A Structure-Preserving GSAV Exponential Integrator for Smectic-A Liquid Crystals

Abstract

The modified Landau--de Gennes (mLdG) theory provides a powerful continuum framework for modeling smectic-A (SmA) liquid crystals by coupling the tensorial orientational order parameter $\mathbf{Q}$ with the scalar positional order parameter $u$. In this paper, we develop and analyze a structure-preserving generalized scalar auxiliary variable exponential integrator (GSAV-EI) scheme for the fully coupled mLdG system. The key ingredient is a backward Euler-type reformulation of the exponential integrator, which reveals a coercive discrete structure suitable for energy estimates and error analysis. Moreover, it eliminates the mesh-dependent time-step restriction ($\tau \lesssim h^2$) required in the existing GSAV-EI error analysis and provides a transparent connection between exponential time-differencing and implicit time-stepping methods. Building on this reconstructed structure, we prove unconditional modified-energy stability and establish optimal-order fully discrete error estimates. Numerical experiments in two and three dimensions validate the theoretical convergence rates, verify the discrete energy-dissipation law, and illustrate the self-assembly dynamics of the SmA phase.

31 pages, 9 figures