Symplectic And Multi-Symplectic Integrators For Hamiltonian Systems

Classical mechanics has three equivalent frameworks:Newtonian, Lagrangian and Hamiltonian mechanics, which are different mathematical formulations describing the same physical problems. Each of them supplies an independent approach to solving practical problems. It follows that equivalent mathematical forms may yield different results in practice.Hamiltonian systems form a significant category of dynamical systems. Any real physical process with negligible dissipation can be described by a Hamiltonian system. Traditional high order integrators minimize the error brought by approximation, but in general they miss some important structures (symplectic or multisymplectic structure, for example) of the systems. As a result, powerful in simulation of physical phenomena as they are, these methods are usually limited to the scope of short term computation. Therefore, devising numerical methods that preserve the intrinsic physical or geometric properties in long term integration has turned out to be in urgent need, In recent years, symplectic schemes for Hamiltonian systems of ordinary differential equations (ODEs) are systematically and extensively explored. Theoretical and practical research suggests that symplectic integrators behave much better than nonsymplectic integrators for solving the Hamiltonian ODEs, especially over long time. On the other hand, as a natural generalization of Hamiltonian ODEs, Hamiltonian partial differential equations (PDEs) are proposed with multisymplectic structures in both temporal and spatial directions. One of the great challenges in the numerical solution of PDEs is the development of robust and stable algorithms for Hamiltonian PDEs. Similar to symplectic methods for Hamiltonian ODEs, multisymplectic integrators are supposed to preserve the multisymplectic structures of Hamiltonian PDEs numerically.This thesis is divided into three chapters.Chapter1recalls briefly the background of Hamiltonian systems and symplectic/multisymplectic methods. Firstly, it introduces some elementary properties of Hamiltonian systems and symplectic spaces as well as some familiar symplectic schemes for Hamiltonian ODEs including symplectic Runge-Kutta methods, symplectic partitioned Runge-Kutta methods and symplectic Runge-Kutta-Nystrom methods. Several significant conservation laws of multisymplectic Hamiltonian systems and a couple of multisymplectic methods are presented.In chapter2, based on the work of the paper [44], symplectic and asymptotically symplectic ERKN (extended Runge-Kutta-Nystrom) methods for oscillatory Hamiltonian ordinary differential equations are considered. The order conditions for ERKN methods are presented and the concept of asymptotic symplecticness is introduced. Some asymptotically symplectic methods of orders3and5are constructed. The results of numerical experiments for the one-dimensional harmonic oscillator and multidimensional perturbed oscillators show the robustness and competence of the new methods in long run integration.In chapter3, multisymplectic integrators for the Hamiltonian wave equations are investigated. Taking into account the oscillatory character of Hamiltonian wave equation, we show that the discretization to the Hamiltonian wave equation utilizing two symplectic exponentially fitted RKN methods in space and time respectively leads to multisymplectic integrator, and so is the case when a symplectic partitioned Runge-Kutta method is applied in one of the two directions and a symplectic exponentially fitted RKN method in the other. Two schemes are constructed based on exponentially fitted RKN methods and the symplectic Euler method. Numerical examples show the superiority of our new schemes compared with their prototypes and the leap-frog scheme.After a summary of the main conclusions of this thesis, some prospects and challenges from symplectic and multisymplectic algorithms are listed in the end.