Applied Numerical Mathematics最新文献

英文中文

Weak Galerkin spectral element methods for elliptic eigenvalue problems: Lower bound approximation and superconvergence 椭圆型特征值问题的弱Galerkin谱元方法：下界逼近和超收敛

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-22 DOI: 10.1016/j.apnum.2025.07.010

Jiajia Pan , Huiyuan Li

Lower bound approximation and super-convergence of the weak Galerkin spectral element method for second-order elliptic eigenvalue problems are comprehensively investigated in this paper. At first, we establish the approximation spaces with diverse polynomial degrees of weak functions and weak gradients by using the one-to-one mapping from the reference element to each physical element. General weak Galerkin triangular/quadrilateral spectral element approximation schemes are then proposed for the eigenvalue problem of the second-order elliptic operators. A study on the well-posedness of our schemes is carried out, resulting in the constraint conditions on the polynomial degrees of the discrete weak function space and the discrete weak gradient space. Further, qualitative numerical analysis and numerical investigation are performed on a series of polynomial degree configurations for the weak function space and the weak gradient space. We obtain in the sequel the super-convergence of the numerical eigenvalues with the weak Galerkin spectral element methods for the first time, and discover some lower bound approximation scenario that has never been reported before in literature.

本文全面研究了二阶椭圆型特征值问题的弱Galerkin谱元法的下界逼近和超收敛性。首先，利用参考元素到各物理元素的一对一映射，建立了弱函数和弱梯度具有不同多项式次的近似空间；针对二阶椭圆算子的特征值问题，提出了一般的弱Galerkin三角/四边形谱元逼近格式。研究了这些格式的适定性，得到了离散弱函数空间和离散弱梯度空间的多项式次的约束条件。在此基础上，对弱函数空间和弱梯度空间的一系列多项式次构型进行了定性数值分析和数值研究。本文首次用弱伽辽金谱元方法得到了数值特征值的超收敛性，并发现了一些文献中从未报道过的下界近似情形。

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引用次数: 0

Robust globally divergence-free weak Galerkin variational data assimilation method for convection-dominated Oseen equations 对流占优Oseen方程的鲁棒全局无发散弱Galerkin变分数据同化方法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-22 DOI: 10.1016/j.apnum.2025.07.011

Xian Zhang, Ya Min, Minfu Feng

This paper presents a weak Galerkin (WG) finite element method based on the variational approach for data assimilation of the unsteady convection-dominated Oseen equation. The WG scheme uses piecewise polynomials of degrees k(

k \geq 1

) and

k - 1

respectively for the approximations of the velocity and the pressure in the interior of elements, and uses piecewise polynomials of degree k for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and initial value. It is proved that the velocity error in the

L^{2}

-norm has a Reynolds-robust error bound with quasi-optimal convergence order

k + 1 / 2

in the convection-dominated region. To solve the discrete optimality system efficiently, the conjugate gradient iterative algorithm is developed, which also preserves the globally divergence-free property of WG scheme. Numerical experiments are provided to verify the obtained theoretical results.

本文提出了一种基于变分法的弱伽辽金（WG）有限元方法，用于非定常对流占优Oseen方程的数据同化。WG方案分别采用k（k≥1）阶分段多项式和k−1阶分段多项式逼近单元内部的速度和压力，采用k阶分段多项式逼近单元界面上的速度和压力的数值轨迹。该方法可以得到速度和初始值的全局无发散近似。证明了l2范数的速度误差在对流主导区域具有拟最优收敛阶为k+1/2的reynolds -鲁棒误差界。为了有效地求解离散最优性系统，提出了共轭梯度迭代算法，该算法保持了WG格式的全局无发散性。数值实验验证了所得理论结果。

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引用次数: 0

FEM approximation of dynamic contact problem for fracture under fluid volume control using generalized HHT-α and semi-smooth Newton methods 流体体积控制下裂缝动态接触问题的广义HHT-α和半光滑牛顿法有限元逼近

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-22 DOI: 10.1016/j.apnum.2025.07.009

Victor A. Kovtunenko , Yves Renard

A class of elastodynamic contact problems for fluid-driven cracks stemming from hydro-fracking application is considered in the framework of finite element approximation. The dynamic contact problem aims at finding a non-negative fracture opening and a mean fluid pressure which are controlled by the volume of pumped fracturing fluid. Well-posedness of the fully discrete variational problem is proved rigorously by using the Lagrange multiplier and penalty methods for the minimization problem subjected to both: unilateral and non-local constraints. Numerical solution of the dynamic nonlinear equation is computed in 2D experiments using the semi-smooth Newton and the generalized Hilber–Hughes–Taylor α-method.

