| Hamiltonian systems arise widely in the fields of physics, mechanics, engineering, pure and applied mathematics, etc. It is generally accepted that all real physical processes with negligible dissipation could be expressed, in some way, by Hamiltonian formalism, so that the research work for corresponding numerical methods is of important interest. The most important property for Hamiltonian systems is the Poincare and Liuville's conservation law of phase areas, i.e., the phase flow is a one-parameter symplectic transformation. In numerically solving these equations, we hope that the numercial schemes can hold this property, and the corresponding numerical methods are called as symplectic methods. In this thesis, we present the research work about the pseudo-symplecticity of iteration schemes in implicit symplectic RK methods as well as the parallel waveform relaxation implementation theory for symplectic methods.In chapter 1, the author introduces the theory of symplectic geometry for Hamiltonian systems, and summarizes the advantages, the construction ways, and current research situation for symplectic methods.In chapter 2, based on symplectic P_series theory, the concept of the pseudo-symplectic P_series is established, which is an extension to pseudo-symplectic B_series. For a P_series induced from some one-step numerical method, the sufficient and necessary conditions of pseudo-symplectic order q are presented. We get the perturbed system equivalent to the pseudo-symplectic numerical scheme by use of backward error analysis, and prove that this perturbed system is hamiltonian when truncating with less order than pseudo-symplectic order. So, the energy error will keep bounded within the time interval in terms of O(h1"'1) (p is the consistent order of the method). Furthermore, the author proves the characteristic that the global error will grow linearly along the trajectory with time when pseduo-symplectic order q is equal to 2p, which is of importance in the accurate long-term computation.In chapter 3, the author studies mainly the pseudo-symplectic theory of the iteration in implicit symplectic RK methods. The symplectic RK methods are implicit for the non-separable Hamiltonian systems. We have to utilize some iteration method (e.g. Newton iteration) to solve the non-linear equations of internal stages. During the process of implementation, the finite number of iterations must cause the inexactness, which will lead to the loss of exact preservation of symplecticity. We use the pseudo-symplecticity to interpret the inexactness, and pseudo-symplectic order to magnitude of inexactness. By using the pseudo-symplectic B_series and P-series theories, the relation between the pseudo-symplecitc order q and iteration number k can be established. This will also pose an impedient to the stopping criteria depending only on accuracy Tol with additional requirement for pseudo-symplecticity. On the other hand, the author proves that the high-order prediction can not affect the final pseudo-symplectic order, though it can improve the order of the iterative methods. We make a little modification for original iteration scheme and construct a seriesiteration method so as to improve the pseudo-symplecticity, even raise the pseudo-symplectic order.In chapter 4, in view of constructing parallel symplectic methods, the author considers the waveform relaxation implementation for symplectic methods. The waveform relaxation method will converge to some limit method, and naturally we need to make sure that this limit method is symplectic. We get the sufficient condition for the limit method to be symplectic by use of the definition of symplectic transformation. Under some reasonable assumptions, the author proves that this condition is also necessary. We find out that the waveform relaxation implementation for a symplectic method, when requiring corresponding continuous RK method to be as a natural continuous extension(NCE), will hold the symplecticity in the sense of limit, if and only if original symplectic...
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