Results in Applied Mathematics最新文献

英文中文

A comparison of implicit–explicit Runge–Kutta time integration schemes in numerical solvers based on the Galerkin and Petrov–Galerkin spectral methods for two-dimensional magneto-hydrodynamic problems 基于Galerkin和Petrov-Galerkin谱法的二维磁流体动力学问题数值求解中隐式-显式龙格-库塔时间积分格式的比较

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-02-01 DOI: 10.1016/j.rinam.2026.100683

Anna Piterskaya, Mikael Mortensen

The research adopts the Galerkin and Petrov–Galerkin spectral methods to analyze the effects of implicit–explicit Runge–Kutta (IMEX RK) time schemes on the stability, accuracy, precision, and efficiency of computations when used in numerical simulations of the diffusion, Burgers’, and magneto-hydrodynamic (MHD) equations. The maximum time steps suitable for the Galerkin and Petrov–Galerkin spectral methods are determined based on an analysis of the diffusion equation. It is shown that for the Petrov–Galerkin spectral method it is possible to use a time step size that is nearly three times as large as that of the Galerkin spectral method. The results of the relative error analysis of Burgers’ equation show that the IMEX RK schemes become more stable and accurate with an increase in the number of basis functions and smaller time steps, while the effect of higher viscosity leads to a decrease in relative errors. It is found that when the number of basis functions is large, the higher-order IMEX RK schemes in conjunction with the Petrov–Galerkin spectral method efficiently discretize time and space and achieve high accuracy at various viscosities due to their well-conditioned band matrices. In addition, the problem of the influence of magnetic fields on the efficiency of different IMEX RK schemes is considered within the framework of a two-dimensional MHD system by studying the evolution of small perturbations in a conducting fluid flow. The analyses show that the higher-order IMEX RK schemes provide enhanced stability and accuracy, while the lower-order schemes offer computational efficiency at the expense of some accuracy.

本文采用Galerkin和Petrov-Galerkin谱方法，分析了隐式-显式龙格-库塔（IMEX RK）时间格式对扩散方程、Burgers方程和磁流体动力学方程数值模拟的稳定性、准确性、精密度和效率的影响。通过对扩散方程的分析，确定了适用于Galerkin和Petrov-Galerkin谱法的最大时间步长。结果表明，对于Petrov-Galerkin谱方法，可以使用几乎是Galerkin谱方法三倍大的时间步长。Burgers方程的相对误差分析结果表明，随着基函数数量的增加和时间步长的减小，IMEX RK方案的稳定性和精度提高，而高粘度的影响导致相对误差降低。研究发现，当基函数数量较大时，高阶IMEX RK格式与Petrov-Galerkin谱方法结合使用时，由于其条件良好的频带矩阵，可以有效地离散时间和空间，并在不同粘度下获得较高的精度。此外，通过研究导电流体流动中的小扰动演化，在二维MHD系统的框架内考虑了磁场对不同IMEX RK格式效率的影响问题。分析表明，高阶IMEX RK方案提供了更高的稳定性和精度，而低阶方案以牺牲一定的精度为代价提供了计算效率。

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引用次数: 0

On the inversion of polynomials of discrete Laplace matrices 离散拉普拉斯矩阵的多项式反演

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-02-01 DOI: 10.1016/j.rinam.2026.100686

S. Asghar , Q. Peng , F.J. Vermolen , C. Vuik

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.

矩阵多项式的有效反演是计算数学中的一个重大挑战。设计了一个求多维拉普拉斯矩阵的多项式逆的程序。该方法基于特征向量和特征值展开。该方法与先前已知的一维空间离散拉普拉斯逆表达式一致（Vermolen et al., 2022）。进一步推广了该形式，得到了时变问题的封闭形式表达式。

引用次数: 0

Non-dimensional FEM analysis of a functionally graded thermopiezoelectric rod subjected to a moving heat source 运动热源作用下功能梯度热压电棒的无量纲有限元分析

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-02-01 DOI: 10.1016/j.rinam.2026.100684

Vitalii Stelmashchuk

A functionally graded one-dimensional piezoelectric rod excited by a moving heat source is considered. Its dynamic multifield response under this type of thermal load is investigated in the context of coupled Lord-Shulman thermopiezoelectricity theory.

