\(L_2\) 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.
KeywordsRunge–Kutta methods Strong stability Strong-stability preserving
Mathematics Subject Classication65L20 65L06 65M12
The authors would like to thank the anonymous reviewers for their helpful comments.
- 3.Conway, J.B.: A Course in Functional Analysis. Springer Science New York Inc, New York (1997)Google Scholar
- 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
- 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
- 21.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
- 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