BIT Numerical Mathematics

, Volume 52, Issue 3, pp 773–795 | Cite as

Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods



This paper is concerned with multidimensional exponential fitting modified Runge-Kutta-Nyström (MEFMRKN) methods for the system of oscillatory second-order differential equations q″(t)+Mq(t)=f(q(t)), where M is a d×d symmetric and positive semi-definite matrix and f(q) is the negative gradient of a potential scalar U(q). We formulate MEFMRKN methods and show clearly the relationship between MEFMRKN methods and multidimensional extended Runge-Kutta-Nyström (ERKN) methods proposed by Wu et al. (Comput. Phys. Comm. 181:1955–1962, 2010). Taking into account the fact that the oscillatory system is a separable Hamiltonian system with Hamiltonian \(H(p,q)=\frac{1}{2}p^{T}p+ \frac{1}{2}q^{T}Mq+U(q)\), we derive the symplecticity conditions for the MEFMRKN methods. Two explicit symplectic MEFMRKN methods are proposed. Numerical experiments accompanied demonstrate that our explicit symplectic MEFMRKN methods are more efficient than some well-known numerical methods appeared in the scientific literature.


Exponential fitting MEFMRKN methods Symplecticity conditions ERKN integrators Oscillatory systems 

Mathematics Subject Classification (2000)

65L05 65L06 65M20 



The authors sincerely thank Professor Axel Ruhe and the two anonymous reviewers for their valuable suggestions, which help improve this paper significantly.


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Copyright information

© Springer Science + Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingP.R. China
  2. 2.State Key Laboratory for Novel Software TechnologyNanjing UniversityNanjingP.R. China
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

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