标题： 指数积分器在Allen-Cahn方程中的最大值边界保持性 显示英文标题

标题： Maximum bound preservation of exponential integrators for Allen-Cahn equations

摘要： 我们开发并分析了一类任意高阶、最大值边界保持的时间推进格式，用于求解Allen-Cahn方程。 这些格式是在指数积分的迭代框架内构建的，结合了精心选择的数值求积规则，包括高斯-勒让德求积规则和左高斯-拉道求积规则。 值得注意的是，所提出的格式被严格证明在不需要任何额外后处理技术的情况下无条件地保持最大值边界，同时实现任意高阶的时间精度。 提供了在$L^2$范数下的详细误差分析。 数值实验验证了理论结果，展示了所提出方法的有效性，并强调了不恰当的求积规则选择可能会违反最大值边界原理，导致错误的动力学行为。 显示英文摘要

摘要： We develop and analyze a class of arbitrarily high-order, maximum bound preserving time-stepping schemes for solving Allen-Cahn equations. These schemes are constructed within the iterative framework of exponential integrators, combined with carefully chosen numerical quadrature rules, including the Gauss-Legendre quadrature rule and the left Gauss-Radau quadrature rule. Notably, the proposed schemes are rigorously proven to unconditionally preserve the maximum bound without requiring any additional postprocessing techniques, while simultaneously achieving arbitrarily high-order temporal accuracy. A thorough error analysis in the $L^2$ norm is provided. Numerical experiments validate the theoretical results, demonstrate the effectiveness of the proposed methods, and highlight that an inappropriate choice of quadrature rules may violate the maximum bound principle, leading to incorrect dynamics.