Quasi‐optimal error estimates are derived for the continuous‐time orthogonal spline collocation (OSC) method and also two discrete‐time OSC methods for approximating the solution of 1D parabolic singularly perturbed reaction–diffusion problems. OSC with C¹ splines of degree r ≥ 3 on a Shishkin mesh is employed for the spatial discretization while the Crank–Nicolson method and the BDF2 scheme are considered for the time‐stepping. The results of numerical experiments validate the theoretical analysis and also exhibit additional quasi‐optimal results, in particular, superconvergence phenomena.

中文翻译：

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奇摄动一维抛物线反应扩散问题的高阶离散时间正交样条搭配方法

对于连续时间正交样条搭配（OSC）方法以及两种离散时间OSC方法，都得出了最佳误差估计，这些方法用于逼近一维抛物线奇异摄动反应扩散问题。OSC与Ç ^1个花键度- [R 上的Shishkin网格≥3被用于空间离散而曲柄Nicolson方法和BDF2方案被认为是在时间步。数值实验的结果验证了理论分析的正确性，并且还表现出了更多的准最优结果，特别是超收敛现象。