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

Yaru Liu, Chaoyu Quan, Dong Wang

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.

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

Abstract

This paper introduces a generalized matrix-valued Allen--Cahn model, where the unknown matrix-valued field belongs to with dimension . By taking different values of and , 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.

Paper Structure

This paper contains 8 sections, 11 theorems, 125 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Existence, uniqueness, maximum principle, and energy dissipation law of the solution
  3. High-order rescaled ETDRK schemes
  4. Maximum bound principles
  5. Original energy dissipation
  6. Convergence analysis
  7. Numerical Experiments
  8. Conclusion

Key Result

Lemma 2.1

Let $m_1 \geq m_2$ be two positive integers. For any $U\in \mathbb{R}^{m_1\times m_2}$ with $\|U\|_{\rm F}\leq \sqrt{m_2}$, if $\kappa\geq \max\{\frac{3}{2} m_2-1,2\}$, then we have Furthermore, for any $U_1,U_2\in \mathbb{R}^{m_1\times m_2}$ with $\|U_1\|_{\rm F},\|U_2\|_{\rm F}\leq \sqrt{m_2}$, if $\kappa\geq 3m_2+1$, we have

Figures (9)

  • Figure 1: Evolution of the supremum norm $\|\cdot\|_{\mathcal{X}}$ (first row) and of the discrete energy (second row) for the rescaled ETDRK$r$ schemes ($r=3,4,5$) with initial condition \ref{['eq:4.1']}. The dashed line in the first row indicates the maximum bound $1$, while the dashed line in the second row represents the initial energy level.
  • Figure 2: Evolution of the vector field at $t=0,50,200,500,800,1000$. The initial field is given in \ref{['eq:4.1']}.
  • Figure 3: Evolution of the supremum norm $\|U\|_{\mathcal{X}}$ (first row) and the discrete energy (second row) for the rescaled ETDRK$r$ schemes ($r=3,4,5$) with initial condition \ref{['eq:4.2']}. The dashed line in the first row represents the theoretical maximum bound $\sqrt{2}$, and the dashed line in the second row represents the initial energy.
  • Figure 4: Evolution of the matrix-valued field and interface at $t=0,50,100,200,500,700$. The initial field is given in \ref{['eq:4.2']}.
  • Figure 5: Evolution of the supremum norm $\|U\|_{\mathcal{X}}$ (first row) and the discrete energy (second row) for the rescaled ETDRK$r$ schemes ($r=3,4,5$) with initial condition \ref{['eq:4.3']}. The dashed line in the first row represents the theoretical maximum bound $\sqrt{m_2}$, and the dashed line in the second row represents the initial energy.
  • ...and 4 more figures

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Existence, uniqueness and maximum principle
  • proof
  • Theorem 2.4: Energy dissipation law
  • proof
  • Theorem 3.1: Discrete MBP
  • proof
  • ...and 15 more