一种用于Allen-Cahn型梯度流的二阶最大值保持且能量稳定的指数时间差分方法
A Second-Order Maximum-Bound-Preserving and Energy-Stable Exponential Time-Differencing Method for Allen--Cahn-Type Gradient Flows
- Beijing Normal University（北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种基于GSAV和ETDRK2的二阶线性数值格式，用于Allen-Cahn型梯度流，无条件保持最大值原理和能量稳定性，并具有时空二阶精度。
AI中文摘要：
能量耗散定律和最大值保持原理（MBP）是著名的Allen-Cahn方程的两个重要物理特性。在本文中，我们针对一类Allen-Cahn型梯度流，开发并分析了一种新颖的二阶线性数值格式。我们的格式基于广义标量辅助变量（GSAV）方法和一种新颖的二阶指数时间差分Runge--Kutta（ETDRK2）方法。所得到的公式克服了将这两种技术结合时长期存在的困难，同时保留了MBP和能量稳定性。我们证明了所提出的格式无条件地保持MBP和能量稳定性。此外，我们对所提出的格式进行了严格的误差分析，建立了时间和空间上的二阶精度，而无需在时间步长$\ au$和空间网格尺寸$h$之间施加任何耦合条件。我们还进行了一些数值实验，以证明所提出格式的效率及其对理论性质的保持。
英文摘要：
The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen--Cahn equation. In this paper, we develop and analyze novel second-order linear numerical schemes for a class of Allen--Cahn type gradient flows. Our scheme is based on the generalized scalar auxiliary variable (GSAV) approach and a novel second-order exponential time-differencing Runge--Kutta (ETDRK2) method. The resulting formulation overcomes a longstanding difficulty in combining these two techniques while retaining both the MBP and energy stability. We prove that the proposed scheme unconditionally preserves both the MBP and energy stability. In addition, rigorous error analysis is carried out for the proposed scheme, establishing second-order accuracy in both time and space without imposing any coupling condition between the time step $τ$ and the spatial mesh size $h$. We also present some numerical experiments to demonstrate the efficiency of the proposed scheme and its preservation of the theoretical properties.
