A Generalized Matrix-Valued Allen--Cahn Model and Its Numerical Solution
This work provides a theoretical and numerical framework for matrix-valued phase-field models, which is incremental but extends applicability to broader scenarios in materials science and related fields.
The authors tackled the generalization of the Allen-Cahn model to matrix-valued fields, proving it maintains key properties like maximum bound principle and energy dissipation, and developed high-order numerical schemes that preserve these properties unconditionally or under constraints.
This paper introduces a generalized matrix-valued Allen--Cahn model, where the unknown matrix-valued field belongs to $\mathbb{R}^{m_1\times m_2}$ with dimension $m_1\geq m_2$. By taking different values of $m_1$ and $m_2$, this model covers the classical scalar-valued, vector-valued, and square-matrix-valued Allen--Cahn equations. At the continuous level, the proposed model is proven to admit a unique solution satisfying the maximum bound principle (MBP) and the energy dissipation law. At the discrete level, a class of arbitrarily high-order exponential time differencing Runge-Kutta (ETDRK) schemes is investigated that preserve the MBP unconditionally. Moreover, we prove that the first- and second-order ETDRK schemes satisfy the discrete energy dissipation unconditionally, while third- and higher-order schemes preserve the discrete energy dissipation under suitable time-step constraints. The proof of sharp convergence order in time is provided. Numerical experiments are carried out to confirm our theoretical results.