Maximum bound principle preserving and energy decreasing exponential time differencing schemes 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. 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.

Figures (23)

• Figure 4.1: Evolution of the supremum norm $\|\cdot\|_{\mathcal{X}}$ and energy computed by the ETD1 scheme in Example 1. The dashed line in the left figure is the maximum bound $\sqrt m$ while the dashed line in the right figure is the initial energy.
• Figure 4.2: Evolution of the supremum norm $\|\cdot\|_{\mathcal{X}}$ and energy computed by the ETDRK2 scheme in Example 1. The dashed line in the left figure is the maximum bound $\sqrt m$ while the dashed line in the right figure is the initial energy.
• Figure 4.3: Evolution of the matrix-valued field and interface at $t=0,50,100,150,200,500$. The initial field is given in \ref{['eq:4.2']} with $\alpha(x,y)=0$.
• Figure 4.4: Evolution of the supremum norm $\|\cdot\|_{\mathcal{X}}$ and energy with initial condition \ref{['eq:4.2']} and $\alpha(x,y)=0$. The dashed line in the left figure is the maximum bound $\sqrt m$ while the dashed line in the right figure is the initial energy.
• Figure 4.5: Evolution of the matrix-valued field and interface at $t=0,50,100,150,200,500$. The initial field is given in \ref{['eq:4.2']} with $\alpha(x,y)=\frac{\pi}{2}\sin(2\pi(x+y))$.

Theorems & Definitions (21)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 2.5: Maximum bound principle, existence, and uniqueness of solution to \ref{['eq:mac_ka']}
• proof
• Theorem 3.1: MBP of ETD1 scheme