Applied Numerical Mathematics最新文献

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A mixed-order compact generalized finite difference method for stable seismic wavefield simulation 稳定地震波场模拟的混合阶紧致广义有限差分法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.apnum.2026.01.005

Zhili Chen, Yong Wang, Zhixian Gui

The Generalized Finite Difference Method (GFDM) is a promising meshless approach for seismic wave simulation, offering superior flexibility in modeling complex geological structures. However, its critical weakness lies in the sensitivity of numerical stability to the node distribution within the computational stencil. Asymmetric or irregular point clouds often lead to ill-posed systems and unstable simulations, especially when using the theoretical minimum number of nodes required for a given order of accuracy. This instability severely limits GFDM's practical application in high precision seismic modeling. To address this challenge, we propose a novel Mixed-Order Compact GFDM (MOCGFDM) inspired by the principles of the Compact Finite Difference Method (CFDM). Our key innovation is the incorporation of derivative information from neighboring stencil points directly into the objective function used to compute the difference coefficients. This introduces a compact constraint that significantly enhances the robustness of the system. Crucially, the proposed method modifies only the pre-computation of the difference coefficients, leaving the core wave propagation algorithm unchanged and thus preserving computational efficiency. Numerical experiments across various models demonstrate that MOCGFDM achieves markedly higher stability in high order simulations compared to the conventional GFDM. It effectively enables stable computations with fewer nodes in irregular point clouds and allows for larger critical time steps. Consequently, this method not only improves reliability but also indirectly boosts simulation efficiency, providing a robust and practical meshless solution for high fidelity seismic wavefield simulation.

广义有限差分法（GFDM）是一种很有前途的无网格地震波模拟方法，在复杂地质构造建模方面具有优越的灵活性。然而，它的关键弱点在于数值稳定性对计算模板内节点分布的敏感性。不对称或不规则的点云通常会导致不适定的系统和不稳定的模拟，特别是当使用给定精度顺序所需的理论最小节点数时。这种不稳定性严重限制了GFDM在高精度地震建模中的实际应用。为了解决这一挑战，我们提出了一种受紧凑有限差分法（CFDM）原理启发的新型混合阶紧凑GFDM （MOCGFDM）。我们的关键创新是将相邻模板点的导数信息直接纳入用于计算差系数的目标函数中。这引入了一个紧凑的约束，大大增强了系统的鲁棒性。关键是，该方法只修改了差系数的预计算，使核心波传播算法保持不变，从而保持了计算效率。各种模型的数值实验表明，与传统的GFDM相比，MOCGFDM在高阶模拟中具有更高的稳定性。它有效地实现了不规则点云中较少节点的稳定计算，并允许更大的临界时间步长。因此，该方法不仅提高了可靠性，而且间接提高了仿真效率，为高保真度地震波场仿真提供了鲁棒实用的无网格解决方案。

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引用次数: 0

Linearized and high-order accurate one-parameter compact difference schemes for solving the Gray-Scott reaction-diffusion model 求解Gray-Scott反应扩散模型的线性化和高阶精确单参数紧致差分格式

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2026-01-04 DOI: 10.1016/j.apnum.2026.01.001

Hao Han, Zengqiang Tan

The Gray-Scott reaction-diffusion (GS-RD) model plays an important role to study the pattern formation dynamics in nature. This paper studies linear and high-order accurate schemes for numerically solving the GS-RD model, where the schemes are built on the one-parameter methods in time and the compact difference scheme in space. The error estimates and stability of the proposed schemes are analysed and it is proved that the fully discrete schemes have second-order accuracy in temporal direction and four-order accuracy in spatial direction. Specifically, for two- and three-dimensional cases, the alternating direction implicit (ADI) technique is adopted to reduce computational cost of the schemes. Several numerical experiments are conducted to validate the theoretical results as well as the computational effectiveness and accuracy of the derived schemes.

