Energy stability and error estimates for a second-order structure-preserving exponential integrator method for smectic-A liquid crystals
Abstract
In this work, we develop a second-order, linear, decoupled, and structure-preserving numerical scheme for the modified Landau--de Gennes model of smectic-A (SmA) liquid crystals. The main contributions are threefold. First, to the best of our knowledge, we propose the first integration of the generalized scalar auxiliary variable (GSAV) approach with a second-order exponential time-differencing Runge--Kutta (ETDRK2) discretization, leading to a second-order GSAV--ETD2 scheme. Second, we prove that the proposed scheme satisfies an unconditional energy-dissipation law, thereby closing the theoretical gap in the energy-stability analysis of second-order GSAV exponential integrators of this class. Third, by deriving a coercive discrete reformulation, we establish a fully discrete error estimate without imposing any coupling condition between τ and h, achieving the optimal convergence rate 𝒪(τ2+h2). Numerical experiments are presented to verify our theoretical results and to simulate the self-assembly dynamics of the SmA phase.
Related
- Data-Driven Free-Energy Learning from Trajectories via Structure-Preserving Exponential Time Differencing
- Structure-preserving full- and low-rank exponential methods for the optimal control of Lindblad equations
- Linearize or Discretize First? Error Estimates under Mesh-Uniform Regularity and Exponential Stability for the KdV Equation with Boundary Dissipation
- AREX: Affine-Residual Exponential Integrator for Few-Step Sampling in Flow Matching
- Strong error analysis of a stochastic exponential integrator for SPDEs driven by fractional Brownian motion with H<1/2
BibTeX
@misc{hu2026energy,
title = {{Energy stability and error estimates for a second-order structure-preserving exponential integrator method for smectic-A liquid crystals}},
author = {Wenshuai Hu and Guanghua Ji and Xiao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.21175},
eprint = {2609.21175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21175},
}
RIS
TY - UNPB
TI - Energy stability and error estimates for a second-order structure-preserving exponential integrator method for smectic-A liquid crystals
AU - Wenshuai Hu
AU - Guanghua Ji
AU - Xiao Li
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.21175
UR - https://arxiv.org/abs/2609.21175
AB - In this work, we develop a second-order, linear, decoupled, and structure-preserving numerical scheme for the modified Landau--de Gennes model of smectic-A (SmA) liquid crystals. The main contributions are threefold. First, to the best of our knowledge, we propose the first integration of the generalized scalar auxiliary variable (GSAV) approach with a second-order exponential time-differencing Runge--Kutta (ETDRK2) discretization, leading to a second-order GSAV--ETD2 scheme. Second, we prove that the proposed scheme satisfies an unconditional energy-dissipation law, thereby closing the theoretical gap in the energy-stability analysis of second-order GSAV exponential integrators of this class. Third, by deriving a coercive discrete reformulation, we establish a fully discrete error estimate without imposing any coupling condition between τ and h, achieving the optimal convergence rate 𝒪(τ^2+h^2). Numerical experiments are presented to verify our theoretical results and to simulate the self-assembly dynamics of the SmA phase.
ER -