Computers & Mathematics with Applications最新文献

英文中文

Two-stage fourth-order Hermite weighted compact nonlinear scheme for hyperbolic conservation laws 双曲型守恒律的两阶段四阶Hermite加权紧非线性格式

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-10 DOI: 10.1016/j.camwa.2025.11.026

Weihao Xie , Zhifang Du , Guangxue Wang , Huaibao Zhang

In this work, we propose a fifth-order Hermite weighted compact nonlinear scheme (HWCNS) within the framework of the two-stage fourth-order Lax-Wendroff-type time discretization by Li and Du (SIAM J Sci Comput 38 (5):A3046-A3069, 2016). Unlike the traditional weighted compact nonlinear scheme (WCNS), which uses the Runge-Kutta method for high-order temporal integration, and solves only for nodal values, this new HWCNS simultaneously evolves both nodal and midpoint values within the same time discretization. These solutions are then used to compute first-order spatial derivatives at the nodes. By incorporating both nodal values and their derivatives, the scheme enables a high-order nonlinear Hermite interpolation. Furthermore, we introduce a “polynomial stencil selection procedure” derived from the targeted essentially non-oscillatory (TENO) scheme to improve the performance of the nonlinear interpolation. A variety of benchmark cases are addressed in one- and two-dimensional dimensions. The proposed scheme, based on the Lax-Wendroff time discretization, exhibits promising characteristics of minimal dissipation and dispersion errors for fine-scale features in smooth flow regions, and demonstrates robust shock-capturing capabilities with high resolutions, benefiting from its compact stencil in both time and space. Moreover, integration of the TENO technique into the nonlinear interpolation yields a further reduction in numerical dissipation, as shown in the numerical tests.

本文在Li和Du的两阶段四阶lax - wendroff型时间离散化框架内提出了一种五阶Hermite加权紧化非线性格式（HWCNS）（SIAM J .计算机学报，38 (5):A3046-A3069, 2016）。与传统加权紧化非线性格式（WCNS）采用龙格-库塔方法进行高阶时间积分，只求解节点值不同，该算法在同一时间离散化过程中同时演化节点和中点值。然后使用这些解来计算节点上的一阶空间导数。通过结合节点值及其导数，该方案实现了高阶非线性埃尔米特插值。此外，我们引入了一种“多项式模板选择程序”，该程序源于目标基本非振荡（TENO）方案，以提高非线性插值的性能。在一维和二维维度中处理各种基准测试案例。该方法基于Lax-Wendroff时间离散化，在光滑流动区域具有最小耗散和最小色散误差的特点，并具有高分辨率的强大激波捕获能力，这得益于其紧凑的时间和空间模板。此外，如数值试验所示，将TENO技术集成到非线性插值中可以进一步减少数值耗散。

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引用次数: 0

Numerical analysis of the second-order fully discrete schemes for parabolic problem based on serendipity virtual element method 基于偶然性虚元法的抛物型问题二阶全离散格式的数值分析

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-10 DOI: 10.1016/j.camwa.2025.11.016

Jianjun Wan, Yuanjiang Xu, Jiaxin Wei, Shilei Xu, Chunyan Niu

In this paper, the Crank-Nicolson and BDF2 schemes, based on the serendipity virtual element method, are applied to solve parabolic problems. The main content of this article is to analyze the fully discrete error in zero and energy norm for these two second-order fully discrete schemes and to conduct numerical experiments. We derive optimal zero and energy norm error estimates for the Crank-Nicolson and BDF2 schemes. In addition, the errors in zero norm and energy norm of the semi-discrete scheme of serendipity virtual elements are estimated. Through numerical experiments conducted on rectangular, convex polygonal, and non-convex polygonal meshes, discrete errors are presented in the spatial and temporal directions, respectively. Finally, we demonstrated the computational complexity advantage of serendipity virtual elements by calculating the total degrees of freedom.

