In this paper we generalize the method of constructing sympl ctic schemes by generating function in the case of autonomous Hamiltonian system to that of nonautonomous system.
The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied.An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desi...The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied.An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order.In general the proposed symplectic schemes are fully implicit,and they become computationally expensive for mean square orders greater than two.However,for stochastic Hamiltonian systems preserving Hamiltonian functions,the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes.A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.展开更多
In this paper, we will prove by the help of formal energies only that one can improve the order of any symplectic scheme by modifying the Hamiltonian symbol H, and show through examples that this action exactly and di...In this paper, we will prove by the help of formal energies only that one can improve the order of any symplectic scheme by modifying the Hamiltonian symbol H, and show through examples that this action exactly and directly simplifies Feng's way of construction of higher-order symplectic schemes by using higher-order terms of generating functions.展开更多
The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symp...The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conser- vation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Stormer-Verlet scheme is first constructed in a Hamiltonian frame- work. The conservation law of the StSrmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Stormer-Verlet scheme associated with the con- servation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the StSrmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the StSrmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Stormer-Verlet scheme.展开更多
In this paper,we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems.The one class is the symplectic scheme,which contains two new fa...In this paper,we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems.The one class is the symplectic scheme,which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method,respectively.Each member in these schemes is symplectic for any fixed parameter.A more general form of generating functions is introduced,which generalizes the three classical generating functions that are widely used to construct symplectic algorithms.The other class is a novel family of energy and quadratic invariants preserving schemes,which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step.The existence of the solutions of these schemes is verified.Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.展开更多
In this note, we will give a proof for the uniqueness of 4th-order time-reversible sym- plectic difference schemes of 13th-fold compositions of phase flows φtH(1) , φtH(2) , φtH(3) with different temporal parameter...In this note, we will give a proof for the uniqueness of 4th-order time-reversible sym- plectic difference schemes of 13th-fold compositions of phase flows φtH(1) , φtH(2) , φtH(3) with different temporal parameters for splitable hamiltonian H = H(1) + H(2) + H(3).展开更多
In this paper,the symplectic perturbation series methodology of the non-conservative linear Hamiltonian system is presented for the structural dynamic response with damping.Firstly,the linear Hamiltonian system is bri...In this paper,the symplectic perturbation series methodology of the non-conservative linear Hamiltonian system is presented for the structural dynamic response with damping.Firstly,the linear Hamiltonian system is briefly introduced and its conservation law is proved based on the properties of the exterior products.Then the symplectic perturbation series methodology is proposed to deal with the non-conservative linear Hamiltonian system and its conservation law is further proved.The structural dynamic response problem with eternal load and damping is transformed as the non-conservative linear Hamiltonian system and the symplectic difference schemes for the non-conservative linear Hamiltonian system are established.The applicability and validity of the proposed method are demonstrated by three engineering examples.The results demonstrate that the presented methodology is better than the traditional Runge–Kutta algorithm in the prediction of long-time structural dynamic response under the same time step.展开更多
文摘In this paper we generalize the method of constructing sympl ctic schemes by generating function in the case of autonomous Hamiltonian system to that of nonautonomous system.
文摘The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied.An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order.In general the proposed symplectic schemes are fully implicit,and they become computationally expensive for mean square orders greater than two.However,for stochastic Hamiltonian systems preserving Hamiltonian functions,the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes.A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.
文摘In this paper, we will prove by the help of formal energies only that one can improve the order of any symplectic scheme by modifying the Hamiltonian symbol H, and show through examples that this action exactly and directly simplifies Feng's way of construction of higher-order symplectic schemes by using higher-order terms of generating functions.
基金the National Natural Science Foundation of China(Nos.11672241,11372253,and 11432010)the Astronautics Supporting Technology Foundation of China(No.2015-HT-XGD)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment(Nos.GZ1312 and GZ1605)
文摘The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conser- vation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Stormer-Verlet scheme is first constructed in a Hamiltonian frame- work. The conservation law of the StSrmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Stormer-Verlet scheme associated with the con- servation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the StSrmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the StSrmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Stormer-Verlet scheme.
基金National Key Research and Development Project of China(Grant No.2018YFC1504205)National Natural Science Foundation of China(Grant No.11771213,11971242)+1 种基金Major Projects of Natural Sciences of University in Jiangsu Province of China(Grant No.18KJA110003)Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘In this paper,we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems.The one class is the symplectic scheme,which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method,respectively.Each member in these schemes is symplectic for any fixed parameter.A more general form of generating functions is introduced,which generalizes the three classical generating functions that are widely used to construct symplectic algorithms.The other class is a novel family of energy and quadratic invariants preserving schemes,which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step.The existence of the solutions of these schemes is verified.Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.
基金Special Funds for Major State Basic Research Projects of China (No.G1999032801-10 and No. G19999032804), and by the knowledge in
文摘In this note, we will give a proof for the uniqueness of 4th-order time-reversible sym- plectic difference schemes of 13th-fold compositions of phase flows φtH(1) , φtH(2) , φtH(3) with different temporal parameters for splitable hamiltonian H = H(1) + H(2) + H(3).
基金This work was supported by the National Nature Science Foundation of China(Grant 11772026)Defense Industrial Technology Development Program(Grants JCKY2017208B001 and JCKY2018601B001)Beijing Municipal Science and Technology Commission via project(Grant Z191100004619006),and Beijing Advanced Discipline Center for Unmanned Aircraft System.
文摘In this paper,the symplectic perturbation series methodology of the non-conservative linear Hamiltonian system is presented for the structural dynamic response with damping.Firstly,the linear Hamiltonian system is briefly introduced and its conservation law is proved based on the properties of the exterior products.Then the symplectic perturbation series methodology is proposed to deal with the non-conservative linear Hamiltonian system and its conservation law is further proved.The structural dynamic response problem with eternal load and damping is transformed as the non-conservative linear Hamiltonian system and the symplectic difference schemes for the non-conservative linear Hamiltonian system are established.The applicability and validity of the proposed method are demonstrated by three engineering examples.The results demonstrate that the presented methodology is better than the traditional Runge–Kutta algorithm in the prediction of long-time structural dynamic response under the same time step.
