This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.

翻译：本文提出了一种数值方法，用于求解具有小正参数{\epsilon}乘以最高阶导数的二阶常微分方程的奇异摄动对流扩散边值问题。我们特别研究了Dirichlet边界条件。为求解该微分方程，我们提出了一种迎风有限差分方法，并结合Shishkin网格方案来捕捉边界层附近的解。我们的求解器既直接又具有高精度，且计算时间随网格点数线性增长。数值公式的MATLAB代码已公开提供。我们通过数值结果验证了理论分析，并评估了所提方法的精度。文中包含的表格和图形展示了数值结果，表明所提方法能够高精度地逼近精确解。