Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well. The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near \(t = 0\). In this paper, we propose a modification. We first split the time interval [0, T] into \([0, T_0]\) and \([T_0, T]\), where \(T_0\) (\(0< T_0 < T\)) is reasonably small. Then, the graded L1 scheme is applied in \([0, T_0]\), while the uniform one is used in \([T_0, T]\). Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.

具有弱奇异核的Volterra子扩散问题很好地描述了子扩散过程的动力学。通常选择分级L 1方案来离散化此类问题，因为它可以处理\（t = 0 \）附近的奇异性。在本文中，我们提出了一种修改。我们首先将时间间隔[0，  T ]分为\（[0，T_0] \）和\（[T_0，T] \），其中\（T_0 \）（\（0 <T_0 <T \））为相当小。然后，将分级L 1方案应用于\（[0，T_0] \），而将统一的L 1方案应用于\（[T_0，T] \）。我们的一次性系统就是基于这种策略而得出的。为了有效地解决出现的系统，我们将其分为两个子问题并设计了两个前置条件。还研究了这两个预处理器的一些特性。此外，我们将方法扩展为解决半线性扩散问题。数值结果表明该方法是有效的。