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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the averaged vector field method


Authors: Elena Celledoni, Brynjulf Owren and Yajuan Sun
Journal: Math. Comp. 83 (2014), 1689-1700
MSC (2010): Primary 65P10, 65L05; Secondary 65L06, 37M99
Published electronically: January 24, 2014
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Abstract: No Runge-Kutta method can be energy preserving for all Hamiltonian systems. But for problems in which the Hamiltonian is a polynomial, the averaged vector field (AVF) method can be interpreted as a Runge-Kutta method whose weights $ b_i$ and abscissae $ c_i$ represent a quadrature rule of degree at least that of the Hamiltonian. We prove that when the number of stages is minimal, the Runge-Kutta scheme must in fact be identical to the AVF scheme.


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Additional Information

Elena Celledoni
Affiliation: Department of Mathematical Sciences, NTNU, N-7491 Trondheim, Norway
Email: elenac@math.ntnu.no

Brynjulf Owren
Affiliation: Department of Mathematical Sciences, NTNU, N-7491 Trondheim, Norway
Email: bryn@math.ntnu.no

Yajuan Sun
Affiliation: LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences (CAS), P.O. Box 2719, Beijing 100190, China
Email: sunyj@lsec.cc.ac.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-2014-02805-6
PII: S 0025-5718(2014)02805-6
Keywords: Energy preservation, Runge--Kutta methods, polynomial Hamiltonian systems
Received by editor(s): November 5, 2012
Published electronically: January 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.