A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps

TL;DR

This paper introduces a novel preconditioning approach for efficiently solving all-at-once systems from discretized Volterra subdiffusion equations, combining graded and uniform schemes to handle solution singularities.

Contribution

It proposes a hybrid discretization strategy and develops two preconditioners for the all-at-once system, extending the method to semilinear problems.

Findings

Preconditioners improve solver efficiency.

Hybrid scheme handles singularities effectively.

Numerical results confirm method's effectiveness.

Abstract

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.