This work delves into the exponential time differencing (ETD) schemes for the matrix-valued Allen-Cahn equation. In fact, the maximum bound principle (MBP) for the first- and second-order ETD schemes is presented in a prior publication [SIAM Review, 63(2), 2021], assuming a symmetric initial matrix field. Noteworthy is our novel contribution, demonstrating that the first- and second-order ETD schemes for the matrix-valued Allen-Cahn equation -- both being linear schemes -- unconditionally preserve the MBP, even in instances of nonsymmetric initial conditions. Additionally, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen-Cahn equation. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields.

翻译：本文深入研究矩阵值Allen-Cahn方程的指数时间差分（ETD）格式。实际上，一阶和二阶ETD格式的最大界原则（MBP）已在先前文献[SIAM Review, 63(2), 2021]中提出，但仅针对对称初始矩阵场。本文的创新贡献在于，证明了矩阵值Allen-Cahn方程的一阶和二阶ETD格式——均为线性格式——即使在非对称初始条件下也能无条件保持MBP。此外，我们证明这两种ETD格式无条件保持矩阵值Allen-Cahn方程的能量耗散律。文中通过数值算例验证理论结果，并模拟相应矩阵场的演化过程。