Applied Numerical Mathematics最新文献

英文中文

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

High-level convergence order accelerators of iterative methods for nonlinear problems 非线性问题迭代法的高收敛阶加速器

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-08 DOI: 10.1016/j.apnum.2025.07.003

Alicia Cordero , Renso V. Rojas-Hiciano , Juan R. Torregrosa , Maria P. Vassileva

We present an efficient strategy to increase, under certain conditions, the order of convergence of iterative methods to solve nonlinear systems of equations. We analytically compare the new accelerator with others and establish the conditions under which this technique is more efficient. We perform an analysis of the efficiency of some one-step accelerators that increase the convergence order by two units. New concepts about efficiency are introduced which allow us to compare different iterative schemes from other points of view. We demonstrate that our proposal is a good alternative to the existing ones. As a consequence, we propose two new maximally efficient, damped Newton-Traub type schemes of order 5 and 6. These are an improvement of two other maximally efficient methods. Their numerical performance is better than that of known methods of the same order, and we find that it is a very economical way to achieve high order. Some numerical examples confirm the theoretical results.

在一定条件下，我们提出了一种有效的策略来提高求解非线性方程组的迭代方法的收敛阶。通过与其他加速器的分析比较，确定了该技术更有效的条件。我们分析了一些将收敛阶提高两个单位的单步加速器的效率。介绍了有关效率的新概念，使我们能够从其他角度比较不同的迭代方案。我们证明我们的建议是现有建议的一个很好的替代方案。因此，我们提出了两种新的最有效的5阶和6阶阻尼Newton-Traub型格式。这是对另外两种效率最高的方法的改进。它们的数值性能优于已知的同阶方法，是实现高阶的一种非常经济的方法。一些数值算例证实了理论结果。

引用次数: 0

Low-storage exponentially fitted explicit Runge-Kutta methods 低存储指数拟合显式龙格-库塔方法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-05 DOI: 10.1016/j.apnum.2025.06.017

I. Higueras , J.I. Montijano , L. Rández

In this paper, we study explicit Runge-Kutta (RK) methods for solving high-dimensional systems of ordinary differential equations (ODEs), with oscillatory or periodic solutions, that can be implemented with a few memory registers. We will refer to these schemes as Low-Storage Exponentially Fitted explicit Runge-Kutta methods (LSEFRK).

In order to obtain them, we first study second-order and third-order low-storage (LS) schemes that can be implemented with two memory registers per step of the van der Houwen- and Williamson-type. Next, we construct optimal LSEFRK methods by imposing exponential fitting conditions along with accuracy and stability properties. In this way, new optimal three-stage third-order and five-stage fourth-order LSEFRK schemes are constructed for each type of LS method.

The performance of these new schemes is tested by solving some high-dimensional differential systems with periodic solutions. Comparison with other non-LS exponentially fitted and low-storage non-EF RK methods from the literature shows that the new LSEFRK schemes outperform the efficiency of RK methods that only satisfy either the LS or the EF condition.

本文研究了求解具有振荡或周期解的高维常微分方程系统的显式龙格-库塔（RK）方法，这种方法可以用少量内存寄存器实现。我们将这些方案称为低存储指数拟合显式龙格-库塔方法（LSEFRK）。为了获得它们，我们首先研究了二阶和三阶低存储（LS）方案，这些方案可以用van der Houwen和williamson型的每步两个存储寄存器来实现。接下来，我们通过施加指数拟合条件以及精度和稳定性来构造最优LSEFRK方法。在此基础上，针对每种LS方法构造了新的最优三阶和五阶LSEFRK方案。通过求解一些具有周期解的高维微分系统，验证了这些新格式的性能。与文献中其他非LS指数拟合和低存储非EFRK方法的比较表明，新的LSEFRK方案优于仅满足LS或EF条件的RK方法的效率。

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引用次数: 0

High-order structure-preserving approaches for constrained conservative or dissipative systems 约束保守或耗散系统的高阶结构保持方法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-05 DOI: 10.1016/j.apnum.2025.06.018

Jiaxiang Cai , Yushun Wang

We propose a class of high-order schemes preserving original conservation/dissipation energy law and constraints for the constrained conservative/dissipative system. These schemes are efficient, i.e., only require solving linear system with constant coefficients at each time step, plus an algebraic optimization problem which consumes negligible cost. The proposed schemes are applied to conservative semiclassical nonlinear Schrödinger equation, as well as dissipative three-component ternary Cahn-Hilliard phase-field model. Some numerical experiments are conducted to validate applicability, effectiveness and accuracy of the proposed schemes.

