Conjugate symplecticity of second-order linear multi-step methods

We review the two different approaches for symplecticity of linear multi-step methods (LMSM) by Eirola and Sanz-Serna, Ge and Feng, and by Feng and Tang, Hairer and Leone, respectively, and give a numerical example between these two approaches. We prove that in the conjugate relation G 3 λ τ ∘ G 1 τ = G 2 τ ∘ G 3 λ τ with G 1 τ and G 3 τ being LMSMs, if G 2 τ is symplectic, then the B-series error expansions of G 1 τ , G 2 τ and G 3 τ of the form G τ ( Z ) = ∑ i = 0 + ∞ ( τ i / i ! ) Z [ i ] + τ s + 1 A 1 + τ s + 2 A 2 + τ s + 3 A 3 + τ s + 4 A 4 + O ( τ s + 5 ) are equal to those of trapezoid, mid-point and Euler forward schemes up to a parameter θ (completely the same when θ = 1 ), respectively, this also partially solves a problem due to Hairer. In particular we indicate that the second-order symmetric leap-frog scheme Z 2 = Z 0 + 2 τ J - 1 ∇ H ( Z 1 ) cannot be conjugate-symplectic via another LMSM.