在有限元逼近的框架下，研究了水力压裂过程中流体驱动裂纹的弹动力接触问题。动态接触问题的目的是找到一个非负的裂缝开口和平均流体压力，这是由泵送压裂液的体积控制的。利用拉格朗日乘子和惩罚方法，对单边约束和非局部约束下的最小化问题进行了严格证明，证明了完全离散变分问题的适定性。利用半光滑牛顿法和广义Hilber-Hughes-Taylor α-法在二维实验中计算了动态非线性方程的数值解。

引用次数: 0

H(div)-conforming IPDG FEM with pointwise divergence-free velocity field for the micropolar Navier-Stokes equations 微极Navier-Stokes方程的H（div）型无点发散速度场IPDG有限元

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-18 DOI: 10.1016/j.apnum.2025.07.007

Xinran Huang, Haiyan Su, Xinlong Feng

The mass-conservative finite element method (FEM) is considered for the micropolar Navier-Stokes equations (MNSE), which couple the Navier-Stokes equations (NSE) with the angular momentum equation. A fully divergence-free algorithm is proposed for the MNSE. The Raviart-Thomas element is employed for discretizing the velocity field, ensuring that its divergence-free property is maintained. Furthermore, the interior penalty discontinuous Galerkin (IPDG) method is utilized in order to guarantee the

H^{1}

-continuity of velocity. Some implicit-explicit treatments are used to address the convection terms. We also provide energy stability proof and pressure robust error estimation for the proposed scheme. Finally, the accuracy and effectiveness of the proposed algorithm are validated through several 2D/3D numerical experiments.

考虑将Navier-Stokes方程（NSE）与角动量方程耦合的微极Navier-Stokes方程（MNSE）的质量守恒有限元法。提出了一种完全无发散的MNSE算法。采用Raviart-Thomas单元对速度场进行离散，保证了速度场的无散度特性。为了保证速度的h1连续性，采用了内罚不连续伽辽金（IPDG）方法。采用隐式显式处理来处理对流项。我们还提供了能量稳定性证明和压力鲁棒误差估计。最后，通过若干2D/3D数值实验验证了该算法的准确性和有效性。

引用次数: 0

A novel amplifying methodology in Gauss-Legendre IRK integrations to cope with high-frequency stiff problems 高斯-勒让德IRK积分中处理高频刚性问题的一种新的放大方法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-17 DOI: 10.1016/j.apnum.2025.07.006

Sanaz Hami Hassan Kiyadeh , Hosein Saadat , Ramin Goudarzi Karim , Ali Safaie , Fayyaz Khodadosti

This work presents a new amplification methodology based on the widely used Gauss-Legendre implicit Runge-Kutta integrations by addressing the phase lag and amplification factor. The novel methodology focuses on these two elements, which are the complex amplifiers associated with the GLIRK integrations.

To enhance the amplifier capabilities of the GLIRK integrations, we introduce two novel equations that clarify the relationships between the amplification factor and phase lag. This paper culminates in the improvement of two well-defined GLIRK integrations, each carefully designed to eliminate both the phase lag and the amplification factor in practical applications. The examination of absolute stability regions in the complex plane, as well as stability regions in the z-v plane, is relevant to the new GLIRK integrations presented.

To satisfy the admissibility of the new methodology, we establish a competitive environment alongside the classical GLIRK integration.

This competitive space includes numerical examples that demonstrate the low cost of the new amplified GLIRK integrations in addressing stiff problems with high frequency. Ultimately, this cost-effectiveness and superiority become increasingly evident as the frequency of the stiff problems increases.

本文提出了一种新的放大方法，基于广泛使用的高斯-勒让德隐式龙格-库塔积分，通过解决相位滞后和放大因子。新颖的方法侧重于这两个元素，即与GLIRK集成相关的复杂放大器。为了提高GLIRK集成电路的放大能力，我们引入了两个新的方程来阐明放大因子和相位滞后之间的关系。本文最终改进了两个定义良好的GLIRK集成，每个集成都经过精心设计，以消除实际应用中的相位滞后和放大因子。复平面上的绝对稳定区域以及z-v平面上的稳定区域的检验与提出的新的GLIRK积分有关。为了满足新方法的可接受性，我们在经典GLIRK集成的基础上建立了一个竞争环境。这个竞争空间包括数值例子，证明了新的放大GLIRK集成在解决高频棘手问题方面的低成本。最终，随着棘手问题出现的频率增加，这种成本效益和优越性变得越来越明显。

