Generalization And Convergence Analysis Of Structure-preserving Algorithms For Partial Differential Equations

All real physical processes where the dissipation can be neglected can be formulated as Hamiltonian systems.Devising structure-preserving algorithms of Hamilto-nian systems has a great significance in accurate and efficient numerical simulations for long-term computation.Nowadays,structure-preserving algorithms for finite dimensional Hamiltonian systems are well established,and have successfully made many applications.In comparison,the structure-preserving algorithms for infinite dimensional Hamiltonian systems are still in the early stage,where there are many issues about fundamental theories waiting to be solved.For instance,one of the great challenges in scientific computation is the construction of high order schemes and the standard numerical analyses for structure-preserving algorithms of partial differential equations.In addition,structure-preserving algorithms exhibit superior properties in solving conservative systems and a series of results on the theoretical analysis were obtained.However,the research for nonconservative systems is under development.Specifically,there are few theoretical results on the special treatment for nonconservative partial differential equations.Thus,we devote the present thesis to the following two aspects.1.The construction and convergence analysis of high order structure-preserving algorithms for infinite dimensional Hamiltonian systems.2.The construction and convergence analysis of structure-preserving algorithms for nonconservative partial differential equations.The main results are listed as follows.1.We propose a sixth order energy-conserved scheme for the three-dimensional time-domain Maxwell's equations.We first show that the scheme can preserve all of the desired structures.Then,based on the energy method,an optimal error estimate,without any restrictions on the grid ratio,is established for the scheme in discrete L2-norm.Finally,we analyze some properties of this scheme from the point of the numerical dispersion relation.2.Based on the splitting idea,we propose a conservative Fourier pseudo-spectral scheme for the damped nonlinear Schrodinger equation in one dimension.We first show that the scheme can preserve the discrete total energy and conformal mass conservation laws.Then,the existence,the uniqueness and the stability of the solution of the proposed scheme are investigated.Finally,an a priori estimate,without any restrictions on the grid ratio,is established for the scheme in discrete L?-nonn.3.We propose a linearized and conservative Fourier pseudo-spectral scheme for the damped nonlinear Schrodinger equation in three dimensions.We first prove that the scheme can preserve the two discrete total conservation laws.Then,the existence and uniqueness of the solution of the proposed scheme are investigated.Finally,based on the mathematical induction and the energy method,an unconditionally optimal error estimate is established for the scheme in discrete L2-norm.