Applied Numerical Mathematics最新文献

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A uniformly convergent analysis for multiple scale parabolic singularly perturbed convection-diffusion coupled systems: Optimal accuracy with less computational time 多尺度抛物线奇异扰动对流扩散耦合系统的均匀收敛分析：用更少的计算时间获得最佳精度

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-25 DOI: 10.1016/j.apnum.2024.09.020

This study addresses time-dependent multiple-scale reaction-convection-diffusion initial boundary value systems characterized by strong coupling in the reaction matrix and weak coupling in the convection terms for a locally optimal accurate solution. The discrete problem, which typically loses its tridiagonal structure, expands its bandwidth to four in such coupled systems, resulting in a substantial computational load. Our objective is to mitigate this computational burden through a splitting approach that transforms the non-tridiagonal matrix into a tridiagonal form while maintaining the consistency, local optimal accuracy in space, and stability of the numerical scheme. We employ equidistributed non-uniform grids, guided by a carefully chosen monitor function, to approximate the continuous space domain. The discretization strategy targets local optimal linear accuracy across space and time on the domain's interior points. In addition, we have also provided the global convergence analysis of the present splitting approach, mathematically. The mathematical evidence is also obtained from the numerical experiments by comparing the splitting approach (either diagonal or triangular forms) of the reaction matrix to its coupled form. The results strongly confirm the effectiveness of this approach in delivering uniform linear accuracy, based on the present problem discretizations while significantly reducing the computational costs.

本研究探讨了以反应矩阵强耦合和对流项弱耦合为特征的时变多尺度反应-对流-扩散初始边界值系统的局部最优精确解。离散问题通常会失去其三对角结构，在这种耦合系统中，其带宽会扩大到四个，从而造成巨大的计算负荷。我们的目标是通过拆分方法减轻这种计算负担，将非对角矩阵转化为对角线形式，同时保持数值方案的一致性、空间局部最优精度和稳定性。我们采用等分布非均匀网格，以精心选择的监控函数为指导，逼近连续空间域。离散化策略的目标是域内点在空间和时间上的局部最优线性精度。此外，我们还从数学角度对目前的分割方法进行了全局收敛分析。通过比较反应矩阵的拆分方法（对角线或三角形形式）和耦合形式，我们还从数值实验中获得了数学证据。结果有力地证实了这种方法在提供统一线性精度方面的有效性，基于目前的问题离散化，同时显著降低了计算成本。

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引用次数: 0

Stability analysis and error estimates of implicit-explicit Runge-Kutta least squares RBF-FD method for time-dependent parabolic equation 时变抛物方程的隐式-显式 Runge-Kutta 最小二乘 RBF-FD 方法的稳定性分析和误差估计

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.apnum.2024.09.018

In this paper, for the time-dependent parabolic equations defined on complex geometries domain, we develop and analyze the least-squares radial basis function finite difference method (RBF-FD) coupled with the implicit-explicit Runge-Kutta (IMEX-RK) time discretization up to third order accuracy, which improves stability and accuracy. We derive the absolute stability region and the optimal time-step constraint for four kinds of IMEX-RK schemes. Compared to the traditional explicit or implicit time discretization, these are not trivial. Under a wide time-step constraint, the stability and the error estimates in

l_{2}

-norm are established. Finally, several numerical experiments on the regular domain and non-convex domain are performed to validate the theoretical analysis.

本文针对定义在复杂几何域上的时变抛物线方程，开发并分析了最小二乘径向基函数有限差分法（RBF-FD）与隐式-显式 Runge-Kutta （IMEX-RK）时间离散法（最高可达三阶精度），从而提高了稳定性和精度。我们推导了四种 IMEX-RK 方案的绝对稳定区域和最佳时间步长约束。与传统的显式或隐式时间离散化相比，这些并不简单。在宽时间步长约束下，建立了 l2 准则的稳定性和误差估计。最后，在规则域和非凸域上进行了若干数值实验，以验证理论分析。

引用次数: 0

A weak Galerkin finite element method for fourth-order parabolic singularly perturbed problems on layer adapted Shishkin mesh 层适配 Shishkin 网格上四阶抛物线奇异扰动问题的弱 Galerkin 有限元方法

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.apnum.2024.09.019

In this paper, we propose a weak Galerkin finite element approximation for a class of fourth-order singularly perturbed parabolic problems. The problem exhibits boundary layers and so we have considered layer adapted triangulations, in particular Shishkin triangular mesh in the spatial domain. For temporal discretization, we utilize the Crank-Nicolson scheme on a uniform mesh. Stability and error estimates along with the uniform convergence of the method has been proved. Numerical examples are included which verifies our analysis.

