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

TL;DR

This paper introduces two energy-stable, maximum-bound-principle-preserving numerical schemes for Q-tensor flow in liquid crystals, ensuring energy dissipation and stability in 2D and 3D simulations with proven error estimates.

Contribution

It develops and analyzes fully-discrete stabilized exponential scalar auxiliary variable schemes that preserve energy dissipation and maximum principles for liquid crystal Q-tensor flow.

Findings

Unconditionally satisfies discrete energy dissipation laws.

Successfully preserves maximum-bound principles in 2D and some 3D cases.

Provides rigorous error estimates for the second-order scheme.

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.