Abstract In this article, a parameter-uniform implicit scheme is constructed for a class of parabolic singularly perturbed reaction-diffusion initial-boundary value problems with large delay in the spatial direction. In general, the solution of these problems exhibits twin boundary layers and an interior layer (due to the presence of the delay in the reaction term). Crank-Nicolson difference formula (on a uniform mesh) is used in time to semi-discretize the given PDE, and then the standard finite difference scheme (on a piecewise-uniform mesh) is used for the system of ordinary differential equations obtained in the semi-discretization. The convergence analysis shows that the method is e-uniformly convergent of order two in the temporal direction and almost first-order in the spatial direction. Two test examples are encountered to show the efficiency of the method, validate the computational results, and to confirm the predicted theory.

中文翻译：

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大时滞奇异摄动初边值问题的参数一致数值处理

摘要 本文针对一类在空间方向上具有大延迟的抛物线奇异摄动反应扩散初边界值问题，构造了参数一致隐式格式。通常，这些问题的解决方案表现出双边界层和一个内层（由于存在反应项中的延迟）。及时使用 Crank-Nicolson 差分公式（在均匀网格上）对给定的偏微分方程进行半离散化，然后将标准有限差分格式（在分段均匀网格上）用于在半离散化。收敛分析表明，该方法在时间方向上为二阶e-均匀收敛，在空间方向上几乎一阶收敛。