Computers & Mathematics with Applications最新文献

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MS-PINN: A physics-informed neural network for multi-field coupled evolution modeling in metal solidification MS-PINN：用于金属凝固多场耦合演化建模的物理信息神经网络

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-20 DOI: 10.1016/j.camwa.2026.01.015

Chen Bai , Yunhu Zhang , Hongxing Zheng , Quan Qian

Simulating the metal solidification process is crucial for improving product quality, optimizing manufacturing processes, and developing new materials. Traditional numerical methods like the Finite Element Method (FEM) and Physics-Informed Neural Networks (PINNs) face significant challenges when applied to metal solidification simulations due to inefficiencies and inaccuracies in dealing with multiphysics coupling, nonlinearity, and spatio-temporal complexity. Despite their potential, PINNs require further optimization to accurately capture complex physical phenomena in practical simulations. In this study, we propose a novel method based on PINNs, termed MS-PINN, which integrates Fourier Feature Embedding (FFE), Residual-based Adaptive Resampling (RAD), and Self-adaptive Loss Balanced methods (SAL) to significantly enhance simulation accuracy and efficiency. FFE improves the model’s ability to capture high-frequency features, RAD increases learning efficiency in high-gradient regions, and SAL dynamically adjusts loss function weights to optimize the training process. Experimental results show that MS-PINN outperforms traditional PINNs and other advanced approaches, achieving average error reductions of approximately 81.00% compared to Conv-LSTM, 77.11% compared to TCN, and 61.56% compared to PINN in reconstruction experiments. In predictive experiments, MS-PINN reduces errors by 53.23%, 68.81%, and 72.54% compared to PINN, TCN, and CONV-LSTM methods, respectively. Additionally, we developed a general PDE-solving software, NeuroPDE, based on this method. NeuroPDE has demonstrated success not only in the solidification process of Cu-1wt.%Ag alloy but also in solving Burgers, diffusion, and Navier-Stokes (NS) equations, including turbulent datasets characterized by high Reynolds numbers, and their inverse problems. This highlights NeuroPDE’s versatility and broad applicability in solving complex forward and inverse problems in fluid dynamics and other fields.

模拟金属凝固过程对于提高产品质量、优化制造工艺和开发新材料至关重要。传统的数值方法，如有限元法（FEM）和物理信息神经网络（pinn），由于在处理多物理场耦合、非线性和时空复杂性方面的低效率和不准确性，在应用于金属凝固模拟时面临着重大挑战。尽管具有潜力，但在实际模拟中需要进一步优化才能准确捕获复杂的物理现象。在这项研究中，我们提出了一种基于pinn的新方法，称为MS-PINN，它集成了傅里叶特征嵌入（FFE），基于残差的自适应重采样（RAD）和自适应损失平衡方法（SAL），以显着提高仿真精度和效率。FFE提高了模型捕获高频特征的能力，RAD提高了高梯度区域的学习效率，SAL动态调整损失函数权值以优化训练过程。实验结果表明，MS-PINN优于传统的PINN和其他先进的方法，在重建实验中，与convl - lstm相比，MS-PINN的平均误差降低约81.00%，与TCN相比，MS-PINN的平均误差降低约77.11%，与PINN相比，MS-PINN的平均误差降低约61.56%。在预测实验中，MS-PINN比PINN、TCN和convl - lstm方法分别降低了53.23%、68.81%和72.54%的误差。此外，我们还基于该方法开发了通用的pde求解软件NeuroPDE。NeuroPDE不仅在Cu-1wt的凝固过程中取得了成功。%Ag合金，但也在解决汉堡，扩散，和Navier-Stokes （NS）方程，包括湍流数据集特征的高雷诺数，和他们的反问题。这突出了NeuroPDE在解决流体动力学和其他领域复杂的正反问题方面的多功能性和广泛适用性。

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引用次数: 0

Mean-square contractivity and convergence rate of stochastic theta Milstein schemes for non-autonomous SDEs with non-globally Lipschitz diffusion coefficients 具有非全局Lipschitz扩散系数的非自治SDEs随机theta Milstein格式的均方收缩性和收敛速度

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-20 DOI: 10.1016/j.camwa.2026.01.007

