The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.

翻译：能量耗散律和最大值原理是Allen-Cahn方程的两个关键物理性质。虽然许多现有时间步进方法已知能够保持能量耗散律，但大多数方法适用于修正形式的能量。本文证明，当Allen-Cahn方程的非线性项满足Lipschitz连续时，一类任意高阶指数时间差分龙格-库塔（ETDRK）格式在温和步长约束下能够保持原始能量耗散性质。此外，我们通过应用重整化后处理技术确保非线性项满足Lipschitz条件，从而使数值解无条件满足最大值原理。因此，所提出的格式同时保持了原始能量耗散律和最大值原理，并可实现任意高阶精度。我们还建立了所提格式的最优误差估计。通过数值实验验证了我们的理论结果。