arXiv subjects

Wenshuai Hu

Publications and source records attributed to Wenshuai Hu.

8 recordsLinked to original sources

arXiv2026

Energy stability and error estimates for a second-order structure-preserving exponential integrator method for smectic-A liquid crystals

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 $\mathcal{O}(τ^2+h^2)$. Numerical experiments are presented to verify our theoretical results and to simulate the self-assembly dynamics of the SmA phase.

math.NA↗

arXiv2026

A Second-Order Maximum-Bound-Preserving and Energy-Stable Exponential Time-Differencing Method for Allen--Cahn-Type Gradient Flows

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen--Cahn equation. In this paper, we develop and analyze novel second-order linear numerical schemes for a class of Allen--Cahn type gradient flows. Our scheme is based on the generalized scalar auxiliary variable (GSAV) approach and a novel second-order exponential time-differencing Runge--Kutta (ETDRK2) method. The resulting formulation overcomes a longstanding difficulty in combining these two techniques while retaining both the MBP and energy stability. We prove that the proposed scheme unconditionally preserves both the MBP and energy stability. In addition, rigorous error analysis is carried out for the proposed scheme, establishing second-order accuracy in both time and space without imposing any coupling condition between the time step $τ$ and the spatial mesh size $h$. We also present some numerical experiments to demonstrate the efficiency of the proposed scheme and its preservation of the theoretical properties.

math.NA↗

arXiv2026

Maximum bound principle for Q-tensor gradient flow with low regularity integrators

The Landau-de Gennes (LdG) theory is a widely used thermodynamic continuum framework for modeling the behavior of ordered states and defects in liquid crystals with a tensor-order parameter $Q$. In this study, we develop and analyze first- and second-order low-regularity integrator (LRI) schemes for the $Q$-tensor gradient flow and prove the maximum bound principle. In particular, through the reformulation of the LRI schemes, we establish rigorous modified energy dissipation laws for the LRI1a and LRI1b schemes, thereby filling a significant theoretical gap in the existing literature on LRI methods. Moreover, this reformulation establishes a structural bridge between the LRI schemes and backward differentiation formula (BDF) methods, which opens up new possibilities for the construction and analysis of LRI-type methods. We then establish first- and second-order temporal convergence under $H^1$ and $H^2$ regularity assumptions, respectively. Several numerical experiments are presented to validate our theoretical results and to simulate the evolution of defect dynamics.

math.NA↗

arXiv2026

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

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 ($τ\lesssim h^2$) 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.

math.NA↗

arXiv2026

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

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.

math.NA↗

arXiv2026

Identification and Structural Characterization of Twisted Atomically Thin Bilayer Materials by Deep Learning

Two-dimensional materials are expected to play an important role in next-generation electronics and optoelectronic devices. Recently, twisted bilayer graphene and transition metal dichalcogenides have attracted significant attention due to their unique physical properties and potential applications. In this study we describe the use of optical microscopy to collect the color space of chemical vapor deposition (CVD) molybdenum disulfide ($\mbox{MoS}_2$), and the application of a semantic segmentation convolutional neural network (CNN) to accurately and rapidly identify thicknesses of $\mbox{MoS}_2$ flakes. A second CNN model is trained to provide precise predictions on the twist angle of CVD-grown bilayer flakes. This model harnessed a dataset comprising over 10,000 synthetic images, encompassing geometries spanning from hexagonal to triangular shapes. Subsequent validation of the deep learning predictions on twist angles was executed through the second harmonic generation and Raman spectroscopy. Our results introduce a scalable methodology for automated inspection of twisted atomically thin CVD-grown bilayer.

cond-mat.mtrl-sci↗

arXiv2025

On the maximum bound principle and energy dissipation of exponential time differencing methods for the chiral liquid crystal blue phases

The blue phases are fascinating and complex states of chiral liquid crystals which can be modeled by a comprehensive framework of the Landau-de theory, satisfying energy dissipation and maximum bound principle. In this paper, we develop and analyze first and second order exponential time differencing numerical schemes for the gradient flow of the chiral liquid crystal blue phases, which preserve the maximum bound principle and energy dissipation unconditionally at the semi-discrete level. The fully discrete schemes are obtained coupled with the Fourier spectral method in space. And we propose a novel matrix-form Helmholtz basis transformation method to diagonalize the combined operator of the Laplacian and the curl operator, which is a key step in the implementation of the proposed schemes. Then by constructing auxiliary functions, we drive the $L^\infty$ boundedness of the numerical solutions and obtain the energy dissipation and the error estimates in $L^2$ and $L^\infty$ norm. Various numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed methods in simulating the dynamics of blue phases in chiral liquid crystals.

math.NA↗

arXiv2023

A new mixed finite element method for arbitrary element pair for a quasi-static nonlinear permeability thermo-poroelasticity model

In this paper, we develop a multiphysics finite element method for solving the quasi-static thermo-poroelasticity model with nonlinear permeability. The model involves multiple physical processes such as deformation, pressure, diffusion and heat transfer. To reveal the multi-physical processes of deformation, diffusion and heat transfer, we reformulate the original model into a fluid coupled problem that is general Stokes equation coupled with two reaction-diffusion equations. Then, we prove the existence and uniqueness of weak solution for the original problem by the $B$-operator technique and by sequence approximation for the reformulated problem. As for the reformulated problem we propose a fully discrete finite element method which can use arbitrary finite element pairs to solve the displacement $\bu$ pressure $τ$ and variable $\varpi,ς$, and the backward Euler method for time discretization. Finally, we give the stability analysis of the above proposed method, also we prove that the fully discrete multiphysics finite element method has an optimal convergence order. Numerical experiments show that the proposed method can achieve good results under different finite element pairs and are consistent with the theoretical analysis.

math.NA↗