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arXiv 2025年

作者： Yang, Jun Yi, Nianyu Yin, Peimeng 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 Department of Mathematical Sciences The University of Texas at El Paso El PasoTX79968 United States

In this paper, we propose a novel second-order dynamical low-rank mass-lumped finite element method for solving the Allen-Cahn (AC) equation, a semilinear parabolic partial differential equation. The matrix differential equation of the semi-discrete mass-lumped finite element scheme is decomposed into linear and nonlinear components using the second-order Strang splitting method. The linear component is solved analytically within a low-rank manifold, while the nonlinear component is discretized using a second-order augmented basis update & Galerkin (BUG) integrator, in which the S-step matrix equation is solved by the explicit 2-stage strong stability-preserving Runge-Kutta method. The algorithm has lower computational complexity than the full-rank mass-lump finite element method. The dynamical low-rank finite element solution is shown to conserve mass up to a truncation tolerance for the conservative Allen-Cahn equation. Meanwhile, the modified energy is dissipative up to a high-order error and is hence stable. Numerical experiments validate the theoretical results. Symmetry-preserving tests highlight the robustness of the proposed method for long-time {1.s and demonstrate its superior performance compared to existing *** Codes 35K58, 65F55, 65M60, 65Y20 © 2025, CC BY-NC-ND.

关键词： Runge Kutta methods

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Fractional coupled Halanay inequality and its applications

arXiv

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arXiv 2025年

作者： Van Thinh, La Tuan, Hoang The Wang, Dongling Yang, Yin Academy of Finance Le Van Hien Street Duc Thang Ward Bac Tu Liem District Hanoi Viet Nam Institute of Mathematics Vietnam Academy of Science and Technology 18 Hoang Quoc Viet Hanoi10307 Viet Nam Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper introduces a generalized fractional Halanay-type coupled inequality, which serves as a robust tool for characterizing the asymptotic stability of diverse time fractional functional differential equations, particularly those exhibiting Mittag-Leffler type stability. Our main tool is a sub-additive property of Mittag-Leffler function and its optimal asymptotic decay rate estimation. Our results further optimize and improve some existing results in the literature. We illustrate two significant applications of this fractional Halanay-type inequality. Firstly, by combining our results in this work with the positive representation method positive representation of delay differential systems, we establish an asymptotic stability criterion for a category of linear fractional coupled systems with bounded delays. This criterion extends beyond the traditional boundaries of positive system theory, offering a new perspective on stability analysis in this domain. Secondly, through energy estimation, we establish the contractility and dissipativity of a class of time fractional neutral functional differential equations. Our analysis reveals the typical long-term polynomial decay behavior inherent in time fractional evolutionary equations, thereby providing a solid theoretical foundation for subsequent numerical investigations. © 2025, CC BY.

关键词： Asymptotic stability

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A BDDC method for three-dimensional advection-diffusion problems with an adaptive coarse space

arXiv

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arXiv 2025年

作者： Peng, Jie Shu, Shi Wang, Junxian Zhong, Liuqiang College of Mathematics and Statistics Hengyang Normal University Hengyang421010 China School of Mathematical Sciences South China Normal University Guangzhou510631 China School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

The solution of nonsymmetric positive definite (NSPD) systems for advection-diffusion problems is an important research topic in science and {1.. The adaptive BDDC method is a significant class of non-overlapping domain decomposition methods, which is often used for solving symmetric positive definite problems. In this paper, we apply the adaptive BDDC method to solve the NSPD systems of te advection-diffusion problems. Moreover, by designing a class of edge generalized eigenvalue problems based on prior selected primal constraints, the number of primal unknowns is further reduced. Numerical experiments have verified the effectiveness of the proposed methods. Copyright © 2025, The Authors. All rights reserved.