Jinran Yao, Zhengwei Yin, Luozhong Gong

This study is dedicated to investigating the mean-square contractivity and convergence rate of stochastic theta Milstein (STM) schemes for non-autonomous stochastic differential equations (SDEs) in the non-globally Lipschitz setting. More precisely, we begin by analyzing the mean-square contractivity of the numerical discretization for the STM schemes with θ ∈ [1/2, 1] under the coupled monotonicity condition along with the polynomial growth condition with respect to the state variable. Additionally, we employ an iterative strategy to deduce the strong convergence order of the STM schemes under non-globally Lipschitz conditions. It is demonstrated that the numerical approximations of the STM schemes with θ ∈ [1/2, 1] converges strongly with order one in the mean-square sense to the exact solution of the considered SDEs, under Assumptions 2.1 and 4.1 with respect to both the spatial and time variables. In the present study, the convergence analysis specifically addresses the case where

L \in R

in the coupled monotonicity condition, rather than L ≥ 0, as commonly presented in the literature. This analysis does not count upon a priori high-order moment estimates of the considered numerical schemes in establishing the convergence theorem. Finally, numerical experiments are performed to empirically substantiate the theoretical findings.

本文研究了非全局Lipschitz环境下非自治随机微分方程（SDEs）的随机theta Milstein （STM）格式的均方收缩性和收敛率。更准确地说，我们首先分析了关于状态变量的耦合单调性条件下，具有θ ∈ [1/2,1]的STM格式的数值离散化的均方收缩性。此外，我们采用迭代策略推导了非全局Lipschitz条件下STM格式的强收敛阶。证明了在假设2.1和4.1条件下，具有θ ∈ [1/2,1]的STM格式的数值逼近在均方意义上强收敛于所考虑的SDEs的空间和时间变量的精确解，且收敛速度为1阶。在本研究中，收敛性分析专门针对耦合单调性条件下L∈R的情况，而不是文献中常见的L ≥ 0。这种分析不依赖于在建立收敛定理时所考虑的数值格式的先验高阶矩估计。最后，通过数值实验对理论结果进行了实证验证。

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引用次数: 0

Unconditional uniform superconvergent error estimates of anisotropic nonconforming energy-dissipative SAV-CN FEM for fourth-order singular perturbation Bi-flux diffusion model 四阶奇异摄动双通量扩散模型各向异性非协调能量耗散SAV-CN有限元的无条件一致超收敛误差估计

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-20 DOI: 10.1016/j.camwa.2026.01.012

Dongyang Shi , Sihui Zhang

This investigation aims to delve into the unconditional uniform superconvergence behavior of the scalar auxiliary variable (SAV) Crank-Nicolson (CN) fully discrete scheme for the fourth-order singular perturbation Bi-flux diffusion equation with anisotropic constrained nonconforming rotated

Q_{1}

(

C N Q_{1}^{r o t}

) finite element method (FEM). Innovative high-accuracy estimates tailored specifically for the

C N Q_{1}^{r o t}

element on anisotropic meshes are established as the cornerstone for achieving superconvergent outcomes. Following this, a novel SAV-CN scheme is formulated, with the energy dissipation theoretically given to guarantee that the numerical solutions remain bounded in the broken H¹-norm. Then, by leveraging the aforementioned novel estimates and sidestepping the reliance on inverse inequalities, the unconditional uniform superclose result is rigorously derived, which transcends the constraints of the negative power of the perturbation parameter

A = \sqrt{β (1 - β) R}

and the limitations imposed by the ratio between the temporal step τ and spatial partition parameter h. Furthermore, an application of the economic interpolation post-processing technique enables the procurement of superconvergence estimates for the devised numerical schemes. Ultimately, the theoretical analysis is strengthened through the conduct of numerical experiments.

