We consider singularly perturbed parabolic reaction–diffusion problems of scalar and vector types. We construct a discrete Schwarz waveform relaxation method of higher order for their numerical solution. The method is shown to be parameter-uniformly convergent having almost fourth order in space and first order in time. Further, interesting result proven is much faster convergence of iterative process for small perturbation parameter. Numerical results demonstrating the efficiency of the proposed method and validating the theoretically proven convergence results are given.

中文翻译：

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奇异摄动抛物线反应扩散问题的高阶离散Schwarz波形弛豫方法

我们考虑标量和向量类型的奇异扰动抛物线反应扩散问题。我们为其数值解构建了一种高阶离散 Schwarz 波形松弛方法。该方法被证明是参数一致收敛的，在空间上几乎是四阶，在时间上几乎是一阶。此外，有趣的结果证明，对于小扰动参数，迭代过程的收敛速度要快得多。数值结果证明了所提出的方法的效率，并验证了理论证明的收敛结果。