In the literature, the transient response of piezoelectric materials to a moving heat source is extensively studied, but in most cases the solution is obtained using Laplace transform. In this paper we propose a completely numerical approach to solve the problem.

The original initial boundary value problem is transformed into a variational one. The variational problem is then rewritten in a non-dimensional form using appropriate variable substitutions. For space discretization the finite element method is used and for time discretization we utilize a hybrid time integration scheme based on Newmark scheme for hyperbolic equations and generalized trapezoidal rule for parabolic equations. The approximation of Dirac delta function by means of Gaussian probability density function is used to simulate the moving heat source.

The developed numerical scheme is validated against the benchmark solutions available in the literature and our results show great agreement with them. Besides, we empirically analyze the energy conservation properties of proposed time integration scheme and verify spatial and temporal convergence of the obtained solutions. Finally, we study the influence of non-homogeneity index, thermal relaxation time and moving heat source velocity on the solutions of the problem.

研究了一种受运动热源激励的功能梯度一维压电棒。在Lord-Shulman热电耦合理论的背景下，研究了这种类型热负荷下的动态多场响应。在文献中，对压电材料对移动热源的瞬态响应进行了广泛的研究，但在大多数情况下，解是使用拉普拉斯变换得到的。在本文中，我们提出了一个完全的数值方法来解决这个问题。将初始边值问题转化为变分问题。然后用适当的变量替换将变分问题重写为无量纲形式。空间离散采用有限元法，时间离散采用双曲型方程的Newmark格式和抛物型方程的广义梯形规则的混合时间积分格式。用高斯概率密度函数逼近狄拉克函数来模拟运动热源。所开发的数值格式与文献中可用的基准解进行了验证，我们的结果与它们非常吻合。此外，我们还对所提出的时间积分方案的能量守恒特性进行了实证分析，并验证了所得到的解的时空收敛性。最后，研究了非均匀性指数、热松弛时间和移动热源速度对问题解的影响。

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引用次数: 0

A Liouville-type theorem for 3D stationary Navier–Stokes equations 三维平稳Navier-Stokes方程的liouville型定理

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.rinam.2025.100679

Zixuan Shen , Deyi Ma

In this paper, we establish a Liouville-type theorem for smooth solutions of the stationary Navier–Stokes equations under a growth condition on the Lebesgue norms. Based on this condition, we prove a lemma analogous to the Poincaré-type inequality in the curl sense, which serves as a key tool in proving our main result.

在Lebesgue范数的增长条件下，我们建立了平稳Navier-Stokes方程光滑解的liouville型定理。在此基础上，我们证明了旋度意义上的一个类似于poincar型不等式的引理，它是证明我们的主要结果的关键工具。

引用次数: 0

A numerical framework based on piecewise Chebyshev cardinal functions for fractional integro-differential equations 基于分段切比雪夫基数函数的分数阶积分微分方程数值框架

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.rinam.2025.100682

S. Mansoori Aref , M.H. Heydari , M. Bayram

In this study, we develop an operational matrix technique to address a set of fractional nonlinear integro-differential equations with the Caputo–Hadamard derivative. We utilize a family of the piecewise Chebyshev cardinal functions as basis functions in this regard. Some formulas are introduced for calculating the classical and Hadamard fractional integrals of these functions. In the established strategy, the fractional expression is approximated utilizing these piecewise functions. By employing the aforementioned operational matrices and leveraging the cardinal property of the basis functions, we solve an algebraic system to obtain the solution. Convergence is both analytically demonstrated and confirmed through numerical experiments. Additionally, we compare the results obtained using this method with those derived from the hat functions method.

