The Difference Methods For The Initial-Boundary Value Problem Of The Two-Dimensional Allen-Cahn Equations

The phase field model is a mathematical model described by partial differ-ential equations,which has a very rich mathematical significance.The numerical simulation of phase field is beneficial to the development of numerical theory of partial differential equations,numerical algebra theory and Sobolev space theory.At the same time,it is also beneficial to understand the related problems in mate-rials science,and it provides a rich theoretical basis for manufacturing materials and material science research.The Allen-Cahn equation is an important class of equations in phase field models.In 1979,the Allen-Cahn equation was first proposed by Allen and Cahn,which was used to describe the phase separation process in binary alloys and the anti-phase boundary motion in crystalline solids.The Allen-Cahn equation is widely used.However,because of the complexity of the equation,it is difficult to obtain the analytic expression of the true solution.So it is particularly important to use numerical method to solve the equation.There have been many researches on the numerical methods of this kind of model,such as the finite difference method,the finite element method and the spectral method and so on.In this thesis,we mainly study three finite difference schemes of the two-dimensional Allen-Cahn equation with Dirichlet boundary conditions,and they are the bilevel nonlinear difference scheme?the three-level linearized difference scheme and the ADI scheme.At same time,some theoretical analyses of these difference scheme are be discussed,that is,the existence and uniqueness of the solution of the difference scheme,the unconditional convergence under the infinite norm and the maximum principle of the numerical solution are proofed.Finally,the numerical experiments is given to verify the reliability of the method.More precisely,firstly,a two-level nonlinear difference scheme is established for two-dimensional nonlinear Allen-Cahn equation,the truncation error of which is given.The existence of the solution of the finite difference solution is demon-strated with the help of Browder theorem.The scheme is showed to be un-conditionally convergent in_?norm by energy analysis and₂theory with an auxiliary function for the nonlinear term.Secondly,a three-level linearized d-ifference scheme is established,the boundedness and uniqueness of the solution is provided.The unconditional convergence is given by₂analytical method.Finally,an ADI scheme is established,also,the existence and The unconditional convergence of the difference solution is give.Based on this result,it is demon-strated that the difference scheme preserves the maximum principle without any restrictions on spatial step and temporal step sizes.