Efficient Symplectic Methods For Hamiltonian ODEs And Multi-symplectic Methods For Hamiltonian PDEs

This dissertation is a collection of the author's research work during his study pursuing a master degree. The contents are our results on the efficient symplectic methods for Hamiltonian ODEs and multi-symplectic methods for Hamiltonian PDEs, and their applications in practice.For Hamiltonian ODEs, the symplectic method, founded by the late Chinese scientist Feng Kang and American scientist Ruth in the 1980's, has been proved by many theoritical results and numerous numerical experiments to be suitable and efficient for the computation of them, especially on the long-time scale. We analyze the implementation of the symplectic integrators in practice. Due to the well known reasons that the construction of the methods based on generation function may have differential coefficients, so their implementations are largely restricted and the research of such methods are mainly of theoretical interests now. Usually, we resort to symplectic RK(PRK) methods, but a fact is that the implementations of those methods in practice are mainly confined within lower level, for the computation cost of such methods in higher order, which must be implicit for the non-seperable Hamiltonian systems, are prodigious. A technique due to Buthcher and Bickart can remarkably reduce the computation , but it should require the coefficient matrix of the RK(PRK) method used, at the best, has real eigenvalues only. So we consider the symplectic RK method and symplectic PRK method with real eigenvalues only. We prove that an s-stage symplectic RK(PRK) method with real eigenvalues cannot have order more than s + 1, particularly, an s-stage such RK method cannot reach order s +1 when s is even, but in case s is odd this order barrier does not hold. Symplectic PRK method with real eigenvalues does not meet the order barrier mentioned above , and two ways to construct such PRK method are given. Further, we prove that, in higher order level, both symplectic RK methods with real eigenvalues and symplectic PRK methods with real eigenvalues cannot be comparable to composition methods in consideration of the efficiency of these algorithms. However, our recent work showthat composition method may bring the problem as stability and this should be investigated carefully in future. Finally, we recommend some symplectic efficient algorithms of lower order.We also continue the work of Reich on multi-symplectic integrators for multi-symplectic Hamiltonian partial differential equations (PDEs) in parlance of Bridges. We consider PRK methods for Hamiltonian PDEs, and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs. The conservation of energy and momentum is also considered.