arXiv subjects

Guanghua Ji

Publications and source records attributed to Guanghua Ji.

13 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

Mass-Conserving Saddle Dynamics via Generalized Inner Product: Theory, Algorithms, and Applications

To reveal the effect of the inner product choice, we present a unified formulation of saddle dynamics for the functional F with a mass constraint under different inner products. We establish the equivalence between the index-k saddle points and the linearly stable steady states of the corresponding dynamics. Further, we present the dynamics with discrete H^{-1} and L^2 inner products and numerically verify the convergence orders of both dynamics. Finally, we apply the method to a phase field model with driving force under Neumann and periodic boundary conditions. The results uncover previously unreported saddle points and their connectivity, highlighting how the choice of inner product enriches the solution landscape in conservative systems.

math.NA↗

arXiv2026

Efficient Numerical Schemes for a Two-Phase Hydrodynamical Model of Active Liquid Crystals and Solids

We propose several linear, fully decoupled numerical schemes with first- and second-order temporal accuracy for a novel Q-tensor-based two-phase hydrodynamic model describing the coupling of active nematic liquid crystal solutions with isotropic solid substrates. The model is derived from the generalized Onsager principle and includes nontrivial terms that contribute zero to the total free-energy dissipation. We prove that the proposed decoupled linear schemes are thermodynamically consistent at the discrete level. In the passive limit, the SGE-BDF1 and SGE-PDG schemes are unconditionally energy stable, while the SGE-BDF2 scheme is energy stable with respect to a modified energy under a standard boundedness assumption and a sufficiently large stabilization parameter. We perform extensive numerical simulations to investigate how activity and other model parameters affect active nematic fluid-solid interactions. Finally, we analyze the physical mechanisms underlying the observed behaviors, providing deeper insight into the dynamics of soft confined active nematic fluids.

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↗

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↗

arXiv2025

Efficient numerical methods for computing stationary states of spherical Landau-Brazovskii model

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time

math.NA↗

arXiv2024

Error estimate for the first order energy stable scheme of Q-tensor nematic model

We present rigorous error estimates towards a first-order unconditionally energy stable scheme designed for 3D hydrodynamic Q-tensor model of nematic liquid crystals. This scheme combines the scalar auxiliary variable (SAV), stabilization and projection method together. The unique solvability and energy dissipation of the scheme are proved. We further derive the boundness of numerical solution in L^{\infty} norm with mathematical deduction. Then, we can give the rigorous error estimate of order O(δt) in the sense of L2 norm, where δt is the time step.Finally, we give some numerical simulations to demonstrate the theoretical analysis.

math.NA↗

arXiv2024

Vanilla Feedforward Neural Networks as a Discretization of Dynamical Systems

Deep learning has made significant applications in the field of data science and natural science. Some studies have linked deep neural networks to dynamic systems, but the network structure is restricted to the residual network. It is known that residual networks can be regarded as a numerical discretization of dynamic systems. In this paper, we back to the classical network structure and prove that the vanilla feedforward networks could also be a numerical discretization of dynamic systems, where the width of the network is equal to the dimension of the input and output. Our proof is based on the properties of the leaky-ReLU function and the numerical technique of splitting method to solve differential equations. Our results could provide a new perspective for understanding the approximation properties of feedforward neural networks.

cs.LG↗

arXiv2024

Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation

The study of universal approximation properties (UAP) for neural networks (NN) has a long history. When the network width is unlimited, only a single hidden layer is sufficient for UAP. In contrast, when the depth is unlimited, the width for UAP needs to be not less than the critical width $w^*_{\min}=\max(d_x,d_y)$, where $d_x$ and $d_y$ are the dimensions of the input and output, respectively. Recently, \cite{cai2022achieve} shows that a leaky-ReLU NN with this critical width can achieve UAP for $L^p$ functions on a compact domain ${K}$, \emph{i.e.,} the UAP for $L^p({K},\mathbb{R}^{d_y})$. This paper examines a uniform UAP for the function class $C({K},\mathbb{R}^{d_y})$ and gives the exact minimum width of the leaky-ReLU NN as $w_{\min}=\max(d_x,d_y)+Δ(d_x, d_y)$, where $Δ(d_x, d_y)$ is the additional dimensions for approximating continuous functions with diffeomorphisms via embedding. To obtain this result, we propose a novel lift-flow-discretization approach that shows that the uniform UAP has a deep connection with topological theory.

cs.LG↗

arXiv2009

A Prime Number Theorem for Rankin-Selberg L-functions over Number fields

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,π\timesπ')$ over any Galois extension with $π$ and $π'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at least one of the representations subject to a self-contragredient assumption. We then extend these results to two representations $π$ defined on $GL_n/E$ and $π'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic number fields of coprime degree where $π$ and $π'$ admit a base change lift from $\mathbb{Q}$ again given a self-contragredient assumption on at least one of the representations which lift to $π$ or $π'$.

math.NT↗