本文采用基于偶然性虚元法的Crank-Nicolson格式和BDF2格式求解抛物型问题。本文的主要内容是分析这两种二阶全离散格式的全离散零点误差和能量范数，并进行数值实验。我们得到了Crank-Nicolson和BDF2格式的最优零和能量范数误差估计。此外，对偶然性虚元半离散格式的零范数和能量范数误差进行了估计。通过对矩形、凸多边形和非凸多边形网格的数值实验，分别在空间和时间方向上呈现离散误差。最后，通过对总自由度的计算，证明了偶然性虚拟元素在计算复杂度上的优势。

引用次数: 0

Robust adaptive meshing, mesh density functions, and nonlocal observations for ensemble based data assimilation 基于集成的数据同化鲁棒自适应网格划分、网格密度函数和非局部观测

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-09 DOI: 10.1016/j.camwa.2025.11.017

Jeremiah Buenger, Weizhang Huang, Erik S. Van Vleck

Adaptive spatial meshing has proven invaluable for the accurate, efficient computation of solutions of time dependent partial differential equations. In a DA context the use of adaptive spatial meshes addresses several factors that place increased demands on meshing; these include the location and relative importance of observations and the use of ensemble solutions. To increase the efficiency of adaptive meshes for data assimilation, robust look ahead meshes are developed that fix the same adaptive mesh for all ensemble members for the entire time interval of the forecasts and that incorporates the observations at the next analysis time. This allows for increased vectorization of the ensemble forecasts while minimizing interpolation of solutions between different meshes. The techniques to determine these robust, nonuniform meshes are based upon combining metric tensors or mesh density functions. The combination of mesh density functions is achieved through the use of so-called metric tensor intersection, a low dimensional (in the spatial dimension of the PDE) and local (pointwise in the spatial variable) technique. We illustrate the robust ensemble look ahead meshes using traveling wave solutions of a bistable reaction-diffusion equation. Observation operators based on convolution type integrals and their associated metric tensors are derived. These further the goals of making efficient use of adaptive meshes in ensemble based DA techniques, developing and employing robust meshes that are effective for a range of similar behaviors in both the ensembles and the observations, and the integration with advanced numerical PDE techniques (a quasi-Lagrangian moving mesh DG technique employing embedded pairs for time stepping). Numerical experiments with different observation scenarios are presented for a 2D inviscid Burgers’ equation, a multi-component system, a 2D Shallow Water model, and for a coupled system of two 1D Kuramoto-Sivashinsky equations.

自适应空间网格划分已被证明是精确、有效地计算时变偏微分方程解的宝贵方法。在数据数据环境中，自适应空间网格的使用解决了对网格需求增加的几个因素；这些包括观测的位置和相对重要性以及集合解的使用。为了提高自适应网格同化数据的效率，开发了鲁棒的前瞻性网格，在整个预测时间间隔内为所有集成成员固定相同的自适应网格，并在下一次分析时间合并观测结果。这允许增加集合预测的矢量化，同时最小化不同网格之间的解插值。确定这些鲁棒的非均匀网格的技术是基于结合度量张量或网格密度函数。网格密度函数的组合是通过使用所谓的度量张量相交、低维（在PDE的空间维度上）和局部（在空间变量上的点向）技术来实现的。我们用双稳态反应扩散方程的行波解来说明鲁棒系综超前网格。推导了基于卷积型积分的观测算子及其相关度量张量。这些进一步的目标是在基于集成的数据分析技术中有效地使用自适应网格，开发和采用对集成和观测中一系列相似行为有效的鲁棒网格，以及与先进的数值PDE技术（一种使用嵌入对进行时间步进的准拉格朗日移动网格DG技术）的集成。对二维无粘Burgers方程、多组分系统、二维浅水模型和两个一维Kuramoto-Sivashinsky方程耦合系统进行了不同观测情景下的数值实验。

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引用次数: 0

Feedback stabilization and finite element error analysis of viscous Burgers equation around non-constant steady state 非常稳态黏性Burgers方程的反馈镇定及有限元误差分析

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-09 DOI: 10.1016/j.camwa.2025.11.020

Wasim Akram

In this article, we explore the feedback stabilization of a viscous Burgers equation around a non-constant steady state using localized interior controls and then develop error estimates for the stabilized system using finite element method. The system is not only feedback stabilizable but exhibits an exponential decay

- ω < 0

for any ω > 0. The derivation of a stabilizing control in feedback form is achieved by solving a suitable algebraic Riccati equation posed for the linearized system. In the second part of the article, we utilize a conforming finite element method to discretize the continuous system, resulting in a finite-dimensional discrete system. This approximated system is also proven to be feedback stabilizable (uniformly) with exponential decay

- ω + ϵ

for any ϵ > 0. The feedback control for this discrete system is obtained by solving a discrete algebraic Riccati equation. To validate the effectiveness of our approach, we provide error estimates for both the stabilized solutions and the stabilizing feedback controls. Numerical implementations are carried out to support and validate our theoretical results.

