A Generalized Matrix-Valued Allen--Cahn Model and Its Numerical Solution

TL;DR

This paper develops a generalized matrix-valued Allen--Cahn model applicable to various dimensions, proves its mathematical properties, and introduces high-order numerical schemes that preserve key physical principles.

Contribution

It introduces a unified matrix-valued Allen--Cahn model covering scalar, vector, and matrix cases, with proven solution properties and high-order energy-stable numerical schemes.

Findings

Model covers classical Allen--Cahn equations as special cases.

Proposed schemes are unconditionally energy dissipative for first two orders.

Numerical experiments confirm theoretical convergence and stability.

Abstract

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.