基于有限元方法的Cahn-Hilliard方程能量不变二次化数值模拟

Numerical Simulation of Energy Invariant Quadratic Cahn-Hilliard Equation Based on Finite Element Method

摘要: 文章分析了Cahn-Hilliard方程的能量不变二次化法的能量稳定性。首先分析了Cahn-Hilliard方程是满足能量耗散，即能量随时间的推移呈衰减趋势，并且这种随时间衰减的性质能得到保持。其次，对定义的自由能被积函数变换成新的二次函数，即能量不变二次化法同样验证了它的能量衰减性质，并给出了Cahn-Hilliard方程能量不变二次化法的能量稳定性。结果表明，该数值格式是无条件稳定，即能量稳定性与时间步长是无关的。最后我们基于有限元方法给出了一个数值算例，来有效的模拟Cahn-Hilliard方程相位变化情况。

Abstract: The article analyzes the energy stability of the energy-invariant quadratic method of the Cahn-Hilliard equation. Firstly, it is analyzed that the Cahn-Hilliard equation is satisfying energy dissipation, i.e., the energy tends to decay with time, and this decaying property with time can be maintained. Secondly, the defined free energy quadratic function is transformed into a new quadratic function, i.e., the energy-invariant quadratic method is similarly verified for its energy-decaying property, and the energy stability of the energy-invariant quadratic method of the Cahn-Hilliard equation is given. The results show that the numerical format is unconditionally stable, i.e. the energy stability is independent of the time step. Finally we give a numerical example based on the finite element method to effectively simulate the phase variation of the Cahn-Hilliard equation.