在本文中，我们探讨了粘性Burgers方程在非恒定稳态周围的反馈镇定问题，并利用局部内部控制对稳定系统进行了误差估计。该系统不仅是反馈稳定的，而且对任意ω >； 0都表现出指数衰减- ω<；0。通过求解线性化系统的一个合适的代数Riccati方程，推导出反馈形式的稳定控制。在文章的第二部分中，我们利用一致性有限元方法将连续系统离散化，从而得到一个有限维的离散系统。这个近似的系统也被证明是反馈稳定的（一致的），对于任何一个λ >； 0都具有指数衰减- ω+ λ。通过求解离散代数Riccati方程，得到了该离散系统的反馈控制。为了验证我们方法的有效性，我们提供了稳定解决方案和稳定反馈控制的误差估计。数值实现支持和验证了我们的理论结果。

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引用次数: 0

New solvers for coupled flow and transport on quadrilateral meshes: Property-preserving and optimal-order convergence 四边形网格上流动和输运耦合的新解：保性质和最优阶收敛

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-08 DOI: 10.1016/j.camwa.2025.11.014

Boyang Yu , Yonghai Li , Jiangguo Liu

This paper develops two novel numerical solvers on quadrilateral meshes for coupled flow and transport problems. The focus is placed on preserving physical properties such as mass conservation and concentration positivity. Quadrilateral meshes are used due to their flexibility in accommodation of domain geometry. A weak Galerkin (WG) finite element scheme with linear shape functions is utilized to solve the Darcy equation. Mapped bilinear finite volumes on the same mesh are then used to solve the time-dependent convection-diffusion equation, which rely on the numerical velocity obtained from the Darcy scheme. Techniques for positivity-correction are applied to both diffusive and convective fluxes. Global mass conservation, positivity-preserving, and optimal order convergence are carefully examined under appropriate conditions. Numerical tests demonstrate robustness of our new solvers in handling convection dominance and anisotropy/heterogeneity in permeability and/or diffusion.

本文提出了两种新的四边形网格流动与输运耦合问题的数值求解方法。重点是保持物理性质，如质量守恒和浓度正性。由于四边形网格在适应区域几何结构方面具有灵活性，因此采用四边形网格。采用线性形函数的弱伽辽金（WG）有限元格式求解达西方程。然后在同一网格上映射双线性有限体积来求解依赖于Darcy格式的数值速度的随时间的对流扩散方程。正校正技术适用于扩散通量和对流通量。在适当的条件下，仔细检查了全局质量守恒，正守恒和最优阶收敛。数值测试证明了我们的新求解器在处理对流优势和渗透率和/或扩散的各向异性/非均质性方面的鲁棒性。

引用次数: 0

Quadrature error estimates for kernels with logarithmic singularity 具有对数奇异性核的正交误差估计

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-08 DOI: 10.1016/j.camwa.2025.11.022

Ismail Labaali , Maria Rosaria Lancia , Chiara Sorgentone

Boundary integral methods are a powerful tool to solve partial differential equations by reformulating them as integral equations over the boundary of the domain. When dealing with boundary integral methods, and in particular with the numerical integration of layer potentials, it is essential to estimate the magnitude of the error associated with the underlying quadrature rule. As the evaluation point approaches the boundary, the integral becomes nearly-singular and the associated quadrature error increases rapidly. Being able to estimate such error is needed to identify when the accuracy becomes inadequate, and the use of a specialized quadrature method is required. In this work we provide accurate quadrature error estimates for the Gauss-Legendre and the trapezoidal rules in computing two-dimensional layer potentials with logarithmic singularities.

