Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4642777 | Journal of Computational and Applied Mathematics | 2007 | 9 Pages |
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 G3λÏâG1Ï=G2ÏâG3Î»Ï with G1Ï and G3Ï being LMSMs, if G2Ï is symplectic, then the B-series error expansions of G1Ï, G2Ï and G3Ï of the form GÏ(Z)=âi=0+â(Ïi/i!)Z[i]+Ïs+1A1+Ïs+2A2+Ïs+3A3+Ïs+4A4+O(Ïs+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 Z2=Z0+2ÏJ-1âH(Z1) cannot be conjugate-symplectic via another LMSM.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Quan-Dong Feng, Yan-Dong Jiao, Yi-Fa Tang,