一种用于带分级时间步长的Volterra次扩散方程全同时系统的预处理技术
A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps
AI总结：
针对Volterra次扩散方程，提出一种改进的时间离散策略，结合分级与均匀L1格式，并设计两个预处理器高效求解全同时系统，数值实验验证了方法的有效性。
摘要：
具有弱奇异核的Volterra次扩散问题能够很好地描述次扩散过程的动力学特性。分级$L1$格式通常被选用于离散此类问题，因为它可以处理解在$t = 0$附近的奇异性。在本文中，我们提出了一种改进。我们首先将时间区间$[0, T]$分割为$[0, T_0]$和$[T_0, T]$，其中$T_0$（$0 < T_0 < T$）合理较小。然后，在$[0, T_0]$上应用分级$L1$格式，而在$[T_0, T]$上使用均匀格式。我们的全同时系统基于这一策略推导得出。为了高效求解所产生的系统，我们将其分解为两个子问题并设计了两个预处理器。我们还研究了这两个预处理器的若干性质。此外，我们将我们的方法扩展到求解半线性次扩散问题。数值结果报告显示了我们方法的效率。
英文摘要：
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.
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