摘要
非线性Allen-Cahn方程是材料学中相场模拟模型的一类重要方程,用于描述二元合金在一定温度下进行相位分离的过程,在许多科学领域中有广泛的应用。以往的工作主要在矩形区域上考虑求解,本文研究有效的数值方法在圆形区域上求解该方程。首先将Allen-Cahn方程中拉普拉斯算子写为极坐标的形式,然后采用中心差分方法在空间r方向和θ方向分别进行空间离散,得到非线性常微分方程组,并将网格上的数值解以矩阵形式表示。在时间离散过程中,采用积分因子法结合Krylov子空间的方法进行求解。最后给出数值试验。
The nonlinear Allen-Cahn equation is an important phase field simulation model in materials science. It is used to describe the phase separation process of binary alloys at a certain temperature and has been widely used in many scientific fields. The previous work mainly considers the solution on the rectangular region. In this paper, the effective numerical method is used to solve the equation on a circular region. First, the Laplacian operator in Allen-Cahn equation is written as the form of polar coordinates. Then, the central difference method is used to carry out spatial discretization in the space direction r and θ respectively, and the numerical solution on the grid is expressed in matrix form by using large sparse linear system. In the implementation of time discretization, the integral factor method combined with Krylov subspace method was used to solve the problem. Finally, numerical experiments are given.
出处
《应用数学进展》
2021年第1期109-114,共6页
Advances in Applied Mathematics
