Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point

IF 1.5 4区 工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Amit Sharma, Pratima Rai
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引用次数: 0

Abstract

Purpose

Singular perturbation turning point problems (SP-TPPs) involving parabolic convection–diffusion Partial Differential Equations (PDEs) with large spatial delay are studied in this paper. These type of equations are important in various fields of mathematics and sciences such as computational neuroscience and require specialized techniques for their numerical analysis.

Design/methodology/approach

We design a numerical method comprising a hybrid finite difference scheme on a layer-adapted mesh for the spatial discretization and an implicit-Euler scheme on a uniform mesh in the temporal variable. A combination of the central difference scheme and the simple upwind scheme is used as the hybrid scheme.

Findings

Consistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of OΔτ+K2(lnK)2, where Δτ and K denote the step size in the time direction and number of mesh-intervals in the space direction.

Originality/value

Parabolic SP-TPPs exhibiting twin boundary layers with large spatial delay have not been studied earlier in the literature. The presence of delay portrays an interior layer in the considered problem’s solution in addition to twin boundary layers. Numerical illustrations are provided to demonstrate the theoretical estimates.

具有大空间延迟和转折点的抛物线奇异扰动问题的数值逼近
目的本文研究了涉及具有大空间延迟的抛物对流扩散偏微分方程(PDEs)的星状扰动转折点问题(SP-TPPs)。这类方程在计算神经科学等数学和科学的各个领域都很重要,需要专门的数值分析技术。我们设计了一种数值方法,包括在层适应网格上进行空间离散化的混合有限差分方案,以及在时间变量均匀网格上的隐式欧拉方案。研究结果研究了拟议方案的一致性、稳定性和收敛性。结果表明,本方法的参数均匀收敛性为 OΔτ+K-2(lnK)2,其中 Δτ 和 K 分别表示时间方向的步长和空间方向的网格间隔数。除了孪生边界层之外,延迟的存在还在所考虑问题的解中描绘了一个内部层。本文提供了数值图解来证明理论估算。
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
Engineering Computations
Engineering Computations 工程技术-工程:综合
CiteScore
3.40
自引率
6.20%
发文量
61
审稿时长
5 months
期刊介绍: The journal presents its readers with broad coverage across all branches of engineering and science of the latest development and application of new solution algorithms, innovative numerical methods and/or solution techniques directed at the utilization of computational methods in engineering analysis, engineering design and practice. For more information visit: http://www.emeraldgrouppublishing.com/ec.htm
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