本文针对一类四阶奇异扰动抛物线问题提出了一种弱 Galerkin 有限元近似方法。该问题具有边界层，因此我们考虑了与层相适应的三角网格，特别是空间域的 Shishkin 三角网格。在时间离散化方面，我们采用了均匀网格上的 Crank-Nicolson 方案。我们已经证明了该方法的稳定性、误差估计值以及均匀收敛性。其中的数值示例验证了我们的分析。

引用次数: 0

Weak Galerkin finite element method with the total pressure variable for Biot's consolidation model 采用总压力变量的弱伽勒金有限元法用于毕奥特固结模型

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.apnum.2024.09.017

In this work, we develop a weak Galerkin method for the three-field Biot's consolidation model. The key idea is to consider the total pressure variable. We employ the stable pair of weak Galerkin finite elements to discretize the displacement and total pressure, and use totally discontinuous weak functions to approximate pressure in a semi-discrete scheme. Then, we give the fully discrete scheme based on the backward Euler method in time. Furthermore, we prove the well-posedness of the numerical schemes and derive the optimal error estimates for three variables in their nature norms. Our theoretical results are independent of the Lamé constant λ and the storage coefficient

c_{0}

. Finally, some experiments that employ different polynomial degrees and polygonal meshes are presented to demonstrate the efficiency and stability of the weak Galerkin method.

在这项工作中，我们为三场 Biot 固结模型开发了一种弱 Galerkin 方法。其关键思路是考虑总压力变量。我们采用一对稳定的弱 Galerkin 有限元对位移和总压进行离散，并使用完全不连续的弱函数在半离散方案中对压力进行近似。然后，我们给出了基于时间后向欧拉法的全离散方案。此外，我们还证明了数值方案的良好假设性，并推导出三个变量在其性质规范下的最优误差估计。我们的理论结果与拉梅常数 λ 和存储系数 c0 无关。最后，我们介绍了采用不同多项式度和多边形网格的一些实验，以证明弱 Galerkin 方法的效率和稳定性。

引用次数: 0

Supercloseness of the NIPG method on a Bakhvalov-type mesh for a singularly perturbed problem with two small parameters 两个小参数奇异扰动问题的巴赫瓦洛夫型网格上 NIPG 方法的超封闭性

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.apnum.2024.09.016

In this paper, the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh is proposed for a singularly perturbed problem with two small parameters. In order to reflect the behavior of layers more accurately, a balanced norm, rather than the common energy norm, is introduced. By selecting special penalty parameters at different mesh points, we establish the supercloseness of

k + \frac{1}{2}

order, and prove an optimal order of uniform convergence in a balanced norm. Numerical experiments are proposed to confirm our theoretical results.

本文针对具有两个小参数的奇异扰动问题，提出了巴赫瓦洛夫网格上的非对称内部惩罚伽勒金（NIPG）方法。为了更准确地反映层的行为，本文引入了平衡规范而非普通能量规范。通过在不同网格点选择特殊的惩罚参数，我们建立了 k+12 阶的超松性，并证明了平衡规范中均匀收敛的最优阶。我们提出了数值实验来证实我们的理论结果。

引用次数: 0

Error analysis of positivity-preserving energy stable schemes for the modified phase field crystal model 修正相场晶体模型的保正能量稳定方案的误差分析

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.apnum.2024.09.010

In this paper, we introduce second-order numerical schemes for the modified phase field crystal (MPFC) model that are decoupled, linear, positivity-preserving, and unconditionally energy-stable. These schemes adopt a positivity-preserving auxiliary variable method to explicitly handle the nonlinear potential function, resulting in decoupled linear systems with constant coefficients at each time step. We rigorously demonstrate that the auxiliary variables remain positive throughout all time steps and prove the unconditionally energy stability of these schemes. The stability pertains to a discrete modified energy, rather than the original free energy or the pseudo energy of the MPFC system. Moreover, a detailed error analysis is provided. A series of numerical experiments are conducted to validate the accuracy and efficiency of our proposed schemes.

