A class of different schemes for the numerical solving of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect to a perturbation parameter $\varepsilon$ on a modified Shishkin mesh of order 2 has been proven. For such a discrete solution, a global solution based on a linear spline was constructed, also the error of this solution is in expected boundaries. Numerical experiments at the end of the paper, confirm the theoretical results. The global solutions based on a natural cubic spline, and the experiments with Liseikin, Shishkin and modified Bakhvalov meshes are included in the numerical experiments as well.

中文翻译：

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半线性奇异-摄动反应扩散边值问题全局数值解的构造

构造了一类不同的数值求解半线性奇异-摄动反应-扩散边界-值问题的方案。证明了差分格式的稳定性，并证明了数值解的存在唯一性。之后，证明了在 2 阶修正 Shishkin 网格上关于扰动参数 $\varepsilon$ 的一致收敛。对于这样的离散解，构建了基于线性样条的全局解，该解的误差也在预期边界内。文末进行了数值实验，证实了理论结果。基于自然三次样条的全局解，以及 Liseikin、Shishkin 和修正 Bakhvalov 网格的实验也包含在数值实验中。