Gray-Scott反应扩散（GS-RD）模型在研究自然界模式形成动力学中起着重要作用。本文研究了GS-RD模型数值求解的线性和高阶精确格式，在时间上采用单参数方法，在空间上采用紧致差分格式。分析了所提格式的误差估计和稳定性，证明了所提格式在时间方向上具有二阶精度，在空间方向上具有四阶精度。具体而言，对于二维和三维情况，采用交替方向隐式（ADI）技术来降低方案的计算成本。通过数值实验验证了理论结果以及所导出格式的计算有效性和准确性。

引用次数: 0

Two-grid mixed finite element method with backward Euler fully discrete scheme for the nonlinear schrödinger equation 用反向欧拉完全离散格式的两网格混合有限元法求解非线性schrödinger方程

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2026-01-03 DOI: 10.1016/j.apnum.2025.12.009

Zhikun Tian , Jianyun Wang , Jie Zhou

We consider the two-dimensional time-dependent nonlinear Schrödinger equation by the backward Euler fully discrete mixed finite element method and obtain optimal error order in L²-norm. We develop a two-grid algorithm within the backward Euler fully discrete mixed finite element scheme. This algorithm reduces the solution of the nonlinear Schrödinger equation on a fine grid to solving the original nonlinear problem on a much coarser grid, coupled with a linear problem on the fine grid. Moreover, we demonstrate that the two-grid solution achieves the same error order as the standard mixed finite element solution when the coarse and fine mesh sizes satisfy

H = O (h^{\frac{1}{2}})

. Finally, a numerical experiment in the RT₀ space is provided to partly verify theoretical results.

采用倒向欧拉完全离散混合有限元法对二维时变非线性Schrödinger方程进行了分析，得到了l2范数下的最优误差阶。在后向欧拉完全离散混合有限元格式中，提出了一种双网格算法。该算法将非线性Schrödinger方程在细网格上的求解简化为在更粗的网格上求解原非线性问题，并在细网格上求解线性问题。此外，我们还证明了当粗、细网格尺寸满足H=O（h12）时，两网格解与标准混合有限元解具有相同的误差阶。最后，通过RT0空间的数值实验对理论结果进行了部分验证。

引用次数: 0

A separate preconditioned primal-dual splitting algorithm for composite monotone inclusion problems 复合单调包含问题的单独预条件原对偶分裂算法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-27 DOI: 10.1016/j.apnum.2025.12.004

Xiaokai Chang , Xingran Zhao , Long Xu

We propose a separable preconditioned primal-dual splitting (SP-PDS) method for solving composite monotone inclusion problems. The linear subproblem arising in this method can be selected or generated by comprehensively considering factors such as computational complexity and numerical convergence speed. We prove weak convergence in Hilbert space by reformulating the proposed SP-PDS as a decomposed proximal point algorithm, where the preconditioner is decomposed nonsymmetrically. In particular, various efficient preconditioners are introduced in this framework for which only a few inner iterations are needed to implement preconditioning, instead of computing an inexact solution and controlling the error. The performance of separate preconditioning strategy is verified through preliminary numerical experiments on the image denoising and LASSO problems.

提出了一种解复合单调包含问题的可分离预条件原对偶分裂（SP-PDS）方法。该方法产生的线性子问题可以综合考虑计算复杂度和数值收敛速度等因素进行选择或生成。通过将所提出的SP-PDS重构为分解的近点算法，证明了该算法在Hilbert空间中的弱收敛性，其中预条件是非对称分解的。特别地，在该框架中引入了各种有效的预条件，只需要几个内部迭代就可以实现预条件，而不是计算不精确的解并控制误差。通过对图像去噪和LASSO问题的初步数值实验，验证了分离预处理策略的性能。

引用次数: 0

Discrete gradient methods for port-Hamiltonian differential-algebraic equations 波特-哈密顿微分代数方程的离散梯度方法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-27 DOI: 10.1016/j.apnum.2025.12.006

Philipp L. Kinon , Riccardo Morandin , Philipp Schulze

Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient methods to the system class of nonlinear port-Hamiltonian differential-algebraic equations - as they emerge from the port- and energy-based modeling of physical systems in various domains. We introduce a novel numerical scheme tailored for semi-explicit differential-algebraic equations and further address more general settings using the concepts of discrete gradient pairs and Dirac-dissipative structures. Additionally, the behavior under system transformations is investigated and we demonstrate that under suitable assumptions port-Hamiltonian differential-algebraic equations admit a representation which consists of a parametrized port-Hamiltonian semi-explicit system and an unstructured equation. Finally, we present the application to multibody system dynamics and discuss numerical results to demonstrate the capabilities of our approach.