研究了四阶各向异性约束非协调旋转Q1 （CNQ1rot）有限元法四阶奇异摄动双通量扩散方程的标量辅助变量（SAV） Crank-Nicolson （CN）完全离散格式的无条件一致超收敛行为。在各向异性网格上为CNQ1rot元素量身定制的创新高精度估计是实现超收敛结果的基石。在此基础上，提出了一种新的SAV-CN格式，并在理论上给出了能量耗散，以保证数值解在h1 -范数破碎时保持有界。然后，利用上述新估计，避开对逆不等式的依赖，严格推导了无条件均匀超接近结果，该结果超越了扰动参数A=β(1−β)R的负幂约束和时间步长τ与空间分割参数h之比的限制。经济插值后处理技术的应用使所设计的数值格式获得了超收敛估计。最后，通过数值实验加强理论分析。

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引用次数: 0

Unconditionally convergent fourth-order compact difference methods for Quasi-linear convection-diffusion equations 拟线性对流扩散方程的无条件收敛四阶紧致差分方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.camwa.2026.01.018

Mengling Wu , Hongling Hu , Kejia Pan , Wei Wu

In this paper, we propose an unconditionally convergent implicit compact difference method for the quasi-linear convective diffusion equation. First, we use Taylor expansion combined with the truncation error residual correction method to discretize the first-order derivative in the temporal direction, while the first- and second-order derivatives in the spatial direction are approximated by the consistent fourth-order boundary schemes and the fourth-order difference method, respectively. We develop a two-level scheme with

O (τ^{2} + h^{4})

. Secondly, the convergence of the scheme in the L^∞ norm has been proven. To further improve computational efficiency, we apply Richardson extrapolation to improve the temporal accuracy of the scheme to fourth-order, i.e.,

O (τ^{4} + h^{4})

. Finally, we present numerical examples to verify the theoretical analysis and the efficiency of our proposed method.

本文给出了拟线性对流扩散方程的一种无条件收敛隐式紧差分法。首先，采用Taylor展开式结合截断误差残差校正方法对时间方向上的一阶导数进行离散化，而空间方向上的一阶导数和二阶导数分别采用一致的四阶边界格式和四阶差分法进行逼近。我们开发了一个具有O（τ2+h4）的两能级格式。其次，证明了该方案在L∞范数上的收敛性。为了进一步提高计算效率，我们采用Richardson外推将格式的时间精度提高到四阶，即O（τ4+h4）。最后，通过数值算例验证了理论分析和所提方法的有效性。

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引用次数: 0

A corrected Alikhanov scheme for a subdiffusion equation 亚扩散方程的修正Alikhanov格式

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.camwa.2026.01.014

Chaobao Huang , Yujie Yu , Na An , Hu Chen

This paper considers the subdiffusion equation with a weakly singular solution. To achieve the optimal accuracy with a smaller grading parameter r than that of the standard Alikhanov scheme, it’s essential to correct the Alikhanov scheme and the approximation

v^{n - θ} \approx θ v^{n} + (1 - θ) v^{n - 1}

(0 < θ < 1) by investigating the corrected terms

β_{n}^{σ} (v^{1} - v^{0})

and

μ_{n}^{σ} (v^{1} - v^{0})

, respectively. After that, the truncation error of the corrected Alikhanov scheme and the corrected approximation of

v^{n - θ}

are given. By adopting the corrected Alikhanov scheme for the Caputo derivative and the corrected approximation for

v^{n - θ}

, we construct a fully discrete scheme for the subdiffusion equation, employing a standard finite element method in space. Furthermore, the stability analysis and the optimal convergent analysis for the proposed scheme are investigated. Finally, numerical experiments are conducted to verify the theoretical results.

研究一类具有弱奇异解的次扩散方程。为了获得比标准Alikhanov格式更小的分级参数r的最优精度，必须对Alikhanov格式和近似vn−θ≈θvn+(1−θ)vn−1（0 <； θ <； 1）分别通过研究校正项βnσ(v1−v0)和μnσ(v1−v0)进行校正。然后给出了修正后的Alikhanov格式的截断误差和修正后的vn−θ近似值。采用Caputo导数的修正Alikhanov格式和vn−θ的修正近似，利用空间标准有限元方法构造了亚扩散方程的完全离散格式。此外，还研究了该方案的稳定性分析和最优收敛性分析。最后通过数值实验对理论结果进行了验证。

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引用次数: 0

Mixed finite element method for a system of hemivariational inequalities of Navier-Stokes equations coupled with the heat equation Navier-Stokes方程与热方程耦合的半变分不等式系统的混合有限元方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.camwa.2026.01.013

Hailing Xuan , Wensi Wang , Xiaoliang Cheng

This paper explores a mixed finite element method for a system of hemivariational inequalities arising from the stationary Navier-Stokes equations coupled with the heat equation. The hemivariational inequalities system models the motion of a viscous incompressible fluid with thermal effects, where both the boundary conditions for the velocity and temperature fields involve the generalized Clarke gradient. We discuss the equivalence between different variational formulations and establish an existence theorem based on an alternative iterative approach and some results on hemivariational inequalities. The Navier-Stokes hemivariational inequalities system is solved using the mixed finite element method, and corresponding error estimates are derived. Numerical outcomes are presented based on the Uzawa method, which illustrates the optimal convergence rate as indicated by the error analysis.