在这项研究中，我们发展了一种运算矩阵技术来解决一组分数阶非线性积分微分方程的Caputo-Hadamard导数。在这方面，我们利用一组分段切比雪夫基数函数作为基函数。介绍了这些函数的经典积分和阿达玛分数积分的计算公式。在所建立的策略中，分数表达式是利用这些分段函数逼近的。利用上述运算矩阵，利用基函数的基数性质，对一个代数系统进行求解。收敛性得到了解析论证和数值实验的证实。此外，我们还将用这种方法得到的结果与帽函数方法得到的结果进行了比较。

引用次数: 0

Screening effects in Gaussian random fields under generalized spectral conditions 广义谱条件下高斯随机场的筛选效应

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.rinam.2025.100681

Mohammad Meysami , Ali Lotfi

The screening effect is a central idea in spatial prediction: once nearby observations are used, distant ones add little. While Stein’s classical results explain this effect under strong spectral conditions, many models used in practice fall outside those assumptions. In this paper, we extend Stein’s theory to a broader class of Gaussian random fields. We replace strict regular variation by more flexible O-regular variation, resolve the critical borderline case where the spectral exponent equals the dimension, and derive explicit convergence rates for prediction error. Our analysis also shows that screening is robust to mild nonstationarity and anisotropy, and it applies to irregular sampling designs such as Delone sets. To illustrate these results, we provide examples with Matérn, generalized Cauchy, and anisotropic models, along with numerical experiments confirming the theory. These findings clarify when local kriging can safely replace global prediction, and they provide a solid foundation for scalable methods such as covariance tapering and Vecchia approximations.

筛选效应是空间预测的核心思想：一旦使用了近距离的观测结果，远距离的观测结果就没有什么帮助了。虽然斯坦的经典结果在强光谱条件下解释了这种效应，但在实践中使用的许多模型都超出了这些假设。在本文中，我们将Stein的理论推广到更广泛的高斯随机场。我们用更灵活的o规则变化代替严格的规则变化，解决了谱指数等于维数的临界边界情况，并推导出预测误差的显式收敛率。我们的分析还表明，筛选对轻度非平稳性和各向异性具有鲁棒性，并且适用于不规则采样设计，如Delone集。为了说明这些结果，我们提供了mat、广义柯西和各向异性模型的例子，以及证实该理论的数值实验。这些发现阐明了局部克里格何时可以安全地取代全局预测，并为协方差渐缩和维奇亚近似等可扩展方法提供了坚实的基础。

引用次数: 0

Optimal second order covariant derivatives on associated bundles — Application to color image restoration 关联束的最优二阶协变导数。在彩色图像恢复中的应用

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.rinam.2025.100680

Thomas Batard

This paper deals with the application of a geometric setting widely employed in theoretical physics, namely the fiber bundles, to color image restoration. The key idea of this approach is to model an image as a function on a principal bundle satisfying an equivariance property with respect to the action of a Lie group acting on its pixel values, and which can model the lighting changes in a scene. In this context, a natural tool for the differentiation of an image is by means of a covariant derivative. In previous works, optimal covariant derivatives have been constructed as solutions of a variational model consisting of the minimization of the

L^{2}

norm of the covariant derivative of the image, and applied to various tasks in color image restoration through the extension of the Total Variation regularizer to vector bundles. The aim of this paper is to extend these works by constructing optimal second order covariant derivatives as solutions of the minimization of the norm of the second order covariant derivative of the image. Experiments on deblurring and super-resolution corroborate the relevance of the proposed model for color image restoration. More generally, this paper validates the use of the geometric setting of fiber bundles in imaging sciences.

本文讨论了理论物理中广泛使用的几何设置，即光纤束在彩色图像恢复中的应用。这种方法的关键思想是将图像建模为一个主束上的函数，该函数满足李群作用于其像素值的等方差属性，并且可以模拟场景中的照明变化。在这种情况下，对图像进行微分的自然工具是借助协变导数。在以前的工作中，最优协变导数已被构造为由图像协变导数的L2范数的最小化组成的变分模型的解，并通过将总变分正则器扩展到向量束，将其应用于彩色图像恢复的各种任务。本文的目的是通过构造最优二阶协变导数作为图像二阶协变导数范数最小化的解来扩展这些工作。去模糊和超分辨率实验验证了该模型在彩色图像恢复中的适用性。更一般地说，本文验证了纤维束几何设置在成像科学中的应用。

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引用次数: 0

A quantum graph FFT with applications to partial differential equations on networks 量子图FFT及其在网络偏微分方程中的应用

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2026-01-06 DOI: 10.1016/j.rinam.2025.100670

Robert Carlson

The Fast Fourier Transform is extended to functions on finite graphs whose edges are identified with intervals of finite length. Spectral and pseudospectral methods are developed to solve a wide variety of time dependent partial differential equations on domains which are modeled as networks of one dimensional segments joined at nodes.

