Numerical Algorithms

, Volume 65, Issue 3, pp 465–483

Error analysis of explicit TSERKN methods for highly oscillatory systems

Original Paper

DOI: 10.1007/s11075-013-9761-4

Cite this article as:
Li, J. & Wu, X. Numer Algor (2014) 65: 465. doi:10.1007/s11075-013-9761-4


In this paper, we are concerned with the error analysis for the two-step extended Runge-Kutta-Nyström-type (TSERKN) methods [Comput. Phys. Comm. 182 (2011) 2486–2507] for multi-frequency and multidimensional oscillatory systems y″(t) + My(t) = f(t, y(t)), where high-frequency oscillations in the solutions are generated by the linear part My(t). TSERKN methods extend the two-step hybrid methods [IMA J. Numer. Anal. 23 (2003) 197–220] by reforming both the internal stages and the updates so that they are adapted to the oscillatory properties of the exact solutions. However, the global error analysis for the TSERKN methods has not been investigated. In this paper we construct a new three-stage explicit TSERKN method of order four and present the global error bound for the new method, which is proved to be independent of ∥M∥ under suitable assumptions. This property of our new method is very important for solving highly oscillatory systems (1), where ∥M∥ may be arbitrarily large. We also analyze the stability and phase properties for the new method. Numerical experiments are included and the numerical results show that the new method is very competitive and promising compared with the well-known high quality methods proposed in the scientific literature.


Explicit TSERKN methods Error bounds Highly oscillatory systems 

Mathematics Subject Classifications (2010)

Primary 65L05 65L06 65L20 65L70 65M20 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHebei Normal UniversityShijiazhuangChina
  2. 2.Hebei Key Laboratory of Computational Mathematics and ApplicationsHebei Normal UniversityShijiazhuangChina
  3. 3.Department of MathematicsNanjing UniversityNanjingChina
  4. 4.State Key Laboratory for Novel Software TechnologyNanjing UniversityNanjingChina