On the maximum bound principle and energy dissipation of exponential time differencing methods for the matrix-valued Allen–Cahn equation

Abstract 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. Furthermore, we prove that these two ETD schemes preserve the energy dissipation law unconditionally for the matrix-valued Allen–Cahn equation, and their convergence analysis is also provided. Some numerical examples are presented to verify our theoretical results and to simulate the evolution of corresponding matrix fields.