Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

Q1 Mathematics
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引用次数: 0

Abstract

In this paper, we proposed an accurate ϵ-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order η(0,1). The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is ϵ-uniformly convergent of order O((Δt)2η+Δx), where Δt and Δx represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

通过指数拟合算子方案的奇异扰动时间分数对流扩散方程
本文提出了一种精确的ϵ-均匀收敛数值方法,通过指数拟合算子方案求解奇异扰动时间分数对流扩散方程。时间分数导数是在 Caputo 意义上定义的,阶数为η∈(0,1)。在均匀网格上采用 Crank-Nicolson 方法对时间分数导数进行离散化,并采用指数拟合算子方案和标准上风法对空间域进行网格划分。研究了离散化问题的截断误差和均匀稳定性,以证明所提方案的参数均匀收敛性。结果表明,该方案具有 O((Δt)2-η+Δx)阶的ϵ均匀收敛性,其中 Δt 和 Δx 分别代表时域和空间域的步长。为了评估所建议方案的准确性并验证所讨论的理论概念,我们提供了两个数值示例。为了证明所提出的数值方案的效率,我们将其与文献中使用有限差分法获得的数值解进行了比较。结果表明,本方案得到的结果更加精确,收敛速度也更快。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
6.20
自引率
0.00%
发文量
138
审稿时长
14 weeks
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