Applied Numerical Mathematics最新文献

英文中文

High-order numerical solution for solving multi-dimensional Schrödinger-Poisson equation 求解多维Schrödinger-Poisson方程的高阶数值解

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-11-01 Epub Date: 2025-05-27 DOI: 10.1016/j.apnum.2025.05.004

Maedeh Nemati, Mostafa Abbaszadeh, Mehdi Dehghan

This paper explores the numerical solution of the Schrödinger-Poisson equation in one, two, and three dimensions, which has significant applications in quantum mechanics, cosmology, Bose-Einstein condensates, and nonlinear optics. To address the nonlinear aspects of the problem, we employ the split-step method, which decomposes the equation into linear and nonlinear components. The linear part is discretized using the compact finite difference (CFD) method, while the nonlinear component is solved exactly. For temporal discretization, we utilize the Crank-Nicolson method across all dimensions, achieving a second-order convergence rate of

O (τ^{2})

. Spatial discretization is carried out using a CFD scheme, ensuring a fourth-order convergence rate of

O (h^{4})

. In the case of two- and three-dimensional Schrödinger equations, the alternating direction implicit (ADI) method is applied. We establish that the proposed numerical schemes are convergent, unconditionally stable, and maintain the conservation of mass and energy at the discrete level. Numerical experiments in one, two, and three dimensions validate the effectiveness of our approach. Specifically, we compare the split-step CFD scheme with alternative methods, and for higher-dimensional cases, we evaluate the ADI-split-step CFD scheme against the standard split-step CFD method. The results demonstrate that the proposed methods significantly reduce computational time while maintaining accuracy.

本文探讨了一维、二维和三维Schrödinger-Poisson方程的数值解，该方程在量子力学、宇宙学、玻色-爱因斯坦凝聚和非线性光学中具有重要的应用。为了解决问题的非线性方面，我们采用了分步法，将方程分解为线性和非线性分量。采用紧凑有限差分（CFD）方法对线性部分进行离散，对非线性部分进行精确求解。对于时间离散化，我们在所有维度上使用Crank-Nicolson方法，实现O（τ2）的二阶收敛率。采用CFD格式进行空间离散化，保证四阶收敛速率为0 （h4）。在二维和三维Schrödinger方程的情况下，采用交替方向隐式（ADI）方法。我们证明了所提出的数值格式是收敛的，无条件稳定的，并且在离散水平上保持质量和能量的守恒。一维、二维和三维的数值实验验证了该方法的有效性。具体来说，我们将分步CFD方案与其他方法进行了比较，对于高维情况，我们将adi -分步CFD方案与标准分步CFD方法进行了比较。结果表明，该方法在保持精度的前提下显著减少了计算时间。

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引用次数: 0

Numerical simulation of a dynamic human capital model with demographic delays via the local discrete Galerkin method 基于局部离散伽辽金方法的人口统计学时滞动态人力资本模型数值模拟

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-11-01 Epub Date: 2025-06-20 DOI: 10.1016/j.apnum.2025.06.007

Yadollah Ordokhani , Alireza Hosseinian , Pouria Assari

A strong and dynamic economy depends on various factors, with human capital playing a crucial role in fostering resilience and adaptability in an ever-changing world. Human capital, which depends on the past behavior of the system, requires strategic investments in education, health, and skill development. This study presents a numerical approach for solving the human capital model with age-structured delays, formulated as integro-differential equations with double delays and difference kernels. The proposed method employs a local meshless discrete Galerkin approach based on the moving least squares (MLS) technique, which can work with irregular or non-uniform data. The localized nature of the MLS scheme enhances computational efficiency by focusing on small neighborhoods. Moreover, the stabilized MLS framework, achieved by using shifted and scaled polynomial basis functions, enhances numerical stability and reduces sensitivity to the distribution of nodes, thereby transferring these advantageous properties to the method. The simplicity of the proposed algorithm makes it easy to implement on standard personal computers and extend to a wider class of delay integro-differential equations. To assess its reliability, we analyzed its error and determined the convergence order of the presented method. We have applied it to solve several numerical examples, and the obtained results confirm the method's accuracy, stability, and alignment with theoretical findings.

一个强大而充满活力的经济取决于各种因素，而人力资本在培养适应不断变化的世界的韧性和适应性方面发挥着至关重要的作用。人力资本取决于系统过去的行为，需要在教育、卫生和技能发展方面进行战略投资。本文提出了一种求解具有年龄结构延迟的人力资本模型的数值方法，该模型被表述为具有双延迟和差分核的积分-微分方程。该方法采用基于移动最小二乘（MLS）技术的局部无网格离散伽辽金方法，可以处理不规则或非均匀数据。MLS方案的局部特性通过关注小邻域来提高计算效率。此外，通过使用移位和缩放的多项式基函数实现的稳定MLS框架提高了数值稳定性，降低了对节点分布的敏感性，从而将这些优点转移到该方法中。该算法的简单性使其易于在标准个人计算机上实现，并扩展到更广泛的延迟积分微分方程。为了评估其可靠性，我们分析了该方法的误差，并确定了该方法的收敛顺序。应用该方法对若干数值算例进行了求解，得到的结果证实了该方法的准确性、稳定性以及与理论结果的一致性。

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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 : 2025-11-01 Epub 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

Split-step θ-method for stochastic pantograph differential equations: Convergence and mean-square stability analysis 随机受电弓微分方程的裂步θ-法：收敛性和均方稳定性分析

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-11-01 Epub Date: 2025-06-02 DOI: 10.1016/j.apnum.2025.05.010

Fathalla A. Rihan, K. Udhayakumar

This paper introduces a split-step θ-method (SSθ-method) with variable step sizes for solving stochastic pantograph delay differential equations (SPDDEs). We establish the mean-square convergence of the proposed SSθ-method and show that it achieves a strong convergence order of order 1/2. Under certain assumptions, we prove that the SSθ-method is exponentially mean-square stable for

θ \geq 0.5

. Additionally, we analyze the asymptotic mean-square stability of the SSθ-method under a stronger assumption. Finally, numerical examples illustrate the effectiveness of the proposed methods.

