Relaxed Generalized Scalar Auxiliary Variable Exponential Integrator for A Modified Landau-de Gennes Theory for Smectic Liquid Crystals
For researchers in computational physics and liquid crystal modeling, this work provides a more efficient and stable numerical method for simulating the Smectic-A phase, though it is an incremental improvement over existing GSAV-EI methods.
The paper proposes a novel numerical scheme for simulating the Smectic-A phase in liquid crystals, achieving unconditional energy stability and optimal error estimates while eliminating restrictive CFL conditions. Numerical experiments demonstrate accuracy, efficiency, and ability to capture complex defect dynamics.
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.