{"title":"Numerical approximation of parabolic singularly perturbed problems with large spatial delay and turning point","authors":"Amit Sharma, Pratima Rai","doi":"10.1108/ec-09-2023-0534","DOIUrl":null,"url":null,"abstract":"<h3>Purpose</h3>\n<p>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.</p><!--/ Abstract__block -->\n<h3>Design/methodology/approach</h3>\n<p>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.</p><!--/ Abstract__block -->\n<h3>Findings</h3>\n<p>Consistency, stability and convergence are investigated for the proposed scheme. It is established that the present approach has parameter-uniform convergence of <span><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>τ</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"script\">K</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>ln</mml:mi><mml:mi mathvariant=\"script\">K</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:math></span>, where <span><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Δ</mml:mi><mml:mi>τ</mml:mi></mml:math></span> and <span><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mi mathvariant=\"script\">K</mml:mi></mml:mrow></mml:math></span> denote the step size in the time direction and number of mesh-intervals in the space direction.</p><!--/ Abstract__block -->\n<h3>Originality/value</h3>\n<p>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.</p><!--/ Abstract__block -->","PeriodicalId":50522,"journal":{"name":"Engineering Computations","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Engineering Computations","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1108/ec-09-2023-0534","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
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+K−2(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.
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