Energy Structure-preserving Numerical Alogorithms Of Allen-Cahn Equation

This thesis is devoting to constructing efficient energy structure-preserving algorithms for the Allen-Cahn(AC)equation in the phase field equation.The AC equation was originally used to describe the anti-phase boundary motion in crystals,and is now widely used in materials science,image processing,biology and other fields.Due to the existence of nonlinear terms and the variability of the definite solution conditions(boundary and initial conditions)in the AC equation,it is difficult to calculate its exact solution.Therefore,it is of great significance to discuss the numerical solution method of AC equation.This thesis starts with the energy of the AC equation and constructs its structure-preserving numerical algorithm,which aims to keep the numerical format in terms of energy dissipation.The finite difference method and finite element method are used to construct the numerical format of the AC equation,and the stability and energy dissipation of the numerical format are analyzed.Finally,some numerical experiments are given to verify and discuss the format.The thesis is arranged as follows:In Chapter 1,we mainly analyze the current research status at home and abroad,and discuss the significance of the considered problems.In Chapter 2,we present the preliminary knowledge that needs to be used in the process of constructing the numerical algorithms.It mainly includes the average vector field method,the higher-order compact method,and the local discontinuous Galerkin finite element method.In Chapter 3,we use the finite difference method to construct some numerical formats for the AC equation.The basic idea is to use the second-order central difference quotient,fourth-order and sixth-order high-order compact methods to discretize space,and then use the average vector method to discretize time.The stability and energy dissipation of these numerical formats are also analyzed and discussed.Studies have shown that these numerical formats are stable and maintain the dissipation of energy.Then its bound-ary conditions are divided into two categories for discussion:one is the homogeneous Neumann boundary condition;the other is the periodic Neumann boundary condition.We analyze the numerical algorithms of these two types of boundary conditions in detail,and give the corresponding numerical implementation process in detail.In Chapter 4,we use the discontinuous Galerkin finite element method to construct a structure-preserving algorithm for the AC equation.For this reason,the discontinuous Galerkin method is used in the space direction,and the average vector field method is used in the time direction.In order to use the discontinuous Galerkin method for discretization,the intermediate variable q=u_x is introduced to trans-form the AC equation into a first-order form of space.Based on the stability of the numerical format,the numerical flux (?).are selected.Next,it is proved that the energy is kept dissipated in this numerical format.Finally,the implementation procedure of numerical schemes under two boundary conditions is given.In Chapter 5,we give several numerical examples.From the numerical results,we can intuitively see that the four numerical formats given in the thesis can preserve the energy dissipation of the AC equation.Finally,some conclusions are summarized based on the theoretical analysis and numerical illustration.The prospect work to be done is planned.