The energy dissipation is an important and essential property of classical phase-field equations. However, it is still unknown if the phase-field models with Caputo time-fractional derivative preserve this property, which is challenging due to the existence of both nonlocality and nonlinearity. Our recent work shows that on the continuous level, the time-fractional energy dissipation law and the weighted energy dissipation law can be achieved. Inspired by them, we study in this article the energy dissipation of some numerical schemes for time-fractional phase-field models, including the convex-splitting scheme, the stabilization scheme, and the scalar auxiliary variable scheme. Based on a lemma about a special Cholesky decomposition, it can be proved that the discrete fractional derivative of energy is nonpositive, i.e., the discrete time-fractional energy dissipation law, and that a discrete weighted energy can be constructed to be dissipative, i.e., the discrete weighted energy dissipation law. In addition, some numerical tests are provided to verify our theoretical analysis.

中文翻译：

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时间分数相场方程的数值能量耗散

能量耗散是经典相场方程的一个重要且必不可少的性质。然而，具有 Caputo 时间分数阶导数的相场模型是否保留这一性质仍然未知，由于非局域性和非线性的存在，这具有挑战性。我们最近的工作表明，在连续水平上，可以实现时间分数能量耗散定律和加权能量耗散定律。受它们的启发，我们在本文中研究了时间分数相场模型的一些数值方案的能量耗散，包括凸分裂方案、稳定方案和标量辅助变量方案。基于一个关于特殊 Cholesky 分解的引理，可以证明能量的离散分数阶导数是非正的，即，离散时间分数能量耗散定律，并且可以将离散加权能量构造为耗散的，即离散加权能量耗散定律。此外，还提供了一些数值试验来验证我们的理论分析。