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奇异摄动对流扩散边值问题。我们专门研究了Dirichlet边界条件。为求解该微分方程，我们提出了迎风有限差分法，并采用Shishkin网格方案以捕捉边界层附近的解。我们的求解器既直接又具有高精度，其计算时间随网格点数线性增长。该数值方法的MATLAB代码已公开提供。我们通过数值结果验证了理论结果并评估了方法的精度。文中的表格和图形展示了数值结果，表明我们提出的方法能够对精确解提供高精度的近似。