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Asymptotically Compatible Fractional Grönwall Inequality and its Applications

Daopeng Yin, Liquan Mei

TL;DR

This work addresses the challenge of obtaining an accurate and robust time-integration analysis for time-fractional PDEs on non-uniform grids. By deriving sharp asymptotic estimates for discrete complementary convolution kernels and linking them to a Grönwall-type bound, the authors establish an asymptotically compatible fractional Grönwall inequality that bridges the discrete and continuous regimes as the mesh refines. The results unlock rigorous pointwise-in-time error and stability analysis for non-uniform time stepping, particularly the L1 scheme on graded meshes, with explicit error rates dependent on the grading parameter and fractional order. Numerical verifications on growing linear ODEs and reaction-diffusion problems corroborate the theory and demonstrate practical implications for grid design and error control in time-fractional simulations. The paper also provides concise DCS estimates and a necessity result for the Caputo discrete coefficient, informing the stability considerations for higher-order fractional schemes.

Abstract

In this work, we will give proper estimates for the discrete convolution complementary (DCC) kernels, which leads to the asymptotically compatible fractional Grönwall inequality. The consequence can be applied in the analysis of the stability and pointwise-in-time error of difference-type schemes on a non-uniform mesh. The pointwise error is explicitly bound when a non-uniform time grid is given by a specific scale function e.g. graded mesh, can be given directly. Numerical experiments towards the conclusion of this work validate the error analysis.

Asymptotically Compatible Fractional Grönwall Inequality and its Applications

TL;DR

This work addresses the challenge of obtaining an accurate and robust time-integration analysis for time-fractional PDEs on non-uniform grids. By deriving sharp asymptotic estimates for discrete complementary convolution kernels and linking them to a Grönwall-type bound, the authors establish an asymptotