A Numerical Algorithm For Energy Conservation Stability Of Allen-Cahn Equation

Allen-Cahn equation is one of the important phase field models used to describe the diffusion properties of the interface,and it is a kind of nonlinear partial differential equation.Originally used to describe the motion of the antiphase boundary in crystals,it is now widely used in materials science,image processing,biology and other fields.Due to the complexity and variability of the actual problem,the strong nonlinearity of the Allen-Cahn equation,and the small parameter problem of the equation,the exact solution of the equation is not easy to obtain,and the numerical solution can only be obtained in a small time step with the help of numerical methods.If the solution is carried out in a large time step,it will cause the oscillation and energy instability of the numerical solution.The main work of this paper is to start with the energy of the Allen-Cahn equation,construct its efficient and stable numerical algorithm,use the finite difference method and the finite element method to construct the numerical scheme of the Allen-Cahn equation,and analyze the energy stability of the constructed numerical scheme,the proof of the maximum principle,and the analysis of the optimal error estimation.Finally,some specific numerical examples are given,respectively for one and two dimensional cases,It is verified that the solution of the numerical scheme in this paper meets the above theoretical analysis,and the change process of the numerical solution with time is given.