边界积分法是求解偏微分方程的有力工具，它将偏微分方程重新表述为域边界上的积分方程。在处理边界积分方法时，特别是处理层势的数值积分时，估计与基本正交规则相关的误差的大小是必要的。当计算点接近边界时，积分近似奇异，积分误差迅速增大。当精度不足时，需要能够估计这种误差来识别，并且需要使用专门的正交方法。在这项工作中，我们提供了精确的正交误差估计高斯-勒让德规则和梯形规则在计算具有对数奇点的二维层势。

引用次数: 0

Influences of surface effects on bending of nanoplates with complex elastic boundary supports: A BE-RBFs method 表面效应对复杂弹性边界支撑纳米板弯曲的影响：一种be - rbf方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-08 DOI: 10.1016/j.camwa.2025.11.021

Boonme Chinnaboon , Monchai Panyatong , Somchai Chucheepsakul

This paper presents a comprehensive study on bending behavior of shear-deformable nanoplates with elastic boundary supports, emphasizing the critical role of surface effects as described by the Gurtin-Murdoch surface elasticity theory. While previous studies have typically assumed conventional boundary conditions, this research addresses a critical knowledge gap by systematically analyzing the combined effects of surface elasticity and elastic boundary supports. To address the computational challenges posed by complex geometries and boundary conditions, a coupled Boundary Element-Radial Basis Functions (BE-RBFs) method has been developed based on the well-established Analog Equation Method (AEM). The proposed numerical approach efficiently solves the governing equations for nanoplates, providing a powerful alternative to traditional methods that often rely on domain discretization. The results reveal that surface elastic constants and residual surface stress significantly influence nanoplate deflection and bending moments. Furthermore, the analysis of shear deformation and elastic boundary support effects demonstrates that surface properties strongly affect structural stiffness, particularly at the nanoscale. These findings establish a more realistic and comprehensive model for nanoplate analysis, thereby offering critical insights for the design of high-performance NEMS.

本文对具有弹性边界支撑的剪切变形纳米板的弯曲行为进行了全面的研究，强调了Gurtin-Murdoch表面弹性理论所描述的表面效应的关键作用。虽然以前的研究通常假设传统的边界条件，但本研究通过系统地分析表面弹性和弹性边界支撑的综合影响，解决了一个关键的知识空白。为了解决复杂几何和边界条件带来的计算挑战，在模拟方程法（AEM）的基础上，提出了一种边界元-径向基函数（be - rbf）耦合方法。所提出的数值方法有效地求解了纳米板的控制方程，为依赖于区域离散化的传统方法提供了一个强有力的替代方案。结果表明，表面弹性常数和残余表面应力显著影响纳米板的挠度和弯矩。此外，剪切变形和弹性边界支撑效应的分析表明，表面特性强烈影响结构刚度，特别是在纳米尺度上。这些发现为纳米板分析建立了一个更加现实和全面的模型，从而为高性能NEMS的设计提供了重要的见解。

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引用次数: 0

Regularized numerical method for the space-fractional logarithmic Klein-Gordon equation 空间分数阶对数Klein-Gordon方程的正则数值解法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-05 DOI: 10.1016/j.camwa.2025.11.025

Qibo Ma, Xiaoyun Jiang, Junqing Jia

In this paper, we propose a regularized method to resolve the singularity behavior occurring at the origin for the space-fractional logarithmic Klein-Gordon equation (SFLogKGE). A small regularization parameter ε is introduced into the SFLogKGE model to ensure the well-posedness of the logarithmic term. The proposed method preserves the energy, and the energy of the regularized equation approximates that of the original equation with a convergence rate of O(ε). For the convenience of using the time splitting method for numerical solution, we transform the Klein-Gordon equation into an equivalent Schrödinger equation. Then, we use the Lie-Trotter splitting method to derive a time semi-discrete scheme, and combine it with the Fourier spectral method for numerical discretization in the spatial direction. We provide a rigorous proof of the consistency, stability, and the optimal error estimate of the time semi-discrete scheme. Furthermore, we present an original convergence analysis for the regularized time semidiscretization based on the Strang splitting method, which to the best of our knowledge, has not been rigorously established in the existing literature. Finally, numerical experiments verify the first- and second-order temporal convergence of the two splitting methods, demonstrating the sharpness of our theoretical analysis. A series of numerical experiments also illustrate the markedly different evolution behaviors of the fractional equation compared with the classical equation.

