Journal of Scientific Computing

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

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

Article

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.

Keywords

Runge–Kutta methods Strong stability Strong-stability preserving 

Mathematics Subject Classication

65L20 65L06 65M12 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.TU BraunschweigBraunschweigGermany

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