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引用次数: 0

Explicit solution of Lane-Emden type equations via a novel recurrence and Padé approximation approach 用一种新的递归和pad<s:1>近似方法显式解Lane-Emden型方程

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-16 DOI: 10.1016/j.apnum.2025.07.008

Sita Charkrit

This article introduces a novel recursive algorithm for obtaining explicit solutions to initial value problems of Lane-Emden type equations. By combining the traditional power series method with Adomian polynomials, expressed in terms of solution coefficients, the algorithm achieves high accuracy and converges rapidly to the exact solution within only a few iterations. This formulation not only simplifies the solution process but also improves computational efficiency over several existing semi-analytical approaches by requiring fewer iterations to reach a desired level of accuracy. Additionally, the Padé approximation is applied to the power series solution to accelerate convergence and expand the convergence region, allowing the solution to remain accurate over a wider interval. Error analysis using absolute and residual errors confirms that the proposed method, both independently and in combination with Padé approximants, outperforms existing methods in terms of precision and applicability. Several examples illustrate the method’s accuracy, efficiency, and reliability in solving nonlinear singular initial value problems.

本文介绍了一种求解Lane-Emden型方程初值问题显式解的递归算法。该算法将传统的幂级数法与用解系数表示的Adomian多项式相结合，具有较高的精度，只需几次迭代即可快速收敛到精确解。该公式不仅简化了求解过程，而且通过需要更少的迭代来达到所需的精度水平，从而提高了几种现有半分析方法的计算效率。此外，将pad近似应用于幂级数解以加速收敛并扩展收敛区域，使解在更宽的区间内保持精确。使用绝对误差和残差进行误差分析，证实了所提出的方法，无论是单独使用还是与pad近似器结合使用，在精度和适用性方面都优于现有方法。算例说明了该方法在求解非线性奇异初值问题中的准确性、有效性和可靠性。

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引用次数: 0

A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network 具有浅层神经网络的三阶有限差分加权本质非振荡格式

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-14 DOI: 10.1016/j.apnum.2025.07.005

Kwanghyuk Park , Xinjuan Chen , Dongjin Lee , Jiaxi Gu , Jae-Hun Jung

In this work, we develop the finite difference weighted essentially non-oscillatory (WENO) scheme based on the neural network for hyperbolic conservation laws. Supervised learning is employed with the training data consisting of three-point stencils and the corresponding WENO3-JS weights as labels. We design two loss functions, one built on the mean squared error and the other from the mean squared logarithmic error. Each loss function consists of two components, where the first enforces the model to maintain the essentially non-oscillatory behavior while the second reduces the dissipation around discontinuities and improves the performance in smooth regions. We choose the shallow neural network (SNN) for computational efficiency with the Delta layer pre-processing the input. The resulting WENO3-SNN schemes outperform the classical WENO3-JS and WENO3-Z in one-dimensional examples, and show comparable sometimes superior simulations to WENO3-JS and WENO3-Z in two-dimensional examples.

在这项工作中，我们开发了基于神经网络的双曲守恒律的有限差分加权本质非振荡（WENO）格式。采用监督学习，训练数据由三点模板组成，并以相应的WENO3-JS权重作为标签。我们设计了两个损失函数，一个基于均方误差，另一个基于均方对数误差。每个损失函数由两个部分组成，其中第一个部分强制模型保持本质上的非振荡行为，而第二个部分减少了不连续点周围的耗散并提高了平滑区域的性能。为了提高计算效率，我们选择了浅层神经网络（SNN），并使用Delta层对输入进行预处理。所得WENO3-SNN方案在一维示例中优于经典WENO3-JS和WENO3-Z，并且在二维示例中表现出与WENO3-JS和WENO3-Z相当的性能。

引用次数: 0

Error estimates of a two-grid BDF2 virtual element scheme for semilinear parabolic equation 半线性抛物方程双网格BDF2虚元格式的误差估计

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-10 DOI: 10.1016/j.apnum.2025.07.002

Peixuan Wu, Xiaohui Wu, Yang Wang, Ruqing Wang

In this article, we present a new two-grid discretization for the approximation of semilinear parabolic equation found on virtual element method (VEM). The two-step backward differentiation formula (BDF2) is comtemplated in the time dimension, while the VEM is utilized in spatial dimension. The two-grid VEM primarily computes the numerical solution

W_{H}^{n}

from solving a nonlinear system on a coarse mesh with size H and then gets the numerical solution

W_{h}^{n}

to a linear system built by the earlier result

W_{H}^{n}

on a fine mesh with size h (

h ≪ H

). Consequently, our proposed scheme not only reduces total computational expense, but also achieves same accuracy as the single-grid VEM. The convergence analysis in

L^{2}

and semi-

H^{1}

norm for both the VEM and the two-grid VEM methods are provided concretely.

