On conjugate symplecticity of B-series integrators

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.