Three Time Discrete Schemes For Lifshitz-Petrich Phase Field Model

In this paper,we study three classes of time discretization schemes of Lifshitz-Petrich dynamics equations.Including the first-order semi-implicit scheme,splitting scheme,and the scalar auxiliary variable method.The dynamics equations includes the Allen-Cahn and Cahn-Hilliard dynamic equations.The most difficult part of solving such equations is to deal with nonlinear terms.In the semi-implicit scheme,the nonlinear terms are discretized explicitly while linear terms implicitly.The method has first-order precision.The splitting scheme is similar to the semi-implicit scheme.It treats the nonlinear terms explicitly,however the linear terms are splitted by consecutive two steps.By von Neumann linear stability analysis,we can obtain the range of values of the splitting parameters,which guarantees the unconditional stability of the scheme.However numerical results show that the error order depends on the choice of splitting parameters and can not reach the first-order precision.Fi-nally,we use the newly proposed scalar auxiliary variable method to solve our model.We use the idea of this method to construct the first-order,second-order BDF2 and second-order Crank-Nicolson schemes to solve Cahn-Hilliard equation,and verify the efficiency and precision of the method.By studying such numerical discrete schemes,we find that the scalar auxiliary variable method can not only ensure en-ergy dissipation,but also facilitate the construction of first-order and second-order time-discretization schemes conveniently without depending on the specific form of nonlinear terms.Therefore it has very flexible scalability.