表面上Allen-Cahn方程的一种解耦型算法

A Decoupling-Type Strategy for the Allen-Cahn Equation on Curved Surfaces

中文关键词:  FEM-EIEQ  Allen-Cahn方程  表面

英文关键词:FEM-EIEQ  Allen-Cahn equation  surface

中文摘要:

      本研究构建了一种高效解耦型算法用于求解表面上的 Allen-Cahn方程.该算法基于FEM-EIEQ(有限元法和显式不变能量二次化)全离散算法,具有无条件能量稳定性.空间上采用了有限元法,使用可适应复杂区域的三角形网格离散化算法.时间上考虑了显式不变能量二次化法,不仅使非线性势线性化,还设计了一个新的变量,我们将其与非局部分裂算法结合,以实现完全解耦计算.该算法能够成功将Allen-Cahn系统转化为一些完全独立的代数方程和常系数线性椭圆方程,我们只需要在每个时间步长求解这些简单的方程即可.此外,本研究还进行了一些数值实验来证明该算法的有效性.

英文摘要:

      In this paper, we construct an efficient decoupling-type strategy for solving the Allen-Cahn equation on curved surfaces. It is based on an FEM-EIEQ (Finite Element Method and explicit-Invariant Energy Quadratization) fully discrete scheme with unconditional energy stability. Spatially the FEM is adopted, using a triangular mesh discretization strategy that can be adapted to complex regions. Temporally, the EIEQ approach is considered, which not only linearizes the nonlinear potential but also gives a new variable that we combine with the nonlocal splitting method to achieve the fully decoupled computation. The strategy can successfully transform the Allen-Cahn system into some completely independent algebraic equations and linear elliptic equations with constant coefficients, we only need to solve these simple equations at each time step. Moreover, we conducted some numerical experiments to demonstrate the effectiveness of the strategy.