We expand the step-transition operator of any linear multi-step method with order s≥ 2 up to O(Ts+5). And through examples we show how much the perturbation of the step-transition operator caused by the error of init...We expand the step-transition operator of any linear multi-step method with order s≥ 2 up to O(Ts+5). And through examples we show how much the perturbation of the step-transition operator caused by the error of initial value is.展开更多
We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalize...We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalized linear multi-step method G3^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（∑l=0^mγklZl）. We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G3^T is a more general operator.展开更多
We give some formulae for calculation of the expansions for (1) composition of step-transition operators (STO) of any two difference schemes (DS) for ODE's, (2) inverse operator of STO of any DS, and (3) conjugate...We give some formulae for calculation of the expansions for (1) composition of step-transition operators (STO) of any two difference schemes (DS) for ODE's, (2) inverse operator of STO of any DS, and (3) conjugate operator of STO of any DS.展开更多
Using least parameters, we expand the step-transition operator of any linear multi-step method （LMSM） up to O（τ^s＋5） with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 （obtained...Using least parameters, we expand the step-transition operator of any linear multi-step method （LMSM） up to O（τ^s＋5） with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 （obtained by the second author in a former paper）. We prove that in the conjugate relation G3^λτ o G1^τ =G2^τ o G3^λτ with G1 being an LMSM,（1） theorder of G2 can not be higher than that of G1; （2） if G3 is also an LMSM and G2 is a symplectic B-series, then the orders of G1, G2 and G3 must be 2, 2 and 1 respectively.展开更多
基金Special Funds for Major State Basic Research Projects of China (No.G1999032801-10 and No. G1999032804), by the knowledge inn
文摘We expand the step-transition operator of any linear multi-step method with order s≥ 2 up to O(Ts+5). And through examples we show how much the perturbation of the step-transition operator caused by the error of initial value is.
基金Acknowledgements. We would like to thank the editors for their valuable suggestions and corrections. This research is supported by the National Natural Science Foundation of China （Grant Nos. 10471145 and 10672143）, and by Morningside Center of Mathematics, Chinese Academy of Sciences.
文摘We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalized linear multi-step method G3^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（∑l=0^mγklZl）. We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G3^T is a more general operator.
基金This research is supported by Special Funds for Major State Basic Research Projects of China (No. G1999032801-10 and No. G1999032804) by the knowledge innovation program of the Chinese Academy of Sciences and a grant (No. 19801034) from National Natura
文摘We give some formulae for calculation of the expansions for (1) composition of step-transition operators (STO) of any two difference schemes (DS) for ODE's, (2) inverse operator of STO of any DS, and (3) conjugate operator of STO of any DS.
基金This research is supported by the Informatization Construction of Knowledge Innovation Projects of the Chinese Academy of Sciences ＂Supercomputing Environment Construction and Application＂ （INF105-SCE）, and by a grant （No. 10471145） from National Natural Science Foundation of China.
文摘Using least parameters, we expand the step-transition operator of any linear multi-step method （LMSM） up to O（τ^s＋5） with order s = 1 and rewrite the expansion of the steptransition operator for s = 2 （obtained by the second author in a former paper）. We prove that in the conjugate relation G3^λτ o G1^τ =G2^τ o G3^λτ with G1 being an LMSM,（1） theorder of G2 can not be higher than that of G1; （2） if G3 is also an LMSM and G2 is a symplectic B-series, then the orders of G1, G2 and G3 must be 2, 2 and 1 respectively.
