Numerical Methods And Applications Of Nonlinear Allen-Cahn Equations And Cahn-Hilliard Equations

In recent years,with the continuous development of computer technology,the thermodynamic and kinetic data of materials are gradually complete.The phase field model has gradually become an important tool for simulating the microstructure evolution process of various materials.The Allen-Cahn equation and the Cahn-Hilliard equation,as commonly used phase field equations,are widely used in the prediction of the evolution of the mesoscale microstructure and in the fields of materials science,and are gradually applied in astrophysics,fluid mechanics,thermodynamics,chemistry and other research fields.These two types of equations are difficult to obtain analytical solutions,so in general,their approximate solutions are solved by numerical methods.However,it has a usually large amount of calculation,a long time-consuming calculation,and less research on the Allen-Cahn equation in polar coordinates.Therefore,this paper aims to study the above-mentioned issues,and the research contents are as follows :Firstly,the two-dimensional nonlinear Allen-Cahn equation in the plane rectangular coordinate system is solved,the second-order central difference method is applied to the discrete equation,the Kronecker product is used to represent the differential matrix of the two-dimensional Laplace operator,and its spectral decomposition is performed to obtain its Diagonalized form.The integral factor method is applied to the time discretization,combined with the fast discrete cosine transform,the solution is solved in the process of its realization,and the calculation efficiency is improved.The second part solves the nonlinear Allen-Cahn equation in the polar coordinate system.First,for the Laplacian in the Allen-Cahn equation,convert it to the polar form.Then,the grids are divided in the r direction and the q direction respectively,and the central difference method and the integral factor method are used to perform space and time dispersion,and the product of the exponential matrix and the vector.In the time dispersion process,the Krylov subspace method is used.to approximate.The third part studies the fourth-order nonlinear Cahn-Hilliard equation,which is decomposed into two second-order equations by introducing auxiliary variables.Apply the second-order finite difference method to discretize it in space.Combined with Kronecker product,the differential matrix of two-dimensional Laplace operator is written and further diagonalized.The Crank-Nicolson method is used for time discretization,which theoretically proves that the discrete energy has a dissipative property with the development of time.The last part uses the modified Cahn-Hilliard equation to repair and deal with truncated bar graphs and images with watermarks and scribbles.The second-order finite difference method and the Crank-Nicolson method are used to discretize the equation in space and time,respectively.For the fully discretized equation,the fast discrete cosine transform combined with the fixed point iteration method is used to solve the equation.The repair effect and iterative efficiency of the equation are verified by the iterative images.