TY - JOUR
T1 - The Step-Transition Operators for Multi-Step Methods of ODE's
JO - Journal of Computational Mathematics
VL - 3
SP - 193
EP - 202
PY - 1998
DA - 1998/06
SN - 16
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9152.html
KW - Multi-step methods, Explike and loglike function, Fractional and rational approximation, Simplecticity of LMM, Nonexistence of SLMM.
AB - <p style="text-align: justify;">In this paper, we propose a new definition of symplectic multistep methods. This definition differs from the old ones in that it is given via the one step method defined directly on $M$ which is corresponding to the $m$ step scheme defined on $M$ while the old definitions are given out by defining a corresponding one step method on $M\times M \times \cdots \times M=M^m$ with a set of new variables. The new definition gives out a step-transition operator $g: M\longrightarrow M$. Under our new definition, the Leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator $g$ will be constructed via continued fractions and rational approximations.</p>