Second-Order Convergence Scheme for Singularly Perturbed Unsteady Problems with Boundary Turning Points

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Scince College, Bahir Dar University, Bahir Dar, Ethiopia.

2 Department of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia.

Abstract

This study proposes a numerical scheme for solving singularly perturbed unsteady convection-diffusion problems exhibiting boundary turning points. These problems are characterized by the presence of a small perturbation parameter \( \varepsilon \) multiplying the diffusion term, leading to the formation of a sharp boundary layer near the left side of the spatial domain. As \( \varepsilon \to 0 \), the solution undergoes rapid variations within this layer, posing significant challenges to standard numerical methods due to the presence of steep gradients.
To effectively capture the solution behavior, we develop a robust numerical method that combines the Crank--Nicolson scheme for time discretization with a nonstandard finite difference approach for spatial discretization, implemented on uniform meshes. The stability of the proposed scheme is rigorously analyzed using truncation error estimates and the discrete minimum principle. Furthermore, Richardson extrapolation is applied to enhance the spatial order of convergence.
The resulting scheme is shown to be uniformly convergent with respect to the perturbation parameter \( \varepsilon \), and achieves second-order accuracy in both space and time. Numerical experiments on three model problems validate the theoretical findings and demonstrate the efficiency and accuracy of the proposed method.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 15 September 2025

Files

History

  • Receive Date: 22 September 2024
  • Revise Date: 14 August 2025
  • Accept Date: 10 September 2025

Share

How to cite

Statistics