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)(lambda t) o G(1)(t) = G' o G(1)(lambda tau) with G(3)(tau) being LMSMs, if G(2)(tau) is symplectic, then the B-series error expansions of G(1)(tau), G(2)(tau) and G(3)(tau) of the form G(tau) (Z) = Sigma(+infinity)(i=0) (tau(i)/i!)Z(vertical bar i vertical bar) + tau(s+1) dA(1) + tau(s+2)A(2) + tau(s+3) A(3) + tau(s+4) A(4) + O(tau(s+5)) are equal to those of trapezoid, mid-point and Euler forward schemes up to a parameter theta (completely the same when theta = 1), respectively, this also partially solves a problern due to Hairer. In particular we indicate that the second-order symmetric leap-frog scheme Z(2) = Z(0) + 2 tau J(-1) del H(Z(1)) cannot be conjugate-symplectic via another LMSM. (c) 2006 Elsevier B.V. All rights reserved.