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

Yong-Liang Zhao, Xian-Ming Gu, Alexander Ostermann

TL;DR

The paper tackles solving time-fractional Volterra subdiffusion equations with weakly singular kernels by forming an all-at-once system through a two-stage time discretization: a graded L1 scheme on $[0,T_0]$ and a uniform scheme on $[T_0,T]$, yielding a system $\mathcal{M}\bm{u}=\bm{\eta}+\bm{f}$ with $\mathcal{M}=A\otimes I_s-I_t\otimes B$. Two preconditioners, a block lower-triangular $P_1$ and an $\\alpha$-circulant PinT preconditioner $P_\alpha$, are developed to accelerate Krylov solvers for the two subproblems that arise from a natural split of the all-at-once system; their spectral properties indicate rapid convergence and strong parallelizability via FFTs. The method is extended to semilinear subdiffusion with a modified Newton framework, using the same preconditioners to maintain efficiency. Numerical experiments demonstrate substantial reductions in CPU time and iterations and illustrate clustered spectra, underscoring the approach’s potential for parallel computation in time-fractional PDEs.

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.

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

TL;DR

The paper tackles solving time-fractional Volterra subdiffusion equations with weakly singular kernels by forming an all-at-once system through a two-stage time discretization: a graded L1 scheme on and a uniform scheme on , yielding a system with . Two preconditioners, a block lower-triangular and an -circulant PinT preconditioner , are developed to accelerate Krylov solvers for the two subproblems that arise from a natural split of the all-at-once system; their spectral properties indicate rapid convergence and strong parallelizability via FFTs. The method is extended to semilinear subdiffusion with a modified Newton framework, using the same preconditioners to maintain efficiency. Numerical experiments demonstrate substantial reductions in CPU time and iterations and illustrate clustered spectra, underscoring the approach’s potential for parallel computation in time-fractional PDEs.

Abstract

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well.The graded scheme is often chosen to discretize such problems since it can handle the singularity of the solution near . In this paper, we propose a modification. We first split the time interval into and , where () is reasonably small. Then, the graded scheme is applied in , while the uniform one is used in . 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.

Paper Structure

This paper contains 10 sections, 5 theorems, 61 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The all-at-once system
  3. The time-stepping scheme
  4. The all-at-once system
  5. Two preconditioners and their spectral analysis
  6. A block lower tridiagonal preconditioner for Eq. (3.2a)
  7. A PinT preconditioner for Eq. (3.2b)
  8. Extension to the semilinear form of Eq. (1.1)
  9. Numerical experiments
  10. Concluding remarks

Key Result

Theorem 3.1

The degree of the minimal polynomial $p$murphy2000note of $P_1^{-1} \mathcal{M}_{11}$ satisfies that Thus, the dimension of the Krylov subspace $\mathcal{K} \left( P_1^{-1} \mathcal{M}_{11}; \bm{b} \right)$ is at most $\lceil M_0/3 \rceil$.

Figures (3)

  • Figure 1: The decay of the elements of matrix $\mathcal{M}_{11}$, where $\beta = 0.5$ and $N_x = N_y = 11$.
  • Figure 2: Spectra of $\mathcal{M}_{11}$, $P_{1}^{-1} \mathcal{M}_{11}$, $\mathcal{M}_{22}$ and $P_{\alpha}^{-1} \mathcal{M}_{22}$, for $(\beta, r) = (0.9, 3)$ and $M = N = 32$ in Example 1.
  • Figure 3: Spectra of $\mathcal{M}_{11}$, $P_{1}^{-1} \mathcal{M}_{11}$, $\mathcal{M}_{22}$ and $P_{\alpha}^{-1} \mathcal{M}_{22}$, for $(\beta, r) = (0.5, 2)$ and $M = N = 32$ in Example 2.

Theorems & Definitions (7)

  • Theorem 3.1
  • Remark 1
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 3.2
  • Theorem 3.3
  • Remark 2