Numerical Treatment of Uniformly Convergent Method for Convection Diffusion Problem

Year 2022, Issue: 38, 52 - 60, 31.03.2022

Ali Filiz

https://doi.org/10.53570/jnt.1065763

Abstract

In this paper, we will study the convergence properties of the method designed for the convection-diffusion problem. We will prove that the analytical and numerical methods give the same result. Merging the ideas in previous research, we introduce a numerical algorithm on a uniform mesh that requires no exact solution to the local convection-diffusion problem. We display how to obtain the numerical solution of the local Boundary Value Problem (BVP) in a suitable way to ensure that the resulting numerical algorithm recaptures the same convergence properties when using the exact solution of the local BVP. We prove that the proposed algorithm nodally converges to the exact solution.

Keywords

Trapezoidal rule, convection-diffusion problem, boundary value problem, singular points, Green’s function, Lagrange interpolation.

References

• G. I. Shishkin, A Difference Scheme for Singularly Perturbed Equation of Parabolic Type with a Discontinuous Initial Condition, Soviet Mathematics Doklady 37 (1988) 792–796.
• G. I. Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Section, Ekaterinburg, (1992) (in Russian).
• J. Miller, E. Mullarkey, E. O’Riordan, G. Shishkin, A Simple Recipe for Uniformly Convergent Finite Difference Schemes for Singularly Perturbed Problems, Comptes Rendus de l'Académie des Sciences – Mathematics 312 Serie I (1991) 643–648.
• N. S. Bakhvalov, On the Optimization of the Methods for Boundary Value Problems with Boundary Layers, USSR Computational Mathematics and Mathematical Physics 9(4) (1969) 841–859 (in Russian).
• E. C. Gartland, Graded−Mesh Difference Schemes for Singularly Perturbed Two Point Boundary Value Problems, Mathematics of Computation 51 (1988) 631–657.
• R. Vulanovic, Mesh Construction for Discretization of Singularly Perturbed Boundary Value Problems, Doctoral Dissertation, Faculty of Sciences, University of Novisad (1986).
• A. Filiz, A. I. Nesliturk, A. Sendur, A Fully Discrete ε-Uniform Method for Singular Perturbation Problems on Equidistant Meshes, International Journal of Computer Mathematics 89(2) (2012) 190–199.
• A. Sendur, N. Srinivasan, S. Gautam, Error Estimates for a Fully Discrete ε-Uniform Finite Element Method on Quasi-Uniform Meshes, Hacettepe Journal of Mathematics and Statistics 50(5) (2021), 1306–1324.
• H. G. Roos, M. Stynes, L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection Diffusion and Flow Problems, Springer−Verlag, Berlin Heidelberg New York, 1994.
• A. Sendur, A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems, Journal of Applied Mathematics Article ID 4259634 (2018) 16 pages.
• A. Filiz, Analytical Construction of Uniformly Convergent Method for Convection Diffusion Problem, Asian Journal of Fuzzy and Applied Mathematics 9(3) (2021) 32–36.
• M. Ekici, An ε-Uniform Method On Equidistant Meshes, Master’s Thesis, Adnan Menderes University (2010) Aydın, Turkey.
• R. L. Burden, J. D. Faires, Numerical Analysis, Brooks/Cole, Boston, 2011.
• A. Filiz, A. I. Nesliturk, M. Ekici, A Fully Discrete ε-Uniform Method for Convection Diffusion Problem on Equidistant Meshes, Applied Mathematical Sciences 6(17) (2012) 827–842.
• H. J. H. Miller, E. O’Riordan, G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, USA, 1996.
• H. G. Roos, Ten Ways to Generate the Il’in and Related Schemes, Journal of Computational and Applied Mathematics 53 (1994) 43–59.
• H. G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection Diffusion and Flow Problems, Springer, Dresden, 2008.
• M. Stynes, L. Tobiska, A Finite Difference Analysis of a Streamline Diffusion Method on a Shishkin Mesh, Numerical Algorithms 18(3-4) (1998) 337–360.