介绍了求解随机受电弓延迟微分方程（SPDDEs）的变步长裂步θ法（ss θ法）。我们建立了所提出的s θ-方法的均方收敛性，并证明了它具有1/2阶的强收敛阶。在一定的假设条件下，证明了当θ≥0.5时，s θ-方法是指数均方稳定的。此外，在更强的假设下，我们分析了s θ-方法的渐近均方稳定性。最后，通过数值算例验证了所提方法的有效性。

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 : 2025-11-01 Epub 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

Generalized Clenshaw-Curtis quadrature method for systems of linear ODEs with constant coefficients 常系数线性微分方程系统的广义Clenshaw-Curtis正交法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-11-01 Epub Date: 2025-06-09 DOI: 10.1016/j.apnum.2025.06.003

Fu-Rong Lin, Xi Yang, Gui-Rong Zhang

In this paper, we consider high precision numerical methods for the initial problem of systems of linear ordinary differential equations (ODEs) with constant coefficients. It is well-known that the analytic solution of such a system of linear ODEs involves a matrix exponential function and an integral whose integrand is the product of a matrix exponential and a vector-valued function. We mainly consider numerical quadrature methods for the integral term in the analytic solution and propose a generalized Clenshaw-Curtis (GCC) quadrature method. The proposed method is then applied to the initial-boundary value problem for a heat conduction equation and a Riesz space fractional diffusion equation, respectively. Numerical results are presented to demonstrate the effectiveness of the proposed method.

本文研究了常系数线性常微分方程系统初始问题的高精度数值解法。众所周知，这种线性微分方程系统的解析解涉及一个矩阵指数函数和一个积分，其被积是一个矩阵指数函数和一个向量值函数的乘积。本文主要研究了解析解中积分项的数值求积分方法，提出了一种广义的clclenshaw - curtis求积分方法。然后将该方法分别应用于热传导方程和Riesz空间分数扩散方程的初边值问题。数值结果验证了该方法的有效性。

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 : 2025-11-01 Epub 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

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 : 2025-11-01 Epub 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

Numerical method and analysis for fluid-structure model on unbounded domains 无界域流固模型的数值方法与分析

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-11-01 Epub Date: 2025-06-25 DOI: 10.1016/j.apnum.2025.06.009

Hongwei Li, Xinyue Chen

Numerically solving the fluid-structure model on unbounded domains poses a challenge, due to the unbounded nature of the physical domain. To overcome this challenge, the artificial boundary method is specifically applied to numerically solve the fluid-structure model on unbounded domains, which can be used to analyze fluid-structure interactions in various scientific and engineering fields. Drawing inspiration from the artificial boundary method, we employ artificial boundaries to truncate the unbounded domain, subsequently designing the high order local artificial boundary conditions thereon based on the Padé approximation. Then, the initial value problem on the unbounded domain is reduced into an initial boundary value problem on the computational domain, which can be efficiently solved by adopting the finite difference method. Furthermore, a series of auxiliary variables is introduced specifically to address the issue of mixed derivatives arising in the artificial boundary conditions, and the stability, convergence and solvability of the reduced problem are rigorously analyzed. Numerical experiments are reported to demonstrate the effectiveness of artificial boundary conditions and theoretical analysis.

由于物理域的无界性，在无界域上对流固模型进行数值求解是一项挑战。为了克服这一挑战，专门采用人工边界法对无界域上的流固耦合模型进行数值求解，可用于各种科学和工程领域的流固耦合分析。受人工边界法的启发，我们采用人工边界截断无界域，然后基于pad近似在无界域上设计高阶局部人工边界条件。然后，将无界域上的初值问题转化为计算域上的初边值问题，采用有限差分法进行有效求解。此外，针对人工边界条件下的混合导数问题，引入了一系列辅助变量，并对简化后的问题的稳定性、收敛性和可解性进行了严格的分析。数值实验证明了人工边界条件和理论分析的有效性。

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引用次数: 0

A high-order accurate unconditionally stable bound-preserving numerical scheme for the Cahn-Hilliard-Navier-Stokes equations Cahn-Hilliard-Navier-Stokes方程的高阶精确无条件稳定保界数值格式

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-11-01 Epub Date: 2025-06-10 DOI: 10.1016/j.apnum.2025.06.004

Yali Gao , Daozhi Han , Sayantan Sarkar

A high order numerical method is developed for solving the Cahn-Hilliard-Navier-Stokes equations with the Flory-Huggins potential. The scheme is based on the

Q_{k}

finite element with mass lumping on rectangular grids, the second-order convex splitting method and the pressure correction method. The unique solvability, unconditional stability, and bound-preserving properties are rigorously established. The key for bound-preservation is the discrete

L^{1}

estimate of the singular potential. Ample numerical experiments are performed to validate the desired properties of the proposed numerical scheme.

提出了一种求解具有Flory-Huggins势的Cahn-Hilliard-Navier-Stokes方程的高阶数值方法。该方案基于矩形网格质量集总的Qk有限元、二阶凸分裂法和压力修正法。严格地建立了唯一可解性、无条件稳定性和保界性。保持边界的关键是奇异势的离散L1估计。进行了大量的数值实验来验证所提出的数值格式的预期性能。

引用次数: 0

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