Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
{"title":"Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters","authors":"K. Ansari, Mohammad Izadi, S. Noeiaghdam","doi":"10.1515/dema-2023-0144","DOIUrl":null,"url":null,"abstract":"\n <jats:p>This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_dema-2023-0144_eq_001.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</m:mi>\n <m:mrow>\n <m:mo>(</m:mo>\n <m:mrow>\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:msup>\n <m:mrow>\n <m:mi>τ</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mi>s</m:mi>\n </m:mrow>\n </m:msup>\n <m:mo>+</m:mo>\n <m:msup>\n <m:mrow>\n <m:mi>M</m:mi>\n </m:mrow>\n <m:mrow>\n <m:mo>−</m:mo>\n <m:mstyle displaystyle=\"false\">\n <m:mfrac>\n <m:mrow>\n <m:mn>1</m:mn>\n </m:mrow>\n <m:mrow>\n <m:mn>2</m:mn>\n </m:mrow>\n </m:mfrac>\n </m:mstyle>\n </m:mrow>\n </m:msup>\n </m:mrow>\n <m:mo>)</m:mo>\n </m:mrow>\n </m:math>\n <jats:tex-math>{\\mathcal{O}}\\left(\\Delta {\\tau }^{s}+{M}^{-\\tfrac{1}{2}})</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> for <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_dema-2023-0144_eq_002.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>s</m:mi>\n <m:mo>=</m:mo>\n <m:mn>1</m:mn>\n <m:mo>,</m:mo>\n <m:mn>2</m:mn>\n </m:math>\n <jats:tex-math>s=1,2</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>, where <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_dema-2023-0144_eq_003.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi mathvariant=\"normal\">Δ</m:mi>\n <m:mi>τ</m:mi>\n </m:math>\n <jats:tex-math>\\Delta \\tau </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is the time step and <jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_dema-2023-0144_eq_004.png\" />\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\n <m:mi>M</m:mi>\n </m:math>\n <jats:tex-math>M</jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula> is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.</jats:p>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":"42 1","pages":""},"PeriodicalIF":4.6000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/dema-2023-0144","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
Abstract
This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O(Δτs+M−12){\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s=1,2s=1,2, where Δτ\Delta \tau is the time step and MM is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.
期刊介绍:
ACS Applied Bio Materials is an interdisciplinary journal publishing original research covering all aspects of biomaterials and biointerfaces including and beyond the traditional biosensing, biomedical and therapeutic applications.
The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrates knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important bio applications. The journal is specifically interested in work that addresses the relationship between structure and function and assesses the stability and degradation of materials under relevant environmental and biological conditions.