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Mathematics and Computers in simulation 2025年 237卷 70-85页

作者： Bu, Weiping Zheng, Xin School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China

The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay τ based on the uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation error of L1 scheme is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. In order to investigate the error estimate, a novel discrete fractional Grönwall inequality with delay term is proposed, which does not include the increasing Mittag-Leffler function. By applying this Grönwall inequality, we obtain a local error estimate of the above fully discrete scheme without including the Mittag-Leffler function. In particular, the convergence result implies that, for the considered time interval ((i−1)τ,iτ], although the convergence rate in the sense of maximum time error is low for the first interval, i.e. i=1, it will be improved as the increasing i, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing i. Finally, we present some numerical tests to verify the developed theory. © 2025 International Association for Mathematics and Computers in simulation (IMACS)

关键词： Finite element method

RECONSTRUCTION-BASED A POSTERIORI ERROR ESTIMATES FOR THE L1 METHOD FOR TIME FRACTIONAL PARABOLIC PROBLEMS

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Journal of computational Mathematics 2025年第2期43卷 345-368页

作者： Jiliang Cao Aiguo Xiao Wansheng Wang Department of Mathematics Shanghai Normal UniversityShanghai 200234China Hunan Key Laboratory for Computation and Simulation in Science and Engineering&Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan UniversityXiangtan 411105China

In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial *** derive optimal order a posteriori error estimates,the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are *** using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are *** numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results,with the convergence ofαorder for the nonsmooth case on a uniform *** recover the optimal convergence order 2-αon a nonuniform mesh,we further develop a time adaptive algorithm by means of barrier function recently *** numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.

关键词： Time fractional parabolic differential equations A posteriori error estimates Ll method Fractional integral reconstruction Quadratic reconstruction

Hamiltonian boundary value methods applied to KdV-KdV systems

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Applied Numerical Mathematics 2025年 214卷 1-27页

作者： Luo, Qian Xiao, Aiguo Luo, Xuqiong Yan, Xiaoqiang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan 411105 China School of Mathematics and Statistics Changsha University of Science and Technology Changsha410114 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan 411105 China

In this paper, we propose a highly accurate scheme for two KdV systems of the Boussinesq type under periodic boundary conditions. The proposed scheme combines the Fourier-Galerkin method for spatial discretization with Hamiltonian boundary value methods for time integration, ensuring the conservation of discrete mass and energy. By expanding the system in Fourier series, the equations are firstly transformed into Hamiltonian form, preserving the original Hamiltonian structure. Applying the Fourier-Galerkin method for semi-discretization in space, we obtain a large-scale system of Hamiltonian ordinary differential equations, which is then solved using a class of energy-conserving Runge-Kutta methods, known as Hamiltonian boundary value methods. The efficiency of this approach is assessed, and several numerical examples are provided to demonstrate the effectiveness of the method. © 2025 IMACS

关键词： Ordinary differential equations

Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential-algebraic equations

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arXiv 2025年

作者： Wang, Gehao Yu, Yuexin Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This paper is devoted to examining the stability of Runge-Kutta methods for solving stiff nonlinear Volterra delay-integro-differential-algebraic equations (DIDAEs) with constant delay. Hybrid numerical schemes combining Runge-Kutta methods and compound quadrature rules are analyzed for nonlinear stiff DIDAEs. Criteria for ensuring the global and asymptotic stability of the proposed schemes are established. Several numerical examples are provided to validate the theoretical findings. Copyright © 2025, The Authors. All rights reserved.

关键词： Runge Kutta methods

ADAPTIVE RESIDUAL-DRIVEN NEWTON SOLVER FOR NONLINEAR SYSTEMS OF EQUATIONS

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arXiv 2025年

作者： Ding, Renjie Wang, Dongling Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China

Newton-type solvers have been extensively employed for solving a variety of nonlinear system of algebraic equations. However, for some complex nonlinear system of algebraic equations, efficiently solving these systems remains a challenging task. The primary reason for this challenge arises from the unbalanced nonlinearities within the nonlinear system. Therefore, accurately identifying and balancing the unbalanced nonlinearities in the system is essential. In this work, we propose a residual-driven adaptive strategy to identify and balance the nonlinearities in the system. The fundamental idea behind this strategy is to assign an adaptive weight multiplier to each component of the nonlinear system, with these weight multipliers increasing according to a specific update rule as the residual components increase, thereby enabling the Newton-type solver to select a more appropriate step length, ensuring that each component in the nonlinear system experiences sufficient reduction rather than competing against each other. More importantly, our strategy yields negligible additional computational overhead and can be seamlessly integrated with other Newton-type solvers, contributing to the improvement of their efficiency and robustness. We test our algorithm on a variety of benchmark problems, including a chemical equilibrium system, a convective diffusion problem, and a series of challenging nonlinear systems. The experimental results demonstrate that our algorithm not only outperforms existing Newton-type solvers in terms of computational efficiency but also exhibits superior robustness, particularly in handling systems with highly imbalanced nonlinearities. © 2025, CC BY.

关键词： Nonlinear equations

Optimal convergence analysis of an energy dissipation property virtual element method for the nonlinear Benjamin-Bona-Mahony-Burgers equation

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Computers & Mathematics with Applications 2025年 192卷 37-53页

作者： Yanping Chen Wanxiang Liu Fangfang Qin Qin Liang School of Science Nanjing University of Posts and Telecommunications Nanjing 210023 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China

A novel arbitrary high-order energy-stable fully discrete schemes are proposed for the nonlinear Benjamin-Bona-Mahony-Burgers equation based on linearized Crank-Nicolson scheme in time and the virtual element discretization in space. Two skew-symmetric discrete forms are introduced to preserve energy dissipation of the numerical scheme. Furthermore, by utilizing the L 2 projection to approximate the nonlinear term and estimating the error of the discrete bilinear forms carefully, the optimal error estimate of the numerical scheme in the H 1 -norm is obtained. Finally, several numerical examples on various mesh types are provided to demonstrate the energy stability, optimal convergence and high efficiency of the method.