我们提出了一类保留原始守恒/耗散能量律和约束条件的高阶格式。这些方案是高效的，即在每个时间步只需要求解常系数线性系统，加上一个代数优化问题，成本可以忽略不计。将所提出的格式应用于保守的半经典非线性Schrödinger方程，以及耗散的三分量三元Cahn-Hilliard相场模型。通过数值实验验证了所提方案的适用性、有效性和准确性。

引用次数: 0

Piecewise logarithmic Chebyshev cardinal functions: Application for nonlinear integral equations with a logarithmic singular kernel 分段对数切比雪夫基数函数：具有对数奇异核的非线性积分方程的应用

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-04 DOI: 10.1016/j.apnum.2025.06.016

M.H. Heydari , D. Baleanu , M. Bayram

This study introduces a novel class of nonlinear integral equations with a logarithmic singular kernel. The existence and uniqueness of a solution to these equations are rigorously analyzed. To facilitate their solution, we construct the piecewise logarithmic Chebyshev cardinal functions (CCFs), a versatile family of basis functions. In this framework, an operational matrix for the Hadamard fractional integral is derived for the PLCCFs. By employing the connection between this type of logarithmic singularity and the Hadamard fractional integral operator, we develop a straightforward yet powerful numerical approach to solve these equations. In the proposed method, the solution is first approximated using a finite expansion of the piecewise logarithmic CCFs with unknown coefficients. Then, through interpolation and the application of the fractional integral operational matrix, the original integral equation is reformulated as a system of nonlinear algebraic equations, whose solution determines the expansion coefficients. The convergence analysis of the proposed scheme is examined through both theoretical and numerical investigations. The accuracy of the developed method is evaluated by solving some illustrative examples featuring both analytical and non-analytical solutions.

本文介绍了一类具有对数奇异核的非线性积分方程。严格地分析了这些方程解的存在唯一性。为了便于求解，我们构造了分段对数切比雪夫基数函数（CCFs），这是一种通用的基函数族。在此框架下，推导了PLCCFs的Hadamard分数阶积分的运算矩阵。通过利用这类对数奇点与Hadamard分数积分算子之间的联系，我们开发了一种简单而强大的数值方法来求解这些方程。在所提出的方法中，首先使用带未知系数的分段对数ccf的有限展开式来逼近解。然后，通过插值和分数阶积分运算矩阵的应用，将原积分方程重新表述为一个非线性代数方程组，其解决定展开系数。通过理论和数值研究验证了该方案的收敛性分析。通过求解一些具有解析解和非解析解的说明性实例，评价了所开发方法的准确性。

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引用次数: 0

Algebraic multigrid methods for uncertainty quantification of source-type flows through randomly heterogeneous porous media 随机非均质多孔介质中源型流动不确定性量化的代数多重网格方法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-07-03 DOI: 10.1016/j.apnum.2025.06.015

Vincenzo Schiano Di Cola , Salvatore Cuomo , Gerardo Severino , Marco Berardi

We consider steady flow generated by a source through a porous medium where, due to its erratic variations in the space, the conductivity K is regarded as a random field. As a consequence, flow variables become stochastic, and we aim at quantifying their uncertainty. To this purpose, we use Monte Carlo simulations, where for each realization the governing flow equation is solved by a finite volume method. This yields a deterministic linear system solved by algebraic multigrid (AMG) techniques. By leveraging analytical solutions valid for homogeneous (constant K) formations, we first compare different AMG solvers, that are subsequently used as trial in order to extend our approach to heterogeneous porous media. Results demonstrate that AMG methods enable achieving, especially at higher iteration counts, an

L_{2}

-error lower than other, Gaussian-type, approximations.

我们考虑由源通过多孔介质产生的稳定流，其中，由于其在空间中的不规则变化，电导率K被视为随机场。因此，流量变量是随机的，我们的目标是量化它们的不确定性。为此，我们使用蒙特卡罗模拟，其中每个实现的控制流方程都是用有限体积法求解的。这产生了一个由代数多重网格（AMG）技术求解的确定性线性系统。通过利用对均质（恒定K）地层有效的解析解，我们首先比较了不同的AMG求解器，随后将其用作试验，以便将我们的方法扩展到非均质多孔介质。结果表明，AMG方法能够实现比其他高斯型近似更低的l2误差，特别是在更高的迭代次数下。

引用次数: 0

A surface mesh DG-VEM for elliptic membrane shell model 椭圆膜壳模型的表面网格DG-VEM

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-06-30 DOI: 10.1016/j.apnum.2025.06.014

Qian Yang , Xiaoqin Shen , Jikun Zhao , Zhiming Gao

Elliptic membrane shell (EMS), characterized by a system with complex variable coefficients on a surface, poses significant challenges for numerical discretization. In this paper, leveraging the differing regularity of displacement components, we propose a discontinuous Galerkin virtual element method (DG-VEM) for the EMS model. Specifically, we construct

C^{0}

-continuous virtual element spaces for the first two components, whereas the third component is discretized on each element using a polynomial of degree l, with no continuity enforced across element boundaries. This method offers high mesh flexibility, eliminates the need for explicit basis function expressions, and improves accuracy to achieve convergence of any desired order. Furthermore, we establish the existence, uniqueness, stability, and convergence of the numerical solution, along with rigorous error estimates. Several numerical examples are presented to test the convergence and stability of the DG-VEM. Additionally, we demonstrate the method's adaptability to diverse grid subdivisions and show that, for comparable error levels, the DG-VEM for the EMS model requires significantly fewer degrees of freedom than traditional finite element methods.

