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Time stepping adaptation for subdiffusion problems with non-smooth right-hand sides

Sebastian Franz, Natalia Kopteva

TL;DR

The work tackles the numerical solution of time-fractional subdiffusion with non-smooth time dependence in the source term, which can cause locking in adaptive time stepping. It introduces a generalized residual barrier $\mathcal{B}(t)=\sum_{k=0}^K w_k H(t-s_k)\mathcal{R}(t-s_k)$ to account for interior singularities and preserves a posteriori error control via $\|Res(t)\|\le Tol\,\mathcal{B}(t)$, yielding robust time meshes around jumps in $f$. Complementary splitting and shifting strategies decompose the problem into simpler subproblems or local windows, enabling parallelization and reuse of existing adaptive methods. The paper also provides heuristics for automatic detection of problematic times, discusses handling negative $\lambda$, and offers stability guidance and practical considerations for semi-continuity and error evaluation, broadening applicability to a range of linear and semilinear fractional parabolic problems.

Abstract

We consider a time-fractional subdiffusion equation with a Caputo derivative in time, a general second-order elliptic spatial operator, and a right-hand side that is non-smooth in time. The presence of the latter may lead to locking problems in our time stepping procedure recently introduced in [2,4]. Hence, a generalized version of the residual barrier is proposed to rectify the issue. We also consider related alternatives to this generalized algorithm, and, furthermore, show that this new residual barrier may be useful in the case of a negative reaction coefficient.

Time stepping adaptation for subdiffusion problems with non-smooth right-hand sides

TL;DR

The work tackles the numerical solution of time-fractional subdiffusion with non-smooth time dependence in the source term, which can cause locking in adaptive time stepping. It introduces a generalized residual barrier to account for interior singularities and preserves a posteriori error control via , yielding robust time meshes around jumps in . Complementary splitting and shifting strategies decompose the problem into simpler subproblems or local windows, enabling parallelization and reuse of existing adaptive methods. The paper also provides heuristics for automatic detection of problematic times, discusses handling negative , and offers stability guidance and practical considerations for semi-continuity and error evaluation, broadening applicability to a range of linear and semilinear fractional parabolic problems.

Abstract

We consider a time-fractional subdiffusion equation with a Caputo derivative in time, a general second-order elliptic spatial operator, and a right-hand side that is non-smooth in time. The presence of the latter may lead to locking problems in our time stepping procedure recently introduced in [2,4]. Hence, a generalized version of the residual barrier is proposed to rectify the issue. We also consider related alternatives to this generalized algorithm, and, furthermore, show that this new residual barrier may be useful in the case of a negative reaction coefficient.
Paper Structure (6 sections, 15 equations, 3 figures)

This paper contains 6 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized Residual Barrier for Interior Singularities
  3. Splitting and Shifting Approaches
  4. Finding the Problematic Positions
  5. Negative $\lambda$
  6. Computational Stability Considerations

Figures (3)

  • Figure 1: Numerical solution using a collocation method for \ref{['eq:ex1']} (left) and \ref{['eq:ex2']} (right) with $\alpha=0.4$, $\gamma=0.25$, $m=4$, $TOL=10^{-4}$, $Q=1.2$
  • Figure 2: Example \ref{['eq:ex1']} with $\alpha=0.4$, $Q=1.2$. Left: $L_\infty(\Omega)$ value of the residual (blue) and its barrier (red) for a collocation method with $m=4$ and $TOL=10^{-4}$. Right: maximum errors vs. $TOL$.
  • Figure 3: Example with $\lambda=-1$. Left: absolute value of the residual (blue) and its barrier (red) for a collocation method with $m=4$ and $TOL=10^{-4}$. Right: maximum errors for $\alpha=0.4$, $Q=1.2$.