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Maximum bound principle for Q-tensor gradient flow with low regularity integrators

Wenshuai Hu, Guanghua Ji

TL;DR

This work develops four low-regularity integrator schemes for the Q-tensor gradient flow arising from the Landau-de Gennes model of nematic liquid crystals. By deriving the schemes from the Duhamel formula, the authors establish MBP preservation and energy stability under minimal regularity, along with rigorous temporal error estimates showing first- or second-order convergence depending on the scheme. Theoretical results are complemented by extensive 2D and 3D numerical experiments, demonstrating that eigenvalues of the Q-tensor stay within the physical range and that the energy dissipates toward equilibrium, with observed convergence rates aligning with theory. The proposed LRIs offer robust, efficient tools for simulating phase transitions, biaxiality, and temperature-driven dynamics in nematic LC systems, even when solutions lack high regularity.

Abstract

We investigate low-regularity integrator (LRI) methods for the Q-tensor model governing nematic liquid-crystalline semilinear parabolic equation. First- and second-order temporal discretizations are developed using Duhamel's formula, and we rigorously prove that both schemes preserve the maximum bound principle (MBP) and energy dissipation under minimal regularity requirements. Optimal convergence rates are established for the proposed methods. Numerical experiments validate the theoretical findings, demonstrating that the eigenvalues of Q remain strictly confined within the physical range (-1/3},2/3).

Maximum bound principle for Q-tensor gradient flow with low regularity integrators

TL;DR

This work develops four low-regularity integrator schemes for the Q-tensor gradient flow arising from the Landau-de Gennes model of nematic liquid crystals. By deriving the schemes from the Duhamel formula, the authors establish MBP preservation and energy stability under minimal regularity, along with rigorous temporal error estimates showing first- or second-order convergence depending on the scheme. Theoretical results are complemented by extensive 2D and 3D numerical experiments, demonstrating that eigenvalues of the Q-tensor stay within the physical range and that the energy dissipates toward equilibrium, with observed convergence rates aligning with theory. The proposed LRIs offer robust, efficient tools for simulating phase transitions, biaxiality, and temperature-driven dynamics in nematic LC systems, even when solutions lack high regularity.

Abstract

We investigate low-regularity integrator (LRI) methods for the Q-tensor model governing nematic liquid-crystalline semilinear parabolic equation. First- and second-order temporal discretizations are developed using Duhamel's formula, and we rigorously prove that both schemes preserve the maximum bound principle (MBP) and energy dissipation under minimal regularity requirements. Optimal convergence rates are established for the proposed methods. Numerical experiments validate the theoretical findings, demonstrating that the eigenvalues of Q remain strictly confined within the physical range (-1/3},2/3).

Paper Structure

This paper contains 10 sections, 10 theorems, 93 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Numerical schemes
  3. Properties of the semi-discrete numerical solutions
  4. Discrete maximum bound principle
  5. Semi-discrete energy stability
  6. Temporal error estimate
  7. Numerical experiments
  8. Two-dimensional tests
  9. Three-dimensional tests
  10. Conclusion

Key Result

Lemma 3.1

\newlabel1.17 For $\lambda > 0$ and all $W \in \mathcal{Z}$, when it holds that the linear operator $\Delta$ generates a contraction semigroup $\left\{e^{t \Delta}\right\}_{t \geq 0}.$ Similarly, for $\lambda > 0$ and all $W \in \mathcal{X}$, when it holds that the linear operator $\Delta$ generates a contraction semigroup $\left\{e^{t \Delta}\right\}_{t \geq 0}.$

Figures (13)

  • Figure 5.1: Evolutions of the 2-norm and F-norm of the solutions of LRI1a and LRI1b scheme
  • Figure 5.2: Evolutions of the 2-norm and F-norm of the solutions of LRI2a and LRI2b scheme
  • Figure 5.3: Evolutions of the energies of the solutions of LRI schemes
  • Figure 5.4: Evolutions of the principal eigenvectors of $Q$ at t = 0.1, 4.7, 4.9, 5, 5.2 and 50, respectively
  • Figure 5.5: Evolutions of the principal eigenvectors of $Q$ at t = 0.1, 4.7, 4.9, 5, 5.2 and 50, respectively
  • ...and 8 more figures

Theorems & Definitions (23)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.1
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 13 more