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.
翻译:Volterra 以微弱单核内核的反扩散问题描述子扩散过程的动态。 等级 $1 计划通常被选为分解这类问题, 因为它可以处理接近$t=0美元的解决办法的单一性。 在本文中, 我们提出修改。 我们首先将时间间隔[ $0, T] 分成[ $0, T_0, T] 美元, 美元为0美元( $0 < T_0 < T$ ), 合理小。 然后, 等级 $1 计划应用在$[ 10, T_0] 美元, 而制服计划则在$[T_0, T] 美元中使用。 我们的全自动系统以这一战略为基础。 为了高效解决新产生的系统, 我们将其分成两个子问题, 并设计两个先决条件。 此外, 我们扩展了我们的方法来解决半线性亚集问题。 数字结果报告显示我们的方法的效率 。