Structure-preserving Algorithms For Some Partial Differential Equations And Partial Differential Algebraic Equations

A large number of mathematical models in modern sciences and engineering can be described as differential equations,including ordinary differential equations(ODEs),partial differential equations(PDEs),differential algebraic equations(DAEs)and partial differential algebraic equations(PDAEs),etc.The solutions of most of these differential equations are hard to expressed in practical and analytical formu-las with the increasing complexity of problems,and instead numerical simulation-s have been a mainstream to solve such problems,subsequently deriving a series of high-efficient numerical algorithms.This thesis focuses on the construction of structure-preserving algorithms with long-time stability for some infinite dimensional Hamiltonian PDEs,and generalize the methodology to kinds of PDAEs to design new high-precision numerical schemes.The symplectic structure and system energy are two important conservative prop-erties of finite-dimensional Hamiltonian ODEs,and numerical algorithms for preserv-ing these conservative qualities are well established such as the symplectic methods and energy-preserving methods which preserve the symplectic structure and the ener-gy,respectively.Although some related works have been carried out for the infinite-dimensional Hamiltonian PDEs,while comparing with that of the finite-dimensional cases,there still remain many issues to be addressed.For the sake of revealing the conservative properties of Hamiltonian PDEs better,it is usually to reform the sys-tems as multisymplectic Hamiltonian structures so that three local conservation laws can be clearly obtained,including the multisymplectic conservation law,the local en-ergy conservation law as well as the local momentum conservation law.Numerical algorithms which can preserve one or multiple conservation laws will demonstrate an apparent advantage over non-conservative algorithms.Therefore,in the first part of the thesis,we construct some new structure-preserving algorithms for several clas-sic Hamiltonian PDEs with theoretical analysis.Firstly,we propose a fully-explicit symplectic Fourier pseudospectral method for the Klein-Gordon-Schroinger equation,which is much more efficient than the conventional implicit or semi-implicit sym-plectic methods.In addition,we also give the linearized stability analysis and the necessary CFL condition for the scheme.Numerical results show the accuracy and effectiveness of the proposed method in long-time computations.Secondly,based on the multisymplectic form,we present an energy-preserving method for the BB-M equation with the implicit midpoint method and the averaged vector field method(AVF)for the temporal and spatial discretizations,respectively.We also prove that the resulting method can well simulate the propagations of kinds of solitons.As last,we separately give two structure-preserving methods for the RLW equation to preserve the local energy as well as the local momentum.Numerical results demonstrate that the two structure-preserving methods can not only produce good approximations,but also preserve the corresponding conservation laws.In the second part of this thesis,we generalize the methodology and schemes of the aforementioned structure-preserving methods to PDAEs and propose several numerical algorithms for linear PDAEs.Firstly,we obtain a new effective Fourier pseudospectral collocation method to solve the linear PDAEs with periodic boundary conditions,which is constructed by the Fourier pseudospectral method in space and the Crank-Nicolson method in time.Under certain conditions we can prove that the new scheme has spectral convergence rate in space and second order accuracy in time.Secondly,we consider the Galerkin spectral method for the spatial discretization of PDAEs based on the weak form of the system.In particular,we adopt the piecewise linear polynomial as the basis function and the Crank-Nicolson method for temporal discretization.The scheme is second order in both space and time directions under certain conditions.Furthermore,the concept of the differential spatial index and dif-ferential time index are introduced with respect to the fully-discretized scheme.