This study introduces a stable central difference method for solving second-order self-adjoint singularly perturbed boundary value problems. First, the solution domain is discretized. Then, the derivatives in the given boundary value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of equations is developed. The obtained system of algebraic equations is solved by Thomas algorithm. The consistency and stability that guarantee the convergence of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields sixth order convergent. To validate the applicability of the method, two model examples are solved for different values of perturbation parameter ε and different mesh size h. The proposed method approximates the exact solution very well. Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.

中文翻译：

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用Richardson外推法求解二阶自伴随奇异摄动边值问题的四阶稳定中心差分

本研究介绍了一种用于求解二阶自伴随奇异摄动边值问题的稳定中心差分法。首先，解域被离散化。然后，给定边值问题中的导数被有限差分近似代替，并开发了提供代数方程组的数值方案。得到的代数方程组由Thomas算法求解。研究了保证方案收敛的一致性和稳定性。通过应用产生六阶收敛性的理查森外推法进一步加速了该方案的既定收敛性。为了验证该方法的适用性，针对不同的扰动参数 ε 值和不同的网格尺寸 h 求解了两个模型示例。所提出的方法非常接近精确解。此外，本方法是收敛的，并且比文献中报道的一些现有数值方法给出更准确的结果。