On conjugate-symplecticity of B-series integrators
E. Hairer and C.J. Zbinden
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 if the methods are conjugate-symplectic and thus have the same
long-time behavior 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.
Key Words.conjugate-symplecticity; Hamiltonian differential equations; backward error
analysis; modified equations; B-series;
rooted trees; simplifying assumptions; Lobatto IIIA methods,
Lobatto IIIB methods, energy-preserving integrators.