将快速傅里叶变换推广到有限图上的函数，这些图的边用有限长区间标识。谱和伪谱方法用于求解各种时域相关的偏微分方程，这些偏微分方程被建模为在节点处连接的一维段网络。

引用次数: 0

A hybrid numerical method for multi-domain wave propagation problems 多域波传播问题的混合数值方法

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.rinam.2025.100678

Zihui Yan , Xiangran Zheng , Wenzhen Qu , Sheng-Dong Zhao

This study introduces a hybrid numerical methodology designed to address multi-domain wave propagation problems. The temporal discretization is achieved using the Houbolt scheme, a finite-difference-based approach known for its robust stability in time-dependent simulations. Spatially, the computational domain is partitioned into subdomains via a domain decomposition strategy, with continuity and equilibrium constraints imposed along the interfaces to ensure physical fidelity. Within each subdomain, the meshless generalized finite difference method (GFDM) is employed, thereby avoiding the complexity of mesh generation. Through the integration of Taylor series expansion and moving least squares (MLS) approximation, explicit formulations of partial derivatives are constructed. Numerical experiments confirm that the proposed hybrid method provides high accuracy and stability in simulating multi-domain wave propagation phenomena.

本文介绍了一种混合数值方法，旨在解决多域波传播问题。时间离散化是使用Houbolt方案实现的，这是一种基于有限差分的方法，以其在依赖时间的模拟中的鲁棒稳定性而闻名。在空间上，通过域分解策略将计算域划分为子域，并沿界面施加连续性和平衡约束以确保物理保真度。在每个子域中采用无网格广义有限差分法（GFDM），避免了网格生成的复杂性。通过泰勒级数展开和移动最小二乘逼近的积分，构造了偏导数的显式表达式。数值实验结果表明，该方法在模拟多域波传播现象时具有较高的精度和稳定性。

引用次数: 0

A hybrid MPWENO scheme with enhanced accuracy and robustness for the compressible Euler equations 一种提高精度和鲁棒性的混合MPWENO可压缩欧拉方程解

IF 1.3 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.rinam.2025.100674

Yalin Tang , Mingjun Li , Shujiang Tang

In this study, we develop a hybrid high-order monotonicity-preserving Weighted Essentially Non-Oscillatory (MPWENO) finite difference discretization for the compressible Euler equations. The key innovation of this scheme lies in optimizing the numerical flux through the mixed smooth indicator to reduce the numerical oscillations caused by scheme conversion, and adjusting the reference value to modify the MP limiter, thereby enabling the scheme to achieve the required numerical accuracy while maintaining monotonicity. The novelty of this work lies in systematically adjusting the numerical fluxes by using the mixed smooth indicator to distinguish between discontinuities and extrema and improving the accuracy of the reference value to change the MP limiter, which is different from other MP schemes. The proposed method improves the accuracy at the extreme points, exhibits outstanding robustness and excellent resolution, making it particularly suitable for solving complex problems in compressible Euler equations. Moreover, it is easy to implement and applicable to multi-dimensional problems, with significant practical advantages. Numerical results show that this method has high accuracy, strong robustness, and high resolution. These findings highlight the effectiveness and reliability of this method in handling complex compressible flow simulations.

本文研究了可压缩欧拉方程的混合高阶保持单调性加权本质非振荡有限差分离散方法。该方案的关键创新之处在于通过混合光滑指示器优化数值通量，以减少方案转换引起的数值振荡，并通过调整参考值来修改MP限幅器，从而使方案在保持单调性的同时达到所需的数值精度。本工作的新颖之处在于，不同于其他多普勒方案，利用混合光滑指示器区分不连续点和极值点，系统地调整数值通量，提高参考值的精度，从而改变多普勒限制器。该方法提高了极值点的精度，具有较强的鲁棒性和较好的分辨率，特别适用于求解可压缩欧拉方程中的复杂问题。该方法易于实现，适用于多维问题，具有显著的实用优势。数值结果表明，该方法精度高、鲁棒性强、分辨率高。这些发现突出了该方法在处理复杂可压缩流动模拟中的有效性和可靠性。

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