关键词： Domain decomposition methods

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Lattice Boltzmann simulation reveals supercritical bifurcation in flow mode transitions of power-law fluids in the four-roll mill

arXiv

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arXiv 2025年

作者： Yu, Yuan Jiang, Xiao Gu, Qingqing Lin, Chuandong Zhu, Qingyong Yuan, Hai-Zhuan School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China National Center for Applied Mathematics in Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China School of Energy and Power Engineering University of Shanghai for Science and Technology Shanghai200093 China Sino-French Institute of Nuclear Engineering and Technology Sun Yat-sen University Zhuhai519082 China School of Aeronautics and Astronautics Sun Yat-sen University Shenzhen518107 China

The four-roll mill has been traditionally viewed as a device generating simple extensional flow with a central stagnation point. Our systematic investigation using a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model reveals unexpected richness in the flow physics, identifying two previously unreported supercritical bifurcation modes: a quadrifoliate vortex mode featuring four symmetrical counter-rotating vortices, and a dumbbell-shaped quad-vortex mode where vortices detach from but remain symmetric about the stagnation point. The numerical framework, representing the first successful extension of TRT-RLB method to power-law fluid dynamics, enables comprehensive mapping of flow characteristics across Reynolds numbers (1.≤ Re ≤ 50), power-law indices (0.7 ≤ n ≤ 1.3), and geometric configurations. The transition from quadrifoliate vortex mode exhibits distinct pathways depending on the power-law index: at relatively small n, the flow undergoes a direct supercritical bifurcation to simple extensional flow, while at relatively large n, it evolves through an intermediate dumbbell-shaped state. Among geometric parameters, the roller radius r emerges as the dominant factor controlling bifurcation points and vortex dimensions, whereas the roller-container gap δ exerts minimal influence on flow regimes. The transitions between flow modes can be precisely characterized through the evolution of vortex dimensions and velocity gradients at the stagnation point, providing quantitative criteria for flow regime identification. These findings enrich our fundamental understanding of bifurcation phenomena in extensional devices and provide quantitative guidelines for achieving desired flow patterns in four-roll mill applications. Copyright © 2025, The Authors. All rights reserved.

关键词： Vortex flow

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Magnetic Fluid Flows in Porous Media

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Chinese Physics Letters 2011年第8期28卷 198-201页

作者： LI Ming-Jun CHEN Liang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105

The seepage law under a magnetic field is obtained by up-scaling the flow at the pore scale of rigid porous media,and the macroscopic equivalent model is also *** is proved that the macroscopic mass flow depends on the macroscopic magnetic force and the gradients of pressure and of magnetic pressure,as Zahn and Rosensweig have described in their *** permeability tensor is symmetric and positive.

关键词： macroscopic Porous magnetic

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A NEURAL MULTIGRID SOLVER FOR HELMHOLTZ EQUATIONS WITH HIGH WAVENUMBER AND HETEROGENEOUS MEDIA

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SIAM Journal on Scientific Computing 2025年第3期47卷 C655-C679页

作者： Cui, Chen Jiang, Kai Shu, Shi 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 Xiangtan 411105 China

In this paper, we propose a deep learning-enhanced multigrid solver for high-frequency and heterogeneous Helmholtz equations. By applying spectral analysis, we categorize the iteration error into characteristic and noncharacteristic components. We eliminate the noncharacteristic components by a multigrid wave cycle, which employs carefully selected smoothers on each grid. We diminish the characteristic components by a learned phase function and the approximate solution of an advection-diffusion-reaction (ADR) equation, which is solved using another multigrid V-cycle on a coarser scale, referred to as the ADR cycle. The resulting solver, termed Wave-ADR-NS, enables the handling of error components with varying frequencies and overcomes constraints on the number of grid points per wavelength on coarse grids. Furthermore, we provide an efficient implementation using differentiable programming, making Wave-ADR-NS an end-to-end Helmholtz solver that incorporates parameters learned through semisupervised training. Wave-ADR-NS demonstrates robust generalization capabilities for both in-distribution and out-of-distribution velocity fields of varying difficulty. Comparative experiments with other multigrid methods validate its superior performance in solving heterogeneous 2D Helmholtz equations with wavenumbers exceeding 2000. © 2025 Society for Industrial and Applied Mathematics Publications. All rights reserved.

