Conjugate symplecticity of second-order linear multi-step methods

  • Authors:

  • Quan-Dong Feng;Yan-Dong Jiao;Yi-Fa Tang

  • Affiliations:

  • LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, PR China and Graduate School of the Chinese Academy of Sciences, Beijing 10008 ...;LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, PR China and Graduate School of the Chinese Academy of Sciences, Beijing 10008 ...;LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, PR China

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2007

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Abstract

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^@l^@t@?G"1^@t=G"2^@t@?G"3^@l^@t with G"1^@t and G"3^@t being LMSMs, if G"2^@t is symplectic, then the B-series error expansions of G"1^@t, G"2^@t and G"3^@t of the form G^@t(Z)=@?"i"="0^+^~(@t^i/i!)Z^[^i^]+@t^s^+^1A"1+@t^s^+^2A"2+@t^s^+^3A"3+@t^s^+^4A"4+O(@t^s^+^5) are equal to those of trapezoid, mid-point and Euler forward schemes up to a parameter @q (completely the same when @q=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@tJ^-^1@?H(Z"1) cannot be conjugate-symplectic via another LMSM.