Some Numerical Methods And Applications Of The Allen-Cahn Equation Of Phase Field Model

Allen-Cahn(AC) equation is the primary model simulating phase-field in materials science, which describes the process of a binary alloy at a certain temperature phase separation. The analytical solution is generally difficult to obtain. Therefore,it is necessary to explore new numerical methods.In this paper, we mainly study three numerical methods of AC equation, and they are Weighted Essentially Non-Oscillatory(WENO) method, operator splitting method and radial basis function(RBF) method. Firstly, using six order WENO scheme in space and using third order TVD Runge Kutta method for discretization in time, we give the stability proof of the format. Numerical experiments are to obtain the high order numerical solution. Secondly, we divide AC equation into three sub-problems to solve the problem based on the second order operator splitting methods, and also prove the stability of the format. We use Non-oscillation characteristic of this format on AC equation for image restoration applications, and present numerical experiments to show the accuracy and efficiency of the method.Finally, using RBF approaches to solve AC equation and its application in discontinuous detection, then the accuracy and efficiency of the method are verified by some numerical experiments.