Study On Two Classes Of Structure-preserving Methods And Their Applications

At present,the study on the structure-preserving algorithms of the Hamiltonian equation is a major research field of partial differential equations' numerical methods.The Hamiltonian system has two basic preserving characteristics: symplectic or multisymplectic structure in phase space as well as one or more first integrals of the system.This dissertation mainly focuses on how to construct numerical algorithms for the Hamiltonian differential equation,which are able to preserve the first integrals,that is the invariants of Hamiltonian systems.In the process of the specific physical problems,there are many important physical conserved quantities,such as energy,momentum,angular momentum and charge,corresponding to the first integral of differential equation.Thus,it is important for the simulation of physical problems to preserve the first integrals of differential equations in the numerical calculation.In addition,according to the principle,which says ”After discrete,the basic characteristics of the system should be preserved as far as possible after discrete”,Preserving the invariants of Hamiltonian system is beneficial to acquire accurate numerical solution and keep a long time numerical simulation.So it is of important significance to study on the structure-preserving algorithms of the Hamiltonian equation.In this paper,the structure-preserving algorithms of the Hamiltonian partial differential equation are studied,the main work includes:The first and second chapter discusses the practical significance and research status of the structure-preserving algorithms of the Hamiltonian equation.Meanwhile,the characteristics of the conservation of Hamiltonian differential equations and several space discrete methods are introduced.In chapter three,by the Fourier pseudospectral method for spatial discrete,the Hamiltonian boundary value method is used for Hamiltonian partial differential equation.It is the first time to solve the nonlinear Schr?dinger equation,KdV equations and coupled nonlinear Schr?dinger equation by using this method.Numerical results show that,compared with the multi-symplectic method and the average vector field method,the proposed numerical algorithms have better performances in preserving structures of the system,higher numerical accuracy and stronger capacity to achieve numerical simulation for a long time.In chapter four,by combining the discrete variational derivative method with the high precision space discrete method(the Fourier pseudospectral method and the wavelet collocation method),we construct a modified conserving algorithm and a modified linear conserving algorithm.Then the proposed algorithms are applied to solve the DegasperisProcesi equation.According to the numerical results,the modified conserving algorithm has better performances in eliminating the oscillation and modified linear conserving algorithm is more efficient in numerical calculation.The fifth chapter summarizes the whole dissertation and points out the next plan of further work.