Relaxed Generalized Scalar Auxiliary Variable Exponential Integrator for A Modified Landau-de Gennes Theory for Smectic Liquid Crystals

TL;DR

This paper introduces a novel, unconditionally energy-stable numerical scheme for a modified Landau-de Gennes model of smectic liquid crystals, combining GSAV-EI with a relaxed correction strategy for improved efficiency and accuracy.

Contribution

It develops a highly efficient, unconditionally energy-stable numerical method that eliminates CFL restrictions and provides rigorous error analysis for the coupled tensor-scalar system.

Findings

The scheme is unconditionally energy-stable and preserves the discrete energy.

Numerical experiments confirm the method's accuracy and efficiency.

The approach effectively captures complex defect dynamics in smectic liquid crystals.

Abstract

The Smectic-A (SmA) phase is modeled by a modified Landau-de Gennes (mLdG) model proposed by Xia et al. [Phys. Rev. Lett., 126 (2021), 177801], in which a tensor order parameter Q for the orientational order is coupled with a real scalar $u$ characterizing the positional order. In this paper, we propose and analyze a novel, highly efficient, and unconditionally energy-stable numerical scheme for this coupled system by combining the generalized scalar auxiliary variable-exponential integrator (GSAV-EI) approach with a relaxed correction strategy. In particular, we reformulate the exponential time differencing time discretization into an equivalent quasi-implicit backward Euler-type structure, a pivotal step that eliminates the restrictive CFL mesh-ratio conditions of the original GSAV-EI method and enables a rigorous fully discrete error analysis. Theoretically, we rigorously establish the unconditional energy stability with respect to a modified discrete energy and the uniform boundedness of the numerical solutions Q, along with optimal error estimates in both time and space. Comprehensive numerical experiments are presented to demonstrate the accuracy, efficiency, and structural preservation of the algorithm, as well as its capability in capturing complex topological defect dynamics.