本文探讨了求解由稳态Navier-Stokes方程与热方程耦合引起的半变分不等式系统的混合有限元方法。半变分不等式系统模拟了具有热效应的粘性不可压缩流体的运动，其中速度场和温度场的边界条件都涉及广义Clarke梯度。本文讨论了不同变分形式之间的等价性，并基于一种可选迭代方法和关于半变分不等式的一些结果，建立了一个存在性定理。采用混合有限元法求解了Navier-Stokes半变不等式系统，并给出了相应的误差估计。给出了基于Uzawa方法的数值结果，并通过误差分析说明了该方法的最优收敛速度。

引用次数: 0

A hybrid MR-WENO scheme with a simplified troubled-cell indicator for hyperbolic conservation laws 双曲型守恒律的带有简化故障单元指示器的MR-WENO混合方案

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.camwa.2026.01.017

Jinming Zhang , Zhanjing Tao , Jun Zhu , Jianxian Qiu

In this paper, we propose a finite difference hybrid weighted essentially non-oscillatory (WENO) scheme to solve the hyperbolic conservation laws. A simplified troubled-cell indicator is designed for the hybrid scheme. The multi-resolution WENO (MR-WENO) reconstruction is applied in the troubled-cells, while a simple linear reconstruction is used in the remaining regions. The new hybrid scheme inherits the excellent characteristics of the original MR-WENO scheme [1], and can reduce the expensive computational cost of the WENO reconstruction. Compared to the previous hybrid WENO scheme [2] which used the high-degree polynomial in the troubled-cell indicator, the new scheme can reduce the percentage of the troubled-cells, leading to higher computational efficiency. Moreover, our scheme can effectively identify the troubled-cells, and has better resolution for certain problems than the previous scheme. Extensive numerical examples demonstrate the accuracy, efficiency and high resolution of the proposed method.

本文提出了一种有限差分混合加权本质非振荡（WENO）格式来求解双曲守恒律。针对混合方案，设计了一种简化的故障单元指示器。故障单元采用多分辨率WENO （MR-WENO）重建，其余区域采用简单的线性重建。新的混合方案继承了原MR-WENO方案[1]的优良特性，降低了WENO重构的计算成本。与以往在故障单元指标中使用高次多项式的混合WENO方案[2]相比，新方案可以减少故障单元的百分比，从而提高计算效率。此外，该方案能有效地识别出故障单元，对某些问题的解决效果优于原有方案。大量的数值算例证明了该方法的精度、效率和高分辨率。

引用次数: 0

Symmetric direct discontinuous Galerkin method for the biharmonic equation with non-homogeneous boundary condition 非齐次边界条件双调和方程的对称直接不连续Galerkin方法

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-19 DOI: 10.1016/j.camwa.2026.01.010

Hongying Huang , Huanhuan Wang , Lin Zhang

The direct discontinuous Galerkin method is proposed and analyzed for the biharmonic equation with non-homogeneous boundary conditions. Numerical fluxes on all edges of arbitrary triangles are defined, which are concerned with four parameters (β₁, β₂, β₃, β₄), and then the weak variational form of the original problem is derived. The corresponding bilinear form is proved to be bounded and coercive for sufficiently large values of the parameters β₁ and β₂. The well-posedness of the variational problem in finite element space

V_{h}^{k}

and optimal error estimates of the approximate solution under L²-norm and energy norm are obtained. The accuracy and reliability of the proposed method were verified through numerical experiments. Although the theoretical analysis requires the solution u ∈ H⁴, numerical results indicate that the method remains effective for solutions with low regularity.