本文介绍了修正相场晶体（MPFC）模型的二阶数值方案，这些方案是解耦的、线性的、保正的和无条件能量稳定的。这些方案采用了一种正向保留辅助变量方法来明确处理非线性势函数，从而产生了在每个时间步具有恒定系数的解耦线性系统。我们严格证明了辅助变量在所有时间步长内都保持为正，并证明了这些方案的无条件能量稳定性。这种稳定性与离散修正能有关，而不是 MPFC 系统的原始自由能或伪能。此外，还提供了详细的误差分析。我们还进行了一系列数值实验，以验证所提方案的准确性和效率。

引用次数: 0

A novel least squares approach generating approximations orthogonal to the null space of the operator 一种新的最小二乘法，产生与算子空域正交的近似值

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-14 DOI: 10.1016/j.apnum.2024.09.015

We introduce a novel least squares functional specifically formulated to solve linear partial differential equations with operators that have a nonempty null space. Our method involves projecting the solution onto the orthogonal complement of the operator's null space to overcome challenges encountered by conventional numerical methods when nonzero null components are present. We describe the theoretical framework of the proposed method and validate it through numerical examples that show improved accuracy and usability in cases where traditional methods are less effective due to significant null space components. Overall, this approach provides a practical and reliable solution for partial differential equations with substantial null space components.

我们引入了一种新的最小二乘法函数，专门用于求解具有非空空间的算子的线性偏微分方程。我们的方法是将解投影到算子空空间的正交补集上，以克服传统数值方法在出现非零空成分时遇到的难题。我们描述了所提方法的理论框架，并通过数值示例对其进行了验证，结果表明，在传统方法因存在大量空空间成分而效果不佳的情况下，该方法的准确性和可用性得到了提高。总之，这种方法为具有大量空空间分量的偏微分方程提供了实用可靠的解决方案。

引用次数: 0

A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions 带积分边界条件的对流扩散问题弗雷德霍尔积分微分方程数值方法的比较研究

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-13 DOI: 10.1016/j.apnum.2024.09.001

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.

本文对具有小参数和积分边界条件的弗雷德霍尔积分微分方程进行了数值求解。这些方程的解在右边界有一个边界层。中心差分方案逼近二阶导数，后向差分（上风方案）逼近一阶导数，梯形法则用于积分项和 Shishkin 网格。结果表明，从理论上讲，所提出的方案是均匀收敛的，几乎具有一阶收敛性。为了将收敛阶数从一阶提高到二阶，我们使用了后处理和混合方案。为支持理论结果，我们计算了两个数值示例。

引用次数: 0

Exact solution for a discrete-time SIR model 离散时间 SIR 模型的精确解

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.apnum.2024.09.014

We propose a nonstandard finite difference scheme for the Susceptible–Infected–Removed (SIR) continuous model. We prove that our discretized system is dynamically consistent with its continuous counterpart and we derive its exact solution. We end with the analysis of the long-term behavior of susceptible, infected and removed individuals, illustrating our results with examples. In contrast with the SIR discrete-time model available in the literature, our new model is simultaneously mathematically and biologically sound.

我们针对易感-感染-清除（SIR）连续模型提出了一种非标准有限差分方案。我们证明了离散化系统与其连续对应系统在动态上是一致的，并推导出其精确解。最后，我们分析了易感个体、受感染个体和被移除个体的长期行为，并举例说明了我们的结果。与现有文献中的 SIR 离散时模型相比，我们的新模型在数学和生物学上都是合理的。

引用次数: 0

Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods 用准线性化和 Legendre-collocation 方法数值求解非线性 cordial Volterra 积分方程的广义形式

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.apnum.2024.09.013

In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.

在这篇文章中，我们提出了一种针对一般形式的非线性 cordial Volterra 积分方程的数值方法。我们讨论了在这些条件下问题有解的条件。由于问题解的存在取决于标量方程的可解性以及问题的线性形式，因此我们采用了准线性化技术，将非线性问题的求解简化为线性方程序列的求解。我们考虑了线性方程解的存在性及其对非线性问题解的二次收敛性。对于所产生线性方程的数值求解，我们采用了 Legendre-collocation 方法以及正则公式的正则化技术。考虑到 Volterra 积分算子的非紧凑性，我们讨论了配准法的误差分析。为了测试所提方法的效率和准确性，报告了不同情况下的数值示例求解。

引用次数: 0

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