变阶Allen-Cahn方程的有限差分及收敛性分析

Finite Difference and Convergence Analysis of the Variable Order Allen-Cahn Equation

以泊松方程的形式表达的相场模型Allen-Cahn(AC)方程被广泛应用于图像处理、晶体生长、随机扰动等问题的研究中.为分析变阶Allen-Cahn方程的有限差分格式的收敛性,利用一个隐式的有限差分格式来对变阶AC方程进行离散,包含变阶Caputo导数的近似、二阶中心差分以及泰勒展开式的应用.根据盖尔定理和矩阵范数的性质证明了变阶Allen-Cahn方程差分格式的无条件稳定性和收敛性,并通过数值分析验证限差分格式的有效性.结果表明,通过算例的解析解和数值解之间的平面对比图,说明所采用的有限差分格式是可解且准确的.差分格式在时间、空间方向上是收敛的.

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.