椭圆膜壳（EMS）是一个表面上具有复杂变系数的系统，对其数值离散化提出了重大挑战。在本文中，我们利用位移分量的不同规律性，提出了一种不连续Galerkin虚元法（DG-VEM）。具体地说，我们为前两个分量构造了0连续的虚拟元素空间，而第三个分量在每个元素上使用1次多项式离散，没有强制跨元素边界的连续性。该方法提供了高度的网格灵活性，消除了显式基函数表达式的需要，并提高了精度，以实现任何期望的收敛顺序。进一步，我们建立了数值解的存在性、唯一性、稳定性和收敛性，并给出了严格的误差估计。通过数值算例验证了该算法的收敛性和稳定性。此外，我们证明了该方法对不同网格细分的适应性，并表明，对于可比的误差水平，EMS模型的DG-VEM比传统的有限元方法需要更少的自由度。

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引用次数: 0

Exponentially accurate spectral Monte Carlo method for linear PDEs and their error estimates 线性偏微分方程的指数精度谱蒙特卡罗方法及其误差估计

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-06-26 DOI: 10.1016/j.apnum.2025.06.010

Jiaying Feng, Changtao Sheng, Chenglong Xu

This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by α-stable Lévy processes with

α \in (0, 2)

, which was initially proposed and developed by Gobet and Maire in their pioneering works ((2004) [24], and (2005) [25]) for the case

α = 2

. The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-spheres method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both

α \in (0, 2)

and

α = 2

. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.

本文介绍了求解α∈(0,2)α-稳定lsamvy过程驱动的线性泊松方程和抛物方程的谱蒙特卡罗迭代法（SMC），该方法最初是由Gobet和Maire在他们的开创性著作(（2004)[24]和（2005）[25]）中提出和发展的，适用于α=2的情况。该方法有效地集成了多种计算技术，包括基于广义雅可比函数（GJFs）的插值、时空谱法、控制变量技术和一种新的球体行走法（WOS）。通过有限迭代，严格地建立了含分数阶拉普拉斯算子的泊松方程和抛物方程误差界的指数收敛性。值得注意的是，对于α∈(0,2)和α=2，所提出的抛物方程的空时谱蒙特卡罗方法（ST-SMC）是统一的。大量的数值结果证明了该方法的光谱精度和效率，从而验证了理论结果。

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引用次数: 0

Generalization error analysis of deep backward dynamic programming for solving nonlinear PDEs 求解非线性偏微分方程的深度后向动态规划泛化误差分析

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-06-26 DOI: 10.1016/j.apnum.2025.06.013

Du Ouyang, Jichang Xiao, Xiaoqun Wang

We explore the application of the quasi-Monte Carlo (QMC) method in deep backward dynamic programming (DBDP) [1] for numerically solving high-dimensional nonlinear partial differential equations (PDEs). Our study focuses on examining the generalization error as a component of the total error in the DBDP framework, discovering that the rate of convergence for the generalization error is influenced by the choice of sampling methods. Specifically, for a given batch size m, the generalization error under QMC methods exhibits a convergence rate of

O (m^{- 1 + ε})

, where

ε > 0

can be made arbitrarily small. This rate is notably more favorable than that of the traditional Monte Carlo (MC) methods, which is

O (m^{- 1 / 2 + ε})

. Our theoretical analysis shows that the generalization error under QMC methods achieves a higher order of convergence than their MC counterparts. Numerical experiments demonstrate that QMC indeed surpasses MC in delivering solutions that are both more precise and stable.

探讨了准蒙特卡罗（QMC）方法在深度后向动态规划（DBDP）[1]中的应用，用于数值求解高维非线性偏微分方程（PDEs）。我们的研究重点是检查泛化误差作为DBDP框架中总误差的组成部分，发现泛化误差的收敛速度受到采样方法选择的影响。具体而言，对于给定的批大小m， QMC方法下的泛化误差收敛速度为0 (m−1+ε)，其中ε>；0可以任意小。该速率明显优于传统的蒙特卡罗（MC）方法，即O（m−1/2+ε）。理论分析表明，QMC方法的泛化误差比MC方法具有更高的收敛阶。数值实验表明，QMC在提供更精确和稳定的解方面确实优于MC。

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引用次数: 0

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