High Order Accurate Numerical Algorithm For Two Kinds Of Phase Field Equations

Phase field model is a kind of mathematical model which is used mainly to describe the diffusion properties of the interface between two phases.The numerical simulation of phase field equation is very extensive in practical application.However,due to phase field equation has the characteristics of strong nonlinearity,small parameters and high order,and some special physical properties such as energy decline and so on.It is difficult to find the analytical solution.Therefore,it is of great significance to construct stable and effective high-precision numerical methods to solve them.In this paper,we will study high order accurate numerical algorithm for two kinds of classical phase field models that are Allen-Cahn equation and Cahn-Hilliard equation.The specific contents are as follows.Three kinds of high order accurate numerical schemes are presented based on barycentric interpolation collocation method for Allen-Cahn equation,which are the finite difference collocation scheme,operator splitting scheme and scalar auxiliary variable(SAV)scheme,respectively.Firstly,the finite difference collocation scheme is considered based on Crank-Nicolson scheme in time and barycentric interpolation collocation method in space.And,the consistency analysis of the semi-discretized scheme in space is derived.Next,the original equation is split into linear and nonlinear subproblems for the operator splitting scheme.Nonlinear part is solved analytically which can avoid nonlinear iteration.And,the error estimates of the proposed scheme are studied.Then,we will combine barycentric interpolation collocation method with SAV method,and propose an unconditional energy stable SAV scheme.Finally,numerical examples verify the convergence order and energy dissipation properties of the three schemes,respectively.Two kinds of high-precision numerical schemes are developed for Cahn-Hilliard equation,which are barycentric interpolation collocation scheme and stabilized semi-implicit collocation scheme,respectively.At first,barycentric interpolation collocation methods are used both in time and in space and the nonlinear term is discretized by the general iteration method,which derives barycentric interpolation collocation scheme.The scheme has high precision in both spatial and temporal directions.Secondly,a stabilized semi-implicit collocation scheme is constructed based on Crank-Nicolson scheme in time and barycentric interpolation in space.Finally,numerical examples will show the high accuracy and the law of energy decline of the two collocation schemes.