Energy stable and maximum bound principle preserving schemes for the Q-tensor flow of liquid crystals

TL;DR

The paper addresses numerical simulation of the $Q$-tensor flow in nematic liquid crystals by developing two linear, fully discrete schemes based on the stabilized exponential scalar auxiliary variable (sESAV) approach. These schemes achieve unconditional energy stability and maximum bound principle preservation in 2D and certain 3D settings, with a rigorous error analysis provided for the second-order method. The authors demonstrate MBP and energy-dissipation through diverse numerical tests, including convergence studies, disappearing holes, and 3D orientation dynamics, aided by efficient DST-based linear solvers. The results offer robust, scalable tools for accurate simulations of liquid crystal orientation in both two and three dimensions and point to extensions to hydrodynamic Q-tensor models coupled to Navier–Stokes.

Abstract

In this paper, we propose two efficient fully-discrete schemes for Q-tensor flow of liquid crystals by using the first- and second-order stabilized exponential scalar auxiliary variable (sESAV) approach in time and the finite difference method for spatial discretization. The modified discrete energy dissipation laws are unconditionally satisfied for both two constructed schemes. A particular feature is that, for two-dimensional (2D) and a kind of three-dimensional (3D) Q-tensor flows, the unconditional maximum-bound-principle (MBP) preservation of the constructed first-order scheme is successfully established, and the proposed second-order scheme preserves the discrete MBP property with a mild restriction on the time-step sizes. Furthermore, we rigorously derive the corresponding error estimates for the fully-discrete second-order schemes by using the built-in stability results. Finally, various numerical examples validating the theoretical results, such as the orientation of liquid crystal in 2D and 3D, are presented for the constructed schemes.

Figures (4)

• Figure 1: Evolutions of disappearing holes at different time $t$. Snapshots are the major director orientation of the liquid crystal in the $xy$ plane taken at $t=0,\ 0.1,\ 0.2,\ 0.4,\ 0.7, \ 2.0$, respectively (top two rows). The difference of eigenvalues on the $xy$ plane for $Q+ \frac{1}{2}I$ at time $t = 0.1,\ 0.2,\ 0.4$, respectively (bottom row).
• Figure 2: Evolutions of supremum norms (a) and energies (b) of simulated solutions. (c) Comparison of the sIMEX scheme with sESAV2 scheme using $\tau=0.5$ and $\kappa=2$.
• Figure 3: Orientation of liquid crystal at different time $t$. Snapshots are the major director orientation of the liquid crystal in the $xy$ plane taken at $t=0,\ 2,\ 10,\ 18,\ 21, \ 30$, respectively (top two rows). The difference of eigenvalues on the $xyz$ plane for $Q+ \frac{1}{3}I$ at time $t = 2,\ 10,\ 18$, respectively (bottom row).
• Figure 4: Evolutions of the supremum norms (left) and the energies (middle) and the adaptive time steps (right).

Theorems & Definitions (29)

• Lemma 2.1
• Proof 1
• Remark 2.1
• Proposition 3.1
• Theorem 3.2
• Proof 2
• Theorem 3.3
• Remark 3.1
• Lemma 3.4: CMPX19, HWZ22, Lemma 3.1 in STY16
• Lemma 3.5