本文提出了一种正则化方法来求解空间分数阶对数Klein-Gordon方程（SFLogKGE）原点处的奇异行为。在SFLogKGE模型中引入一个较小的正则化参数ε以保证对数项的适定性。该方法保留了能量，正则化方程的能量近似于原方程的能量，收敛速度为0 （ε）。为了便于使用分时法进行数值求解，我们将Klein-Gordon方程转化为等价的Schrödinger方程。然后，利用Lie-Trotter分裂法导出时间半离散格式，并结合傅里叶谱法在空间方向上进行数值离散。我们提供了时间半离散格式的一致性、稳定性和最优误差估计的严格证明。此外，我们提出了一个基于Strang分裂方法的正则化时间半离散化的原始收敛分析，据我们所知，该方法在现有文献中尚未得到严格的建立。最后，数值实验验证了两种分裂方法的一阶和二阶时间收敛性，证明了理论分析的敏锐性。一系列数值实验也说明了分数阶方程与经典方程的演化行为有显著不同。

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引用次数: 0

An efficient reduced-order approximation for the stochastic Allen-Cahn equation 随机Allen-Cahn方程的有效降阶近似

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-05 DOI: 10.1016/j.camwa.2025.11.019

Xiao Qi , Yubin Yan

In this paper, we propose and analyze an efficient numerical method for solving the stochastic Allen-Cahn equation with additive noise. The method combines a stabilized semi-implicit time discretization scheme with a reduced-order finite element spatial discretization method. The core idea is to approximate the original high-dimensional solution space via a low-dimensional subspace, constructed by the Proper Orthogonal Decomposition (POD) method based on an ensemble of snapshots from the full-order model at selected time instances. First, we rigorously establish the spatio-temporal strong convergence rates between the mild solution and the reduced-order solution. Second, in large-sample simulations, the reduced-order basis exhibits a certain generalization capability in capturing the average behavior of the numerical solutions. Numerical experiments are provided to verify the theoretical error estimates and to demonstrate the effectiveness of the proposed method.

本文提出并分析了一种求解具有加性噪声的随机Allen-Cahn方程的有效数值方法。该方法将稳定的半隐式时间离散方法与降阶有限元空间离散方法相结合。其核心思想是通过低维子空间来近似原始高维解空间，该子空间是基于全阶模型在选定时间实例上的快照集合，通过适当正交分解（POD）方法构造的。首先，我们严格地建立了温和解和降阶解之间的时空强收敛速率。其次，在大样本模拟中，降阶基在捕获数值解的平均行为方面表现出一定的泛化能力。数值实验验证了理论误差估计，并证明了所提方法的有效性。

引用次数: 0

A kernel compensation mimetic difference scheme for the grad-div eigenvalue problem 梯度特征值问题的核补偿拟差分格式

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-12-04 DOI: 10.1016/j.camwa.2025.11.015

Chenyang Wang, Yan Xu

We propose a kernel compensation mimetic difference (MD) scheme to solve the grad-div eigenvalue problem. This method utilizes a curl-curl type compensation operator along with carefully selected boundary conditions to effectively manage the infinite-dimensional kernel of the grad-div operator. To ensure high accuracy, we apply stencil-based MD operators to discretize the grad-div operator under Dirichlet boundary conditions. This results in a numerical scheme characterized by a sparse stiff matrix with a narrow bandwidth while achieving high-order accuracy. We construct the compensation operator with a proper boundary condition that is orthogonal to the discrete grad-div operator. A generalized identification method for spurious eigenvalues is presented. The resulting scheme offers several advantages, including high-order accuracy, enhanced computational efficiency with reduced memory usage, and excellent scalability for parallel computation. Numerical tests demonstrate that our approach not only converges at the expected rates but also performs satisfactorily in terms of speed.

针对梯度特征值问题，提出了一种核补偿模拟差分（MD）方案。该方法利用旋旋型补偿算子和精心选择的边界条件，有效地管理了梯度算子的无限维核。为了保证高精度，我们在Dirichlet边界条件下，采用基于模板的MD算子对梯度算子进行离散化。这使得该数值格式具有窄带宽的稀疏刚性矩阵的特点，同时又能获得高阶精度。构造了具有与离散梯度-div算子正交的适当边界条件的补偿算子。提出了一种伪特征值的广义辨识方法。由此产生的方案具有几个优点，包括高阶精度、通过减少内存使用提高计算效率以及并行计算的出色可伸缩性。数值测试表明，该方法不仅能达到预期的收敛速度，而且在速度方面也有令人满意的表现。

引用次数: 0

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