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Second-Order Convergence Scheme for Singularly Perturbed Unsteady Problems with Boundary Turning Points | ||
| Computational Methods for Differential Equations | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 24 شهریور 1404 اصل مقاله (14.08 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22034/cmde.2025.63643.2841 | ||
| نویسندگان | ||
| Yimesgen Mehari Kebede* 1؛ Awoke Andargie Tiruneh2؛ Endalew Getnet Tsega2 | ||
| 1Department of Mathematics, Scince College, Bahir Dar University, Bahir Dar, Ethiopia. | ||
| 2Department of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia. | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Singularly perturbed؛ Boundary Turning points؛ Nonstandard Finite Difference method؛ Crank-Nicolson Method؛ Uniform Convergence | ||
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