Finite Difference and Convergence Analysis of the Variable Order Allen-Cahn Equation
The Allen Cahn(AC)equation,expressed in the form of Poisson's equation,is widely used in the study of image processing,crystal growth,random perturbations,and other problems.To analyze the convergence of the finite difference scheme for the variable order Allen Cahn equation,an implicit finite difference scheme is used to discretize the variable order AC equation,including approximations of the variable order Caputo derivative,second-order central difference,and the application of Taylor expansion.The unconditional stability and convergence of the finite difference scheme for the variable order Allen Cahn equation were proved based on Gale's theorem and the properties of matrix norm,and the effectiveness of the finite difference scheme was verified through numerical analysis.The results indicate that the finite difference scheme used is solvable and accurate,as demonstrated by the planar comparison between the analytical and numerical solutions of the examples.The difference scheme converges in both temporal and spatial directions.
Allen Cahn equationVariable order derivativeDifferential formatConvergence
万方数据
维普
