一类相场方程的能量稳定性分析及数值模拟

Energy Stability Analysis and Numerical Simulation of a Class of Phase-Field Equations

提出一种快速、稳定的数值方法求解具有恒定迁移率的二维Cahn-Hilliard方程.在空间离散上采用二阶有限差分方法,在时间离散上采用Crank-Nicolson方法,从理论上证明离散能量随时间发展具有耗散性质.针对全离散格式下的非线性代数方程组,应用不动点迭代方法求解,并利用快速离散余弦变换(FDCT)以提高计算效率.数值实验结果表明,离散自由能关于时间是非递增的,该方法具有稳定性好、存储量小、计算速度快等优点.

We proposed a fast and stable numerical method to solve the two-dimensional Cahn-Hilliard equation with constant mobility.The second order finite difference method was used in spatial discretization and Crank-Nicolson method was used in time discretization.We proved theoretically that the discrete energy had the property of dissipation with time evolving.The fixed point iteration method was used to solve the nonlinear algebraic equations in the fully discrete scheme,and the fast discrete cosine transform(FDCT)was used to improve the computational efficiency.The numerical results show that the discrete free energy is non increasing with respect to time,and the method has the advantaes of good stability,small storage and fast computation speed.