本文提出了一种新的用虚元法求解半线性抛物方程的两网格离散化方法。在时间维度上考虑了两步后向微分公式（BDF2），而在空间维度上采用了VEM。双网格VEM主要通过在尺寸为H的粗网格上求解非线性系统得到数值解WHn，然后在尺寸为H （H≪H）的细网格上根据先前的结果得到线性系统的数值解WHn。因此，我们的方案不仅减少了总计算费用，而且达到了与单网格VEM相同的精度。具体给出了VEM方法和两网格VEM方法在L2范数和半h1范数上的收敛性分析。

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引用次数: 0

A linear-decoupled and unconditionally energy stable fully discrete scheme for Peterlin viscoelastic model Peterlin粘弹性模型的线性解耦无条件能量稳定全离散格式

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-09 DOI: 10.1016/j.apnum.2025.07.001

Qi Wang , Kun Wang , Guanyu Zhou

In this paper, we design a linear-decoupled and unconditionally energy stable scheme utilizing the ZEC (“zero-energy-contribution”) technique for the diffusion Peterlin viscoelastic model. This model includes a diffusion term with an arbitrary small diffusion coefficient ε for the conformation tensor C. A specific ODE is introduced to deal with the nonlinear coupling terms for velocity u and C satisfying the ZEC property. We approximate the coupled nonlinear terms using the previous time-step results while still maintaining energy stability, allowing us to solve a linear-decoupled system at each time-step. Moreover, each component of C can be computed in parallel. We prove the unique solvability and energy stability of the fully discrete scheme. Additionally, we derive an error bound

C (τ + h^{2})

for the P2/P1/P2 element, where the constant

C

is not explicitly dependent on the reciprocal of ε. Several numerical experiments are presented to demonstrate the accuracy and performance of the proposed scheme. Comparison with a linear-decoupled scheme excluding the ZEC technique indicates that the proposed algorithm offers superior stability and performance.

在本文中，我们利用ZEC（“零能量贡献”）技术为扩散Peterlin粘弹性模型设计了一个线性解耦的无条件能量稳定方案。该模型包含了构象张量C具有任意小扩散系数ε的扩散项，并引入了一种特殊的ODE来处理速度u和C满足ZEC性质的非线性耦合项。我们使用之前的时间步长结果近似耦合非线性项，同时仍然保持能量稳定性，允许我们在每个时间步长解线性解耦系统。此外，C的每个组成部分都可以并行计算。证明了全离散格式的唯一可解性和能量稳定性。此外，我们导出了P2/P1/P2元素的误差界C(τ+h2)，其中常数C不显式依赖于ε的倒数。几个数值实验证明了该方法的准确性和性能。与不含ZEC技术的线性解耦方案的比较表明，该算法具有更好的稳定性和性能。

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引用次数: 0

A higher-order solver for the FitzHugh-Nagumo equation by combining nonstandard and compact finite difference scheme 结合非标准与紧致有限差分格式求解FitzHugh-Nagumo方程的高阶解

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-09 DOI: 10.1016/j.apnum.2025.07.004

Zhi-Chen Li , Yang-Wen Yu , Xiao-Yu Zhang , Qing Fang

This study presents a novel numerical method for solving the FitzHugh-Nagumo equation by combining nonstandard finite difference (NSFD) and high-order compact finite difference schemes. Through rigorous mathematical analysis, we demonstrate the stability and convergence of our approach, revealing that instability arises only under extremely rare conditions. To verify the efficiency of our scheme, we calculated the

l_{2}

and

l_{\infty}

errors as well as the convergence rate by comparing the numerical results with the exact solution. Experiments show that our combined scheme not only ensures stability, but also possesses the lowest error while maintaining high order convergence.

本文提出了一种结合非标准有限差分格式和高阶紧致有限差分格式求解FitzHugh-Nagumo方程的新数值方法。通过严格的数学分析，我们证明了我们的方法的稳定性和收敛性，揭示了不稳定性只在极其罕见的条件下出现。为了验证该方案的有效性，我们通过将数值结果与精确解进行比较，计算了l2和l∞误差以及收敛速度。实验表明，该组合方案在保证稳定性的同时，在保持高阶收敛性的同时，误差最小。

引用次数: 0

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