离散梯度方法是动力系统时间离散化的有力工具，因为无论总能量的形式如何，它们都是结构保持的。在这项工作中，我们讨论了离散梯度方法在非线性端口-哈密顿微分代数方程系统类中的应用，因为它们出现在各个领域的基于端口和能量的物理系统建模中。我们介绍了一种针对半显式微分代数方程的新型数值格式，并进一步使用离散梯度对和狄拉克耗散结构的概念解决了更一般的设置。此外，研究了系统变换下的行为，证明了在适当的假设下，port- hamilton微分代数方程允许由参数化port- hamilton半显式系统和非结构方程组成的表示。最后，我们介绍了该方法在多体系统动力学中的应用，并讨论了数值结果来证明我们方法的能力。

引用次数: 0

Stochastic ADMM with batch size adaptation for nonconvex nonsmooth optimization 随机ADMM批量自适应非凸非光滑优化

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-25 DOI: 10.1016/j.apnum.2025.12.007

Jiachen Jin, Kangkang Deng, Boyu Wang, Hongxia Wang

Stochastic alternating direction method of multipliers (SADMM) is a popular method for solving nonconvex nonsmooth optimization in various applications. However, it typically requires an empirical selection of the static batch size for gradient estimation, resulting in a challenging trade-off between variance reduction and computational cost. This paper proposes adaptive batch size SADMM, a practical method that dynamically adjusts the batch size based on accumulated differences along the optimization path. We develop a simple convergence analysis to handle the dependence of batch size adaptation that matches the best-known complexity with flexible parameter choices. We further extend this adaptive scheme to reduce the overall complexity of the popular variance-reduced methods, SVRG-ADMM and SPIDER-ADMM. Numerical results validate the effectiveness of our proposed methods.

随机交替方向乘法器法（SADMM）是求解非凸非光滑优化问题的一种常用方法。然而，它通常需要对梯度估计的静态批大小进行经验选择，从而导致在方差减少和计算成本之间进行具有挑战性的权衡。本文提出了一种基于优化路径上的累积差值动态调整批大小的实用方法——自适应批大小SADMM。我们开发了一个简单的收敛分析来处理批大小自适应的依赖性，该依赖性与最著名的具有灵活参数选择的复杂性相匹配。我们进一步扩展了这种自适应方案，以降低流行的方差减少方法SVRG-ADMM和SPIDER-ADMM的总体复杂性。数值结果验证了所提方法的有效性。

引用次数: 0

Two ADI compact difference methods for variable-exponent diffusion wave equations 变指数扩散波动方程的两种ADI紧致差分方法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-25 DOI: 10.1016/j.apnum.2025.12.003

Hao Zhang , Kexin Li , Wenlin Qiu

In this work, we study two-dimensional diffusion-wave equations with variable exponent, modeling mechanical diffusive wave propagation in viscoelastic media with spatially varying properties. We first transform the diffusion-wave model into an equivalent form via the convolution method. Two time discretization strategies are then applied to approximate each term in the transformed equation, yielding two fully discrete schemes based on a spatial compact finite difference method. To reduce computational cost, the alternating direction implicit (ADI) technique is employed. We prove that both ADI compact schemes are unconditionally stable and convergent. The error estimates established under reasonable regularity assumption, state that the first scheme achieves α(0)-order accuracy in time and fourth-order accuracy in space, while the second scheme attains second-order accuracy in time and fourth-order accuracy in space. Numerical experiments confirm the theoretical predictions and demonstrate the efficiency of the proposed methods.

在这项工作中，我们研究了具有变指数的二维扩散波方程，模拟了力学扩散波在具有空间变化性质的粘弹性介质中的传播。首先通过卷积法将扩散波模型转换为等效形式。然后应用两种时间离散化策略来近似变换方程中的每一项，得到基于空间紧致有限差分法的两个完全离散格式。为了减少计算量，采用了交替方向隐式（ADI）技术。证明了这两种ADI紧格式都是无条件稳定和收敛的。在合理的正则性假设下建立的误差估计表明，第一种方案在时间上达到α(0)阶精度，在空间上达到四阶精度；第二种方案在时间上达到二阶精度，在空间上达到四阶精度。数值实验验证了理论预测和方法的有效性。

引用次数: 0

A new technique of ADM to improve the precision and stability of numerical solutions 一种提高数值解精度和稳定性的ADM新技术

