In this study, we consider volume-conserved numerical schemes for the volume-conserved time fractional Allen–Cahn equation. We start with the L1 scheme based on a modified exponential scalar auxiliary variable (ESAV) approach for handling the nonlinear potential term. Then we introduce the fast L1 scheme to significantly reduce the CPU time and to make the long-term computation possible. Further to reduce the error between the discrete energy and the original energy of the ESAV approach, we introduce the fast L1 scheme based on a relaxed modified ESAV (R-ESAV) idea. We rigorously show the discrete energy dissipation properties for the above three schemes, including the discrete energy boundedness law, the discrete fractional energy dissipation law, and the discrete weighted energy dissipation law. Theoretical findings are validated by a few numerical experiments.

中文翻译：

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基于 ESAV 和 R-ESAV 方法的时间分数体积守恒 Allen–Cahn 模型的数值能量耗散


在本研究中，我们考虑体积守恒时间分数艾伦-卡恩方程的体积守恒数值格式。我们从基于修正指数标量辅助变量 (ESAV) 方法的 L1 方案开始，用于处理非线性势项。然后我们引入快速L1方案来显着减少CPU时间并使长期计算成为可能。为了进一步减少 ESAV 方法的离散能量与原始能量之间的误差，我们引入了基于宽松修改 ESAV (R-ESAV) 思想的快速 L1 方案。我们严格地展示了上述三种方案的离散能量耗散性质，包括离散能量有界律、离散分数能量耗散律和离散加权能量耗散律。理论研究结果通过一些数值实验得到了验证。