论文标题
时间分数相位方程的降低能量的上限
A decreasing upper bound of energy for time-fractional phase-field equations
论文作者
论文摘要
在本文中，我们研究了时间分数allen-cahn方程的能量耗散特性。我们提出了一种减少的能量上限，相对于时间的减少，与原始能量$ t = 0 $相吻合，并且$ t $倾向于$ \ infty $。由于时间分数衍生物的记忆效应，该上限也可以看作是非局部时间修改能量，原始能量的总和和累积项。特别是，这表明原始能量确实衰减了W.R.T. $ t = 0 $的小社区的时间。我们主要用时间分数的allen-cahn方程来说明理论，但可以应用于其他时间折叠相位模型，例如cahn-hilliard方程。 在离散级别上，用于时间折叠的Allen-CAHN方程的一阶L1和二阶L2方案具有相似的修改能量，因此可以建立稳定性。提供了一些数值结果，以说明这种修改的能量的行为并验证我们的理论结果。
In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. We propose a decreasing upper bound of energy that decreases with respect to time and coincides with the original energy at $t = 0$ and as $t$ tends to $\infty$. This upper bound can also be viewed as a nonlocal-in-time modified energy, the summation of the original energy and an accumulation term due to the memory effect of time fractional derivative. In particular, this indicates that the original energy indeed decays w.r.t. time in a small neighborhood at $t=0$. We illustrate the theory mainly with the time-fractional Allen-Cahn equation, but it could be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation. On the discrete level, the first-order L1 and second-order L2 schemes for time-fractional Allen-Cahn equation have similar decreasing modified energies, so that the stability can be established. Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.
