论文标题
来自Volterra Subdiflusion方程的全面系统的预处理技术
A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps
论文作者
论文摘要
volterra尺寸尺寸的奇异核的尺寸范围很好地描述了次扩散过程的动态。分级$ L1 $方案通常被选择以离散此类问题，因为它可以处理$ t = 0 $的解决方案的奇异性。在本文中，我们提出了修改。我们首先将时间间隔$ [0，t] $分为$ [0，t_0] $和$ [t_0，t] $，其中$ t_0 $（$ 0 <t_0 <t $）很小。然后，分级$ L1 $方案以$ [0，t_0] $应用，而统一的$ l1 $方案则以$ [t_0，t] $使用。我们的全面系统是根据此策略得出的。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两个预处理。还研究了这两个预处理的某些特性。此外，我们扩展了解决半线性次扩散问题的方法。据报道，数值结果显示了我们方法的效率。
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.