提出并分析了具有非齐次边界条件的双调和方程的直接不连续伽辽金方法。定义了任意三角形所有边上与四个参数（β1、β2、β3、β4）有关的数值通量，导出了原问题的弱变分形式。当参数β1和β2的值足够大时，证明了相应的双线性形式是有界的和强制的。得到了有限元空间Vhk中变分问题的适定性，以及在l2范数和能量范数下近似解的最优误差估计。通过数值实验验证了该方法的准确性和可靠性。虽然理论分析需要解u ∈ H4，但数值结果表明，该方法对于低正则性解仍然有效。

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引用次数: 0

Strong convergence analysis of time discretization for stochastic nonlinear diffusion-wave equations driven by fractional Brownian motion 分数阶布朗运动驱动随机非线性扩散波方程时间离散化的强收敛性分析

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-17 DOI: 10.1016/j.camwa.2026.01.011

Xing Liu , Yumeng Yang

This paper constructs a discretization for the stochastic nonlinear diffusion-wave equation involving the Caputo fractional derivative of order α ∈ (1, 2), driven by fractional Brownian motion with Hurst index 0 < H < 1. For time discretization, we propose a quadrature involving Mittag-Leffler functions

E_{β, η} (- t)

. The discretization method combines the integral representation of the solution, the approximation of Mittag-Leffler functions and numerical integration techniques. Two approximation methods for the Mittag-Leffler functions are developed to enhance computational efficiency. The mean square strong convergence order is established by utilizing the confirmed solution regularity. Numerical examples are presented to validate the theoretical results.

本文构造了包含阶α ∈ （1,2）的Caputo分数阶导数的随机非线性扩散波方程的离散化，该方程由Hurst指数为0 <； H <； 1的分数阶布朗运动驱动。对于时间离散，我们提出了一个涉及Mittag-Leffler函数Eβ，η(−t)的正交。离散化方法结合了解的积分表示、Mittag-Leffler函数的逼近和数值积分技术。为了提高计算效率，提出了两种逼近Mittag-Leffler函数的方法。利用确定的解正则性，建立了均方强收敛阶。数值算例验证了理论结果。

引用次数: 0

Linear relaxation schemes with asymptotically compatible energy law for time-fractional phase-field models 具有渐近相容能量律的时间分数相场模型线性松弛方案

IF 2.5 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2026-01-16 DOI: 10.1016/j.camwa.2026.01.008

Hui Yu , Zhaoyang Wang , Ping Lin

In this paper, we propose a variable time-step linear relaxation scheme for time-fractional phase-field equations with a free energy density in general polynomial form. The

L 1^{+}

-CN formula is used to discretize the fractional derivative, and an auxiliary variable is introduced to approximate the nonlinear term by directly solving algebraic equations rather than differential-algebraic equations as in the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. The developed semi-discrete scheme is second-order accurate in time, and the inconsistency between the auxiliary and the original variables does not deteriorate over time. Furthermore, we take the time-fractional volume-conserved Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional Swift-Hohenberg equation as examples to demonstrate that the constructed schemes are energy stable and that the discrete energy dissipation law is asymptotically compatible with the classical one when the fractional-order parameter

α \to 1^{-}

. Several numerical examples demonstrate the effectiveness of the proposed scheme. In particular, numerical results confirm that the auxiliary variable remains well aligned with the original variable, and the error between them does not continue to increase over time before the system reaches steady state.

本文提出了具有一般多项式形式的自由能密度的时间分数相场方程的变时间步长线性松弛格式。采用L1+-CN公式对分数阶导数进行离散化，并引入辅助变量直接求解代数方程来逼近非线性项，而不是像不变能量二次化（IEQ）和标量辅助变量（SAV）方法那样求解微分代数方程。所开发的半离散格式在时间上具有二阶精度，并且辅助变量与原始变量之间的不一致性不随时间的推移而恶化。进一步以时间分数阶体积守恒的Allen-Cahn方程、时间分数阶Cahn-Hilliard方程和时间分数阶Swift-Hohenberg方程为例，证明了当分数阶参数α→1−时，所构造的格式是能量稳定的，且离散能量耗散律与经典能量耗散律渐近相容。算例验证了该方法的有效性。特别是，数值结果证实了辅助变量与原始变量保持良好的对齐，并且在系统达到稳态之前，它们之间的误差不会随着时间的推移而继续增加。

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引用次数: 0

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