关键词：

HTL-WASI preconditioner for finite element discretization of complex ADR equation and its application

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Journal of computational and Applied Mathematics 2025年 470卷

作者： Yang, Ronghua Wang, Junxian Shu, Shi School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan Xiangtan Hunan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan411105 China

The convection diffusion reaction (ADR) equation is widely used in engineering applications. In this paper, we focus on a complex ADR equation arising from Helmholtz equation with impedance boundary conditions, prove the coerceiveness and boundedness of the sesquilinear functional and discuss the fast solver of the discretization. Firstly, we design a Weighted Additive Schwarz with Impedance (WASI) preconditioner, in order to overcome the dependence of the WASI preconditioner on the number of subdomains and the overlapping width, an ideal coarse space is designed. Although it can be proved that the corresponding hybrid two-level preconditioner is the exact inverse of the ADR coefficient matrix, the dimension of this coarse space is too big. In order to overcome this deficiency, a smaller dimensional coarse space was constructed by introducing local generalized eigenvalue problems (GEP) on each subdomain, and the descent rate of the corresponding two-level preconditioned GMRES method was rigorously analyzed which does not depend on mesh size, overlapping width, wave number k, and the absorption parameter. Since the size of the coarse space is sensitive to k and the GEP on each subdomain needs to be solved, the computational cost is too high. Therefore, we design an iterative method based on an economical two-level preconditioner, and establish a heuristic convergence theory that is supported by numerical results. Numerical experiments show the robustness of GMRES with the corresponding economical two-level preconditioner(HTLE-WASI-GMRES). Finally, for the Helmholtz finite element discretization, by using the Shifted Laplace technique and ADR equation, we design a novel fast solver combining with HTLE-WASI-GMRES for ADR equation. Numerical experiments have demonstrated the effectiveness of the algorithm. © 2025

关键词： Convergence of numerical methods

Comments and extensions on "State-equivalent form and minimum-order compensator design for rectangular descriptor systems"

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arXiv 2025年

作者： Shi, Shuo Zhang, Juan 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 Xiangtan411105 China

This technical note presents a counterexample showing that the equivalence conditions proposed by Geng et al. (IEEE Trans. Automat. Control, 2024), which use a minimum-order compensator (MOC) to achieve desired designs, including generalized regularity or both generalized regularity and free of impulse, are sufficient but not necessary. Furthermore, revised equivalence conditions are introduced, along with an equivalence condition ensuring the closed-loop system remains generalized regular, impulse-free, and stable using MOC. Additionally, it is shown that output feedback can replace the MOC, achieving the same design without increasing dimensionality. These findings are validated through a circuit *** Codes 93B05, 93B52, 15A03 Copyright © 2025, The Authors. All rights reserved.

关键词： Feedback

A discrete Perfectly Matched Layer for peridynamic scalar waves in two-dimensional viscous media

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arXiv 2025年

作者： Du, Yu Li, Yonglin Zhang, Jiwei School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan411105 China School of Mathematics and Statistics Hubei Key Laboratory of Computational Science Wuhan University Wuhan430072 China

In this paper, we propose a discrete perfectly matched layer (PML) for the peridynamic scalar wave-type problems in viscous media. Constructing PMLs for nonlocal models is often challenging, mainly due to the fact that nonlocal operators are usually associated with various kernels. We first convert the continua model to a spatial semi-discretized version by adopting quadrature-based finite difference scheme, and then derive the PML equations from the semi-discretized equations using discrete analytic continuation. The harmonic exponential fundamental solutions (plane wave modes) of the semi-discretized equations are absorbed by the PML layer without reflection and are exponentially damped. The excellent efficiency and stability of discrete PML are demonstrated in numerical tests by comparison with exact absorbing boundary *** Codes 65N30, 65R20, 46N20, 45A05, 78A40 Copyright © 2025, The Authors. All rights reserved.

关键词： Boundary conditions

THREE-PRECISION ITERATIVE REFINEMENT WITH PARAMETER REGULARIZATION AND PREDICTION FOR SOLVING LARGE SPARSE LINEAR SYSTEMS

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arXiv 2025年

作者： Ge, Jifeng Zhang, Juan Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan KeyLaboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

This study presents a novel mixed-precision iterative refinement algorithm, GADI-IR, within the general alternating-direction implicit (GADI) framework, designed for efficiently solving large-scale sparse linear systems. By employing low-precision arithmetic, particularly half-precision (FP16), for computationally intensive inner iterations, the method achieves substantial acceleration while maintaining high numerical accuracy. key challenges such as overflow in half-precision and convergence issues for low precision are addressed through careful backward error analysis and the application of a regularization parameter α. Furthermore, the integration of low-precision arithmetic into the parameter prediction process, using Gaussian Process Regression (GPR), significantly reduces computational time without degrading performance. The method is particularly effective for large-scale linear systems arising from discretized partial differential equations and other high-dimensional problems, where both accuracy and efficiency are critical. Numerical experiments demonstrate that the use of mixed-precision strategies not only accelerates computation but also ensures robust convergence, making the approach advantageous for various applications. The results highlight the potential of leveraging lower-precision arithmetic to achieve superior computational efficiency in high-performance computing. © 2025, CC BY.

关键词： Convergence of numerical methods