关键词： advection-diffusion-reaction equation deep learning-enhanced multigrid solver eikonal equation high-frequency and heterogeneous Helmholtz equation iterative error analysis

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Mesh Quality and More Detailed Error Estimates of Finite Element Method

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高等学校计算数学学报（英文版） 2017年第2期10卷 420-436页

作者： Yunqing Huang Liupeng Wang Nianyu Yi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan 411105 China

In this paper,we study the role of mesh quality on the accuracy of linear finite element *** derive a more detailed error estimate,which shows explicitly how the shape and size of elements,and symmetry structure of mesh effect on the error of numerical *** computable parameters Ge and Gv are given to depict the cell geometry property and symmetry structure of the *** compare with the standard a priori error estimates,which only yield information on the asymptotic error behaviour in a global sense,our proposed error estimate considers the effect of local element geometry properties,and is thus more *** certain conditions,the traditional error estimates and supercovergence results can be derived from the proposed error ***,the estimators Ge and Gv are computable and thus can be used for predicting the variation of *** tests are presented to illustrate the performance of the proposed parameters Ge and Gv.

关键词： Finite element methods error estimation cell geometry mesh quality

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Numerical prediction of inner turbulent flow in conical diffuser by using a new five-point scheme and DLR k-ε turbulence model

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Journal of Central South University 2008年第S1期15卷 181-186页

作者：蒋光彪何永森舒适肖映雄 College of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University

The internal turbulent flow in conical diffuser is a very complicated adverse pressure gradient *** k-ε turbulence model was adopted to study *** every terms of the Laplace operator in DLR k-ε turbulence model and pressure Poisson equation were discretized by upwind difference scheme.A new full implicit difference scheme of 5-point was constructed by using finite volume method and finite difference method.A large sparse matrix with five diagonals was formed and was stored by three arrays of one dimension in a compressed *** iterative methods do not work wel1.with large sparse *** algebraic multigrid method(AMG),linear algebraic system of equations was solved and the precision was set at *** computation results were compared with the experimental *** results show that the computation results have a good agreement with the experiment *** precision of computational results and numerical simulation efficiency are greatly improved.

关键词： conical diffuser turbulent flow DLR k-ε turbulence model 5-point scheme algebraic multigrid method(AMG)

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VANISHING VISCOSITY LIMIT FOR THE 3D INCOMPRESSIBLE MICROPOLAR EQUATIONS IN A BOUNDED DOMAIN

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Acta Mathematica Scientia 2023年第2期43卷 959-974页

作者：储洋洋肖跃龙 Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China

In this paper,we investigate the vanishing viscosity limit of the 3D incompressible micropolar equations in bounded domains with boundary *** is shown that there exist global weak solutions of the micropolar equations in a general bounded smooth *** particular,we establish the uniform estimate of the strong solutions for when the boundary is ***,we obtain the rate of convergence of viscosity solutions to the inviscid solutions as the viscosities tend to zero(i.e.,(ε,χ,γ,κ)→0).

关键词： incompressible micropolar equations initial-and boundary-valuc problcm van-ishingviscositylimit

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Numerical Method for the Time Fractional Fokker-Planck Equation

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Advances in Applied Mathematics and Mechanics 2012年第6期4卷 848-863页

作者： Xue-Nian Cao Jiang-Li Fu Hu Huang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityHunan 411105China

In this paper,a new numerical algorithm for solving the time fractional Fokker-Planck equation is *** analysis of local truncation error and the stability of this method are *** analysis and numerical experiments show... 详细信息

关键词： Fractional Fokker-Planck equation Riemann-Liouville fractional derivative truncation error stability

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