A Maximum-principle Preserving And Unconditionally Energy-stable Linear Second-order Finite Difference Scheme For Allen-Cahn Equations

Allen-Cahn equation is a classical high-order nonlinear partial differential equations describing non-conservative field variables in the phase field model.It was originally proposed by Allen and Cahn to describe the anti-phase boundary motion of crystalline solids.Due to the complexity and variability of practical problem,the analytical solution of this equation is generally difficult to obtain,so it is necessary to construct new stable and efficient numerical methods to solve it.In this paper,we propose a new linear second-order finite difference scheme for Allen-Cahn equations with Dirichlet boundary.Firstly,it is shown that the discrete maximum principle holds under reasonable constrains on time step size and coefficient of stabilized term.Secondly,based on the maximum stability,the maximum-norm error is analyzed.Next,it is proved that the modified discrete energy is unconditionally decreasing.Finally,1D and 2D numerical experiments are performed to verify the theoretical results.