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-24 DOI: 10.1016/j.apnum.2025.12.005

Chun Wang

In this paper, a new technique of Adomian decomposition method (ADM) for the numerical solutions of nonlinear differential equations and mathematical models is presented. Different from the traditional ADM and LADM (Laplace Adomian decomposition method), the main idea of this new technique is to properly divide the interval where the problem is to be into several subintervals and then apply ADM on each subinterval, respectively. When applying ADM on the latter subinterval, we take the approximate solution on the former subinterval as the initial value so that we can get the numerical solutions at the nodes of these subintervals by calculating in turn. The new technique of ADM has higher precision than the original and traditional ADM and LADM. By using this technique, the accuracy of the numerical solutions is greatly improved. An obvious advantage of this technique is that the precision of the numerical solutions can be finely adjusted according to the actual needs. On the other hand, compared with the traditional method, the errors of the numerical solutions obtained by this technique have a very strong stability.

本文提出了一种求解非线性微分方程和数学模型数值解的新方法——Adomian分解法。与传统的拉普拉斯Adomian分解方法（ADM）和拉普拉斯Adomian分解方法（LADM）不同，该方法的主要思想是将问题所在的区间适当划分为若干个子区间，然后分别对每个子区间应用ADM。在对后一个子区间应用ADM时，我们将前一个子区间的近似解作为初始值，通过依次计算得到这两个子区间节点处的数值解。与传统的ADM和LADM相比，新技术具有更高的精度。采用这种方法，可以大大提高数值解的精度。这种方法的一个明显优点是，数值解的精度可以根据实际需要进行微调。另一方面，与传统方法相比，该方法得到的数值解的误差具有很强的稳定性。

引用次数: 0

Linear minimum-variance approximants for noisy data 噪声数据的线性最小方差近似

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-09 DOI: 10.1016/j.apnum.2025.12.002

Sergio López-Ureña, Dionisio F. Yáñez

Inspired by recent developments in subdivision schemes founded on the Weighted Least Squares technique, we construct linear approximants for noisy data in which the weighting strategy minimizes the output variance, thereby establishing a direct correspondence with the Generalized Least Squares and the Minimum-Variance Formulas methodologies. By introducing annihilation-operators for polynomial spaces, we derive usable formulas that are optimal for general correlated non-uniform noise. We show that earlier subdivision rules are optimal for uncorrelated non-uniform noise and, finally, we present numerical evidence to confirm that, in the correlated case, the proposed approximants are better than those currently used in the subdivision literature.

受基于加权最小二乘技术的细分方案的最新发展的启发，我们为噪声数据构建了线性近似，其中加权策略使输出方差最小化，从而与广义最小二乘和最小方差公式方法建立了直接对应关系。通过引入多项式空间的湮灭算子，我们推导出了适用于一般相关非均匀噪声的最优公式。我们证明了先前的细分规则对于不相关的非均匀噪声是最优的，最后，我们提供了数值证据来证实，在相关情况下，所提出的近似比目前在细分文献中使用的近似更好。

引用次数: 0

High-order orthogonal spline collocation schemes for two-dimensional nonlinear problems 二维非线性问题的高阶正交样条配置格式

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-12-06 DOI: 10.1016/j.apnum.2025.12.001

Meirong Cheng, Qimin Li, Leijie Qiao

To address the nonlinear control of transverse vibrations in a clamped square plate, we design and analyze an orthogonal spline collocation (OSC) scheme combined with a discrete-time approximation. Two new Crank–Nicolson (CN) OSC variants are introduced for temporal discretization. By applying a Taylor expansion to the nonlinear term, the original fourth-order nonlinear problem is transformed into a linear one, enabling efficient computation. The theoretical investigation is provided. Numerical experiments on several practical examples confirm the effectiveness of the schemes, achieving second-order temporal accuracy and optimal spatial convergence.

为了解决夹持方形板横向振动的非线性控制问题，我们设计并分析了一种结合离散时间近似的正交样条配置（OSC）方案。引入了两个新的Crank-Nicolson (CN) OSC变量进行时间离散化。通过对非线性项进行泰勒展开式，将原来的四阶非线性问题转化为线性问题，实现了高效的计算。并进行了理论研究。几个实例的数值实验验证了该方法的有效性，取得了二阶时间精度和最佳空间收敛性。

引用次数: 0

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