Construction And Analysis Of Some Structure-preserving Algorithms For Hamiltonian Systems

It is undisputed that almost all real physical processes where the dissipation can be neglected can be formulated as Hamiltonian systems. Thus, it is important to solve the Hamiltonian system correctly. Energy conservation law and symplectic structure are two most important properties of Hamiltonian system. A large number of numer-ical integrators, which preserve energy or symplectic conservation laws, have been developed in recent years. In this dissertation, we study the construction and analysis of some structure-preserving algorithms for Hamiltonian systems. The main results are as follows.I. Recently, structure-preserving methods, such as symplectic methods and volume-preserving methods, have been used to solve the Lorentz force system in plasmas. However, there is few energy-preserving methods for solving the system. The Lorentz force system can be written as a non-canonical Hamiltonian system. We apply the Boole discrete line integral to solve the system, and derive a new method which can preserve the energy of the system up to round-off error.II. We consider the AVF methods of order 2,3 and 4 to solve partial differen-tial equations. Applying the three AVF methods in time and Fourier pseudospectral method in space to solve the nonlinear Schrodinger (NLS) equation, we obtain three new energy-preserving schemes. Numerical experiments are presented to demonstrate the accuracy and the energy-preserving property of the three schemes.III. Based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order=5. With the help of the new substitution law, we derive a B-series integrator extending the AVF method for general Hamiltonian system to higher order. The new integrator turns out to be order of six and exactly preserves energy for Hamiltonian systems. Numerical experiments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method.IV. A new semidiscretization based method for solving Hamiltonian partial dif-ferential equations is proposed in this dissertation. Our key idea consists of two ap-proaches. First, the underlying equation is discretized in space via a selected finite element method or spectral element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulted ordinary differential system is solved by a structure-preserving integrator. The relay leads to a fully discretized and structure-preserved scheme. This strategy is fully realized for solving the NLS equation with one dimension through a combina-tion of the Legendre spectral element method in space and the AVF method in time. We can obtain a new energy-preserving method. The 1D NLS equation is also dis-cretized by using the Galerkin finite element method in space and the Crank-Nicolson scheme in time. We can obtain a new energy-preserving and mass-preserving method. We also use the Galerkin spectral element method in space and the Crank-Nicolson scheme in time to solve the NLS equation with two dimensions. We can obtain a new energy-preserving and mass-preserving method for the 2D NLS equation. Similarly, the Klein-Gordon-Schrodinger equation is discretized by using the Galerkin method in space and the symplectic Stomer-Verlet method in time. We can obtain a new explicit symplectic method. The coupled Gross-Pitaevskii equations for spin-1 Bose-Einstein condensate (BEC) are discretized by using the Galerkin method in space and the symplectic mid-point method in time. We can obtain a new method which can preserving the symplectic structure, the mass and the magnetization of the system. The coupled Gross-Pitaevskii equations for spin-orbit BEC are discretized by using the Galerkin method in space and the Crank-Nicolson scheme in time. We can obtain a new energy-preserving and mass-preserving method. Numerical examples are given to further illustrate the conservation properties of the schemes constructed.