An approximation technique for solutions of singularly perturbed one-dimensional convection-diffusion problem


YÜZBAŞI Ş., KARAÇAYIR M.

INTERNATIONAL JOURNAL OF NUMERICAL MODELLING-ELECTRONIC NETWORKS DEVICES AND FIELDS, vol.33, no.1, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 33 Issue: 1
  • Publication Date: 2020
  • Doi Number: 10.1002/jnm.2686
  • Journal Name: INTERNATIONAL JOURNAL OF NUMERICAL MODELLING-ELECTRONIC NETWORKS DEVICES AND FIELDS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, Metadex, zbMATH, DIALNET, Civil Engineering Abstracts
  • Keywords: Galerkin method, numerical solutions, one-dimensional convection-diffusion problem, singular perturbation, weighted residual method
  • Akdeniz University Affiliated: Yes

Abstract

In this study, a weighted residual method is presented in order to numerically solve singularly perturbed one-dimensional parabolic convection-diffusion problem. Assuming an approximate polynomial solution of a prescribed degree N, the method uses the set of bivariate monomials whose degrees do not exceed N as the set of base functions. Then, following Galerkin's path, inner product with the base functions are applied to the residual of the approximate solution polynomial. Incorporation of the initial and boundary conditions are ensured by forcing the approximate solution to satisfy these conditions on equidistant collocation points. The solution of the resulting linear system then yields the approximate polynomial solution. Additionally, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed briefly. The Galerkin-like scheme and the residual correction technique are illustrated with two examples. The obtained results are also compared with other methods presented in the literature.