Some Structure-preserving Algorithms For Conformal Hamiltonian System

The structure-preserving algorithms are an important research area among numerical investigation of differential equations,its aim is to construct numerical integrations to preserve the underlying properties of a continuous dynamical system.All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems,and it is universal in the nature.However,the most classical mechanical systems are non-conservative,so it is difficult to represent these non-conservative ones as classical Hamiltonian formalism or variational formalisms of least action or Lagrangian formalisms.This disadvantage largely restricts the application of structure-preserving algorithms.The dissertation is devoted to construct novel conformal structure-preserving algorithms for Hamiltonian systems with linearly damping.Meanwhile,we investigate the responding discrete conformal conservation law.Main contributions of this dissertation are as follows:1.We mainly propose a conformal momentum-preserving method and a conformal energy-preserving method for damped multi-symplectic formulation.Based on its damped multi-symplectic formulation,the system can be split into a Hamiltonian part and a dissipative part.For the Hamiltonian part,the average vector field(AVF)method and implicit midpoint method are employed in spatial and temporal discretizations respectively,or the implicit midpoint method and AVF method are employed in spatial and temporal discretizations respectively.For the dissipative part,we can solve it exactly.The proposed conformal momentum-preserving method conserves the conformal momentum conservation law in any local time-space region.With appropriate boundary conditions,this method also preserves the corresponding total conformal momentum exactly.Numerical experiments for damped nonlinear Schr?dinger(DNLS)equation indicate that the proposed conformal momentum-preserving method has the conformal properties of structure-preservation,and numerical stability during long-time computations.2.For the coupled damped nonlinear Schr?dinger(CDNLS)system,we introduce two novel conformal structure-preserving algorithms,which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws.The proposed algorithms can preserve corresponding conformal multi-symplectic conservation law and conformal momentum conservation law in any local time-space region,respectively.Moreover,it is further shown that the algorithms admit the conformal charge conservation law,and exactly preserve the dissipation rate of charge under appropriate boundary conditions.Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.3.We develop a novel conformal structure-preserving framework by using Strang splitting method.Based on infinite conformal Hamiltonian system,we derive a conformal symplectic wavelet collocation method for the damped Klein–Gordon(DKG)equation,which conserves the global conformal symplectic conservation law.Based on conformal multi-symplectic formulation and its conformal conservation laws,we derive a novel Preissman method by using implicit midpoint method.The proposed Preissman method not only preserves the conformal multi-symplectic conservation law and conformal momentum conservation law,but also conserves the conformal conservation law arising from linear symmetries.Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.4.For the damped nonlinear Schr?dinger equation(DNLS),we develop a novel conformal structure-preserving framework by using Strang splitting method.Based on damped multi-symplectic formulation and its conformal conservation laws,we derive two numerical methods for the damped nonlinear Schr?dinger equation(DNLSE),including a high order compact conformal multi-symplectic method,a conformal momentumpreserving method and a splitting conformal multi-symplecitc Fourier method.The outstanding advantage of proposed methods is that they conserve these local structures in any time-space region exactly.Under periodic boundary condition,these methods also preserve the dissipation rate exactly.Numerical experiments for bright/dark soliton are presented to demonstrate the conformal properties and effectiveness of the proposed methods during long-time numerical simulations and validate the analysis.