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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
DOI: https://doi.org/10.1090/S0025-5718-2014-02805-6
Published electronically: January 24, 2014
MathSciNet review: 3194126
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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
MR Author ID: 623033
Email: elenac@math.ntnu.no

Brynjulf Owren
Affiliation: Department of Mathematical Sciences, NTNU, N-7491 Trondheim, Norway
MR Author ID: 292686
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

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.