Efficient Structure Preserving Methods For Hamiltonian System

This paper studies the efficient structure preserving algorithm for Hamiltonian system.Con-sider about the finite difference schemes of Klein-Gordon-Schrodinger equation(KGS)and "good"Boussinesq equation(GB),introduce new efficient methods to improve the computational efficien-cy of the algorithm without changing the conservation of the original algorithm.Comparing the numerical accuracy,convergence,conservation,calculation time of original algorithm and new al-gorithm with theoretical analysis and numerical experiments to prove the efficiency of the new algorithm.In the first chapter,the research background and related knowledge are introduced.In the second chapter,an efficient symplectic structure preserving algorithm for KGS equation is proposed.Firstly,the Hamiltonian form and conservation of KGS equation are introduced.Then,the finite difference method and the Euler midpoint method are applied to dispersed the temporal partial derivative and spatial partial derivative of KGS equation,and get a general finite difference symplectic scheme.Next,we study the high order compact method and use it instead of general finite difference method to discretize the spatial partial derivative of KGS equation,and get a efficient high order compact symplectic scheme.The high order compact symplectic scheme preserve conservation of charge and conservation of symplectic structure are proved theoretically.Convergence order of high order compact symplectic scheme higher than the general finite difference symplectic scheme.Finally,numerical experiments show that the theoretical analysis is correct.In the third chapter,an efficient symplectic structure preserving algorithm for GB equation is proposed.Firstly,the Hamiltonian form and conservation of GB equation are introduced.Then,the finite difference method and the Euler midpoint method are applied to dispersed the temporal partial derivative and spatial partial derivative of GB equation,and get a general finite difference scheme.Next,we minutely explained the details of the odd-even split method and use it instead of general finite difference method to discretize the spatial partial derivative of GB equation,and get a efficient odd-even split scheme.The odd-even split scheme preserve conservation of energy or conservation of symplectic structure are proved theoretically,computation cost of odd-even split scheme less than the general finite difference scheme.Finally,numerical experiments show that the theoretical analysis is correct.In the fourth chapter,a brief summary of this paper and the further study on the efficient structure preserving algorithm of Hamilton system are proposed.