%20fourth%20order%20Runge-Kutta%20method%20for%20time-dependent%20partial.pdf, to appear in Annals of Mathematical Sciences and Applications
Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)
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)
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)
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)
Acknowledgements
The authors would like to thank the anonymous reviewers for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ranocha, H., Öffner, P. \(L_2\) Stability of Explicit Runge–Kutta Schemes. J Sci Comput 75, 1040–1056 (2018). https://doi.org/10.1007/s10915-017-0595-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-017-0595-4