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On conjugate symplecticity of B-series integrators

Published in IMA Journal of Numerical Analysis. 2013, vol. 33, no. 1, p. 57-79
Abstract The long-time integration of Hamiltonian differential equations requires special numerical methods. Symplectic integrators are an excellent choice, but there are situations (e.g., multistep schemes or energy-preserving methods), where symplecticity is not possible. It is then of interest to study whether the methods are conjugate symplectic and thus have the same long-time behaviour as symplectic methods. This question is addressed in this work for the class of B-series integrators. Algebraic criteria for conjugate symplecticity up to a certain order are presented in terms of the coefficients of the B-series. The effect of simplifying assumptions is investigated. These criteria are then applied to characterize the conjugate symplecticity of implicit Runge–Kutta methods (Lobatto IIIA and Lobatto IIIB) and of energy-preserving collocation methods.
Keywords Conjugate symplecticityHamiltonian differential equationsBackward error analysisModified equationsB-seriesRooted treesSimplifying assumptionsLobatto IIIA methodsLobatto IIIB methodsEnergy-preserving integrators
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HAIRER, Ernst, ZBINDEN, Christophe. On conjugate symplecticity of B-series integrators. In: IMA Journal of Numerical Analysis, 2013, vol. 33, n° 1, p. 57-79. https://archive-ouverte.unige.ch/unige:26436

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Deposited on : 2013-02-20

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