Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Convolution Quadrature for non smooth sectorial problems

Jing Guo, Maria Lopez-Fernandez

TL;DR

This work develops a stability- and error-aware theory for generalized Convolution Quadrature (gCQ) with variable time steps, tailored to sectorial kernels arising in fractional and subdiffusive problems. By leveraging a real integral representation of the kernel, the authors derive sharp a priori error estimates that remain valid under realistic data regularity, and they show how graded meshes can recover full-order convergence. The paper also introduces fast, oblivious implementations of gCQ for the fractional integral and for linear subdiffusion, achieving substantial memory and computational savings while preserving the convergence properties. Numerical experiments on fractional integrals and diffusion problems with smooth and non-smooth data confirm the predicted rates and demonstrate the practical viability of both the theoretical framework and the fast algorithms. Overall, the results provide a robust pathway for accurate, scalable time integration of memory-dependent problems in applied mathematics.

Abstract

We consider the application of the generalized Convolution Quadrature (gCQ) to approximate the solution of an important class of sectorial problems. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) that allows for variable steps. The available stability and convergence theory for the gCQ requires non realistic regularity assumptions on the data, which do not hold in many applications of interest, such as the approximation of subdiffusion equations. It is well known that for non smooth enough data the original CQ, with uniform steps, presents an order reduction close to the singularity. We generalize the analysis of the gCQ to data satisfying realistic regularity assumptions and provide sufficient conditions for stability and convergence on arbitrary sequences of time points. We consider the particular case of graded meshes and show how to choose them optimally, according to the behaviour of the data. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We describe how the fast and oblivious gCQ can be implemented and illustrate our theoretical results with several numerical experiments.

Generalized Convolution Quadrature for non smooth sectorial problems

TL;DR

This work develops a stability- and error-aware theory for generalized Convolution Quadrature (gCQ) with variable time steps, tailored to sectorial kernels arising in fractional and subdiffusive problems. By leveraging a real integral representation of the kernel, the authors derive sharp a priori error estimates that remain valid under realistic data regularity, and they show how graded meshes can recover full-order convergence. The paper also introduces fast, oblivious implementations of gCQ for the fractional integral and for linear subdiffusion, achieving substantial memory and computational savings while preserving the convergence properties. Numerical experiments on fractional integrals and diffusion problems with smooth and non-smooth data confirm the predicted rates and demonstrate the practical viability of both the theoretical framework and the fast algorithms. Overall, the results provide a robust pathway for accurate, scalable time integration of memory-dependent problems in applied mathematics.

Abstract

We consider the application of the generalized Convolution Quadrature (gCQ) to approximate the solution of an important class of sectorial problems. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) that allows for variable steps. The available stability and convergence theory for the gCQ requires non realistic regularity assumptions on the data, which do not hold in many applications of interest, such as the approximation of subdiffusion equations. It is well known that for non smooth enough data the original CQ, with uniform steps, presents an order reduction close to the singularity. We generalize the analysis of the gCQ to data satisfying realistic regularity assumptions and provide sufficient conditions for stability and convergence on arbitrary sequences of time points. We consider the particular case of graded meshes and show how to choose them optimally, according to the behaviour of the data. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We describe how the fast and oblivious gCQ can be implemented and illustrate our theoretical results with several numerical experiments.
Paper Structure (12 sections, 11 theorems, 152 equations, 11 figures, 4 tables)

This paper contains 12 sections, 11 theorems, 152 equations, 11 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Generalized Convolution Quadrature of the first order
  3. Error analysis in the non smooth case
  4. Application to linear subdiffusion equations
  5. Fast and oblivious gCQ for the fractional integral
  6. Fast and oblivious gCQ for linear subdiffusion equations
  7. Numerical experiments
  8. Computation of the fractional integral
  9. Fractional diffusion equations with non smooth data
  10. Acknowledgements
  11. Data availability
  12. Continuity of the fractional derivative

Key Result

Theorem 4

Let $0<\sigma<\tau_{\max}^{-1}$ with Assume that $K$ satisfies genboundK with $\mu = -\alpha$, for some $\alpha \in (0,1)$. Then if $f\in C^{3}\left( 0,T \right)$, with $f^{\left( \ell\right) }\left( 0\right) =0$, for $\ell = 0,1$, there exist constants $C, \tilde{c}>0$ such that the following error estimate holds

Figures (11)

  • Figure 1: Maximum absolute error in the approximation of \ref{['fracint_exact']} with $\gamma=\frac{1}{\alpha+\beta}$.
  • Figure 2: Maximum absolute error in the approximation of \ref{['fracint_exact']}.
  • Figure 3: Absolute error in the approximation of \ref{['fracint_exact']} with $N=80$.
  • Figure 4: Maximum absolute errors for Example \ref{['ex_ODE']} with $\gamma=\frac{1}{\beta}$.
  • Figure 5: Maximum absolute errors for Example \ref{['ex_ODE']} with different values of $\gamma$.
  • ...and 6 more figures

Theorems & Definitions (26)

  • Definition 2
  • Remark 3
  • Theorem 4: LoSau16
  • Definition 5: Mittag-Leffler function
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Proposition 8
  • proof
  • ...and 16 more