We report a new kind of waveform relaxation (WR) method for general semi-linear fractional sub-diffusion equations, and analyze the upper bound for the iteration errors. It indicates that the WR method is convergent superlinearly, and the convergence rate is dependent on the order of the time-fractional derivative and the length of the time interval. In order to accelerate the convergence, we present the windowing WR method. Then, we elaborate the parallelism based on the discrete windowing WR method, and present the corresponding fast evaluation formula. Numerical experiments are carried out to verify the effectiveness of the theoretic work.

中文翻译：

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分数次扩散方程的波形弛豫

我们报告了一种通用的半线性分数次扩散方程的波形松弛（WR）方法，并分析了迭代误差的上限。这表明WR方法是超线性收敛的，收敛速度取决于时间分数阶导数的阶数和时间间隔的长度。为了加速收敛，我们提出加窗WR方法。然后，我们阐述了基于离散窗口WR方法的并行性，并提出了相应的快速评估公式。进行了数值实验，以验证理论工作的有效性。