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A LINEAR, UNCONDITIONALLY STABLE, SECOND ORDER DECOUPLED METHOD FOR THE NEMATIC LIQUID CRYSTAL FLOWS WITH SAV APPROACH

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arXiv 2025年

作者： Cao, Ruonan Yi, Nianyu School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China

In this paper, we present a second order, linear, fully decoupled, and unconditionally energy stable scheme for solving the Erickson-Leslie model. This approach integrates the pressure correction method with a scalar auxiliary variable technique. We rigorously demonstrate the unconditional energy stability of the proposed scheme. Furthermore, we present several numerical experiments to validate its convergence order, stability, and computational efficiency. © 2025, CC BY.

关键词： computational efficiency

FETD Study of theWave Propagation in Chiral Metamaterials

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Advances in Applied Mathematics and Mechanics 2021年第1期13卷 191-202页

作者： Lei Yang Fuhao Liu Wei Yang Faculty of Information Technology Macao University of Science and TechnologyMacaoChina School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Hunan Key Laboratory for Computation and Simulation in Science and Engineering and School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China

In this paper,we propose a finite element time-domain(FETD)method for the Maxwell’s equations in chiral metamaterials(CMMs).The time-domain model equations are constructed by the auxiliary differential equations(ADEs)*** source excitation method entitled total-field and scattered-field(TF/SF)decomposition technique is applied to FETD method for the first time in simulating the propagation of electromagnetic wave in CMMs,based on which a unified ADE-FETD-UPMLTF/SF scheme is proposed to simulate the wave in *** following properties of CMMs can be observed successfully from the numerical experiments based on our method,i.e.,the ability of the polarization rotation,and the negative phase *** amplitude of reflected wave can effectively be controlled by the physical parameters of CMMs.

关键词： Maxwell’s equations finite element time-domain(FETD) chiral metamaterials(CMMs).

High-accurate and efficient numerical algorithms for the self-consistent field theory of liquid-crystalline polymers

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arXiv 2024年

作者： He, Zhijuan Jiang, Kai Tan, Liwei Wang, Xin Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China

Self-consistent field theory (SCFT) is one of the most widely-used framework in studying the equilibrium phase behaviors of inhomogenous polymers. For liquid crystalline polymeric systems, the main numerical challenges of solving SCFT encompass efficiently solving plenty of six dimensional partial differential equations (PDEs), precisely determining the subtle energy difference among self-assembled structures, and developing effective iterative methods for nonlinear SCF iteration. To address these challenges, this work introduces a suite of high-order and efficient numerical methods tailored for SCFT of liquid-crystalline polymers. These methods include various advaced PDE solvers, an improved Anderson iteration algorithm to accelerate SCFT calculations, and an optimization technique of adjusting the computational domain during the SCF iterations. Extensive numerical tests demonstrate the efficiency of the proposed methods. Based on these algorithms, we further explore the self-assembly behavior of liquid crystalline polymers through simulations in four, five, and six dimensions, uncovering intricate three-dimensional spatial structures. Copyright © 2024, The Authors. All rights reserved.

关键词： Crystalline materials

High-Accurate and Eﬀicient Numerical Algorithms for the Self-Consistent Field Theory of Liquid-Crystalline Polymers

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SSRN

SSRN 2024年

作者： Tan, Liwei He, Zhijuan Wang, Xin Jiang, Kai Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences Shanghai Jiao Tong University Shanghai200240 China

In this paper, we develop and investigate numerical methods for the self-consistent field theory (SCFT) of liquid crystalline polymers. Both the Flory-Huggins interaction potential and the Maier-Saupe orientational interaction are considered, enabling simultaneous exploration of microphase separation and liquid crystalline order in these systems. The main challenge in numerically solving this complex system lies in solving 6-dimensional (3D spatial + 2D orientation + 1D time) partial differential equations and performing nonlinear iterations to optimize the fields. We present ten numerical algorithms for solving high-dimensional PDEs and give an evaluation criterion of choosing the most appropriate PDE solver both from theoretical and numerical performance. Results demonstrate that coupling the fourth-order Runge-Kutta scheme with Fourier pseudo-spectral methods and spherical harmonic transformations is superior compared to other algorithms. We develop two nonlinear iteration schemes, alternating direction iteration method and Anderson mixing method, and design an adaptive technique to enhance the stability of the Anderson mixing method. Moreover, the cascadic multi-level (CML) technique further accelerates the SCFT computation when applied to these iteration methods. Meanwhile, an approach to optimize the computational domain is developed. To demonstrate the power of our approaches, we investigate the self-assembled behaviors of flexible-semiflexible diblock copolymers in 4, 5, 6-dimensional simulations. Numerical results illustrate the eﬀiciency of our proposed method. © 2024, The Authors. All rights reserved.

关键词： Mixing

OPTIMAL L∞(L2) AND L1(L2) A POSTERIORI ERROR ESTIMATES FOR THE FULLY DISCRETE APPROXIMATIONS OF TIME FRACTIONAL PARABOLIC DIFFERENTIAL EQUATIONS

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arXiv 2023年

作者： Cao, Jiliang Wang, Wansheng Xiao, Aiguo Department of Mathematics Shanghai Normal University Shanghai200234 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China

We derive optimal order a posteriori error estimates in the L∞(L2) and L1(L2)-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the L1 methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results. Copyright © 2023, The Authors. All rights reserved.

关键词： Finite element method

IMPLICIT-EXPLICIT RUNGE-KUTTA-ROSENBROCK METHODS WITH ERROR ANALYSIS FOR NONLINEAR STIFF DIFFERENTIAL EQUATIONS

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Journal of computational Mathematics 2021年第4期39卷 599-620页

作者： Bin Huang Aiguo Xiao Gengen Zhang School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China

Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear sti ordinary di erential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta ***,the general order conditions up to order 3 are ***,for the nonlinear sti initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems,the corresponding errors of the implicit-explicit methods are *** last,some numerical examples are given to verify the validity of the obtained theoretical results and the e ectiveness of the methods.

关键词： Sti di erential equations Implicit-explicit Runge-Kutta-Rosenbrock method Order conditions Convergence

A POSTERIORI ERROR ESTIMATORS FOR FOURTH ORDER ELLIPTIC PROBLEMS WITH CONCENTRATED LOADS

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arXiv 2024年

作者： Cao, Huihui Huang, Yunqing Yi, Nianyu Yin, Peimeng Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Department of Mathematical Sciences University of Texas at El Paso El PasoTX79968 United States

In this paper, we study two residual-based a posteriori error estimators for the C0 interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from the model equation without any post-processing technique. We rigorously prove the efficiency and reliability of the estimator by constructing bubble functions. Additionally, we extend this type of estimator to general fourth-order elliptic equations with various boundary conditions. The second estimator is based on projecting the Dirac delta function onto the discrete finite element space, allowing the application of a standard estimator. Notably, we additionally incorporate the projection error into the standard estimator. The efficiency and reliability of the estimator are also verified through rigorous analysis. We validate the performance of these a posteriori estimates within an adaptive algorithm and demonstrate their robustness and expected accuracy through extensive numerical examples. © 2024, CC BY-NC-ND.

关键词： Delta functions

Existence of Solutions for the Continuous Algebraic Riccati Equation via Polynomial Optimization

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arXiv 2024年

作者： Zhang, Juan Zhao, Wenjie Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Hunan First Normal University Hunan Changsha410205 China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper studies the solution existence of the continuous-time algebraic Riccati equation (CARE). We formulate the CARE as two constrained polynomial optimization problems, and then use Lasserre's hierarchy of semi-definite relaxations to solve them. Compared to the existing work, our approaches can obtain the exact positive semi-definite solution of the CARE when the coefficient matrices are symmetric. Moreover, our methods can detect the nonexistence of the positive semi-definite solution for the CARE. Numerical examples show that our approaches are effective compared to the existing methods. Copyright © 2024, The Authors. All rights reserved.

关键词： Riccati equations

NEURAL NETWORK-ENHANCED hr-ADAPTIVE FINITE ELEMENT ALGORITHM FOR PARABOLIC EQUATIONS

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arXiv 2025年

作者： Hao, Jiaxiong Huang, Yunqing Yi, Nianyu Yin, Peimeng Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China National Center for Applied Mathematics in Hunan Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China Department of Mathematical Sciences University of Texas at El Paso El PasoTX79968 United States

In this paper, we present a novel enhancement to the conventional hr-adaptive finite element methods for parabolic equations, integrating traditional h-adaptive and r-adaptive methods via neural networks. A major challenge in hr-adaptive finite element methods lies in projecting the previous step’s finite element solution onto the updated mesh. This projection depends on the new mesh and must be recomputed for each adaptive iteration. To address this, we introduce a neural network to construct a mesh-free surrogate of the previous step finite element solution. Since the neural network is mesh-free, it only requires training once per time step, with its parameters initialized using the optimizer from the previous time step. This approach effectively overcomes the interpolation challenges associated with non-nested meshes in computation, making node insertion and movement more convenient and efficient. The new algorithm also emphasizes SIZING and GENERATE, allowing each refinement to roughly double the number of mesh nodes of the previous iteration and then redistribute them to form a new mesh that effectively captures the singularities. It significantly reduces the time required for repeated refinement and achieves the desired accuracy in no more than seven space-adaptive iterations per time step. Numerical experiments confirm the efficiency of the proposed algorithm in capturing dynamic changes of *** Codes 92B20, 65M60, 35K57 © 2025, CC BY.

关键词： Interpolation