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On the maximum bound principle and energy dissipation of exponential time differencing methods for the chiral liquid crystal blue phases

Wenshuai Hu, Guanghua Ji

TL;DR

The paper tackles robust numerical simulation of chiral blue phases modeled by the Landau–de Gennes $Q$-tensor gradient flow, requiring energy dissipation and a maximum bound principle (MBP). It develops first- and second-order exponential time differencing (ETD) Runge–Kutta schemes, coupled with Fourier spectral spatial discretization, and introduces a Helmholtz Symmetric Trace-Free (HSTF) basis to diagonalize the Laplacian and curl operators for efficient implementation. The authors prove unconditional MBP preservation and energy dissipation at the semi-discrete level, provide $L^2$ and $L^{\infty}$ error estimates, and establish fully discrete stability and convergence, supported by extensive numerical experiments showing MBP retention, energy decay, and realistic blue-phase dynamics. The work delivers a computationally efficient, rigorously stable framework for simulating complex blue-phase morphologies, including BP I and BP III, with practical implications for understanding chiral liquid crystal behavior and guiding experiments.

Abstract

The blue phases are fascinating and complex states of chiral liquid crystals which can be modeled by a comprehensive framework of the Landau-de theory, satisfying energy dissipation and maximum bound principle. In this paper, we develop and analyze first and second order exponential time differencing numerical schemes for the gradient flow of the chiral liquid crystal blue phases, which preserve the maximum bound principle and energy dissipation unconditionally at the semi-discrete level. The fully discrete schemes are obtained coupled with the Fourier spectral method in space. And we propose a novel matrix-form Helmholtz basis transformation method to diagonalize the combined operator of the Laplacian and the curl operator, which is a key step in the implementation of the proposed schemes. Then by constructing auxiliary functions, we drive the $L^\infty$ boundedness of the numerical solutions and obtain the energy dissipation and the error estimates in $L^2$ and $L^\infty$ norm. Various numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed methods in simulating the dynamics of blue phases in chiral liquid crystals.

On the maximum bound principle and energy dissipation of exponential time differencing methods for the chiral liquid crystal blue phases

TL;DR

The paper tackles robust numerical simulation of chiral blue phases modeled by the Landau–de Gennes -tensor gradient flow, requiring energy dissipation and a maximum bound principle (MBP). It develops first- and second-order exponential time differencing (ETD) Runge–Kutta schemes, coupled with Fourier spectral spatial discretization, and introduces a Helmholtz Symmetric Trace-Free (HSTF) basis to diagonalize the Laplacian and curl operators for efficient implementation. The authors prove unconditional MBP preservation and energy dissipation at the semi-discrete level, provide and error estimates, and establish fully discrete stability and convergence, supported by extensive numerical experiments showing MBP retention, energy decay, and realistic blue-phase dynamics. The work delivers a computationally efficient, rigorously stable framework for simulating complex blue-phase morphologies, including BP I and BP III, with practical implications for understanding chiral liquid crystal behavior and guiding experiments.

Abstract

The blue phases are fascinating and complex states of chiral liquid crystals which can be modeled by a comprehensive framework of the Landau-de theory, satisfying energy dissipation and maximum bound principle. In this paper, we develop and analyze first and second order exponential time differencing numerical schemes for the gradient flow of the chiral liquid crystal blue phases, which preserve the maximum bound principle and energy dissipation unconditionally at the semi-discrete level. The fully discrete schemes are obtained coupled with the Fourier spectral method in space. And we propose a novel matrix-form Helmholtz basis transformation method to diagonalize the combined operator of the Laplacian and the curl operator, which is a key step in the implementation of the proposed schemes. Then by constructing auxiliary functions, we drive the boundedness of the numerical solutions and obtain the energy dissipation and the error estimates in and norm. Various numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed methods in simulating the dynamics of blue phases in chiral liquid crystals.
Paper Structure (21 sections, 13 theorems, 179 equations, 7 figures, 1 table)

This paper contains 21 sections, 13 theorems, 179 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Exponential time differencing Runge-Kutta schemes
  3. Maximum bound principle and unconditional energy dissipation
  4. Continuous maximum bound principle
  5. Discrete maximum bound principle
  6. Discrete energy dissipation
  7. Fully discrete exponential time differencing schemes
  8. Spatial discretization
  9. Discrete Laplace and curl operators
  10. Efficient implementations of the ETD schemes
  11. Construction of the Helmholtz Symmetric Trace-Free Basis
  12. Diagonalization of the discrete $\mathcal{L}_N$ operator
  13. Dissipation of the discrete linear operator $\mathcal{L}_{h,\kappa_2}$
  14. Efficient Implementation of the ETD Schemes
  15. Fully discrete energy dissipation and error analysis
  16. ...and 6 more sections

Key Result

Lemma 3.1

\newlabel3.1 Let the operator $\mathcal{L}_{\kappa_2 }$ be defined in Lkappa. For $W \in \mathcal{Z}$, when $L_1 \geq 0, \kappa_2 \geq \frac{ L_4^2}{2 L_1}$, there exists a positive constant $\lambda$ such that Then the linear operator $\mathcal{L}_{\kappa_2 }$ generates a contraction semigroup $\left\{e^{t \mathcal{L}_{\kappa_2 }}\right\}_{t \geq 0},$ and for $t \geq 0$, it holds that

Figures (7)

  • Figure 6.1: Evolutions of the (a) supremum norm $\|Q_h^m\|_\mathcal{Z}$, and (b) energies computed by the ETD1 scheme with blue lines and ETDRK2 scheme with red lines. The dashed lines in (a) represent the bounds of the MBP, the dashed lines in (b) indicate the total energy conservation.
  • Figure 6.2: Evolutions of the (a) maximum eigenvalue of $Q_h^m$ and (b) maximum eigenvalue of $Q_h^m$ computed by the ETD1 scheme with blue lines and ETDRK2 scheme with red lines. The dashed lines in (a,b) represent the bounds of the eigenvalues, which lies in $(-\frac{1}{3},\frac{2}{3})$.
  • Figure 6.3: The evolution of the scalar order parameter $s$ (a1-a4), the biaxiality $\beta_b$ (b1-b4), and the static structure factor $S(\boldsymbol{k})$ (c1-c4) is depicted at simulation times $T=0, 5, 20,$ and $50$.
  • Figure 6.4: (a) displays the isosurface plot of the order parameter $Q_h^n$ with $s=0.3*s_{max}$ at $T=50$ where $s_{max}$ is the maximum value of $s$. (b) displays disclination line network of BP III of the isosurface $s=0.12$ at $\tau_c=-0.25, \kappa=2.5$ from henrich2011structure.
  • Figure 6.5: (a-b) show the static structure factor $S(\mathbf{k})$ plotted on a logarithmic scale onto the $y=0$ plane and $x=0$ plane at $T=50$. (c-d) display the corresponding structure factor of BP III on cuts along $y=0$ and $x=0$ from henrich2011structure.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3: Existence, uniqueness and MBP of the continuous equation \ref{['1.8']}
  • proof
  • Theorem 3.4: MBP of ETD schemes
  • proof
  • Remark 3.1
  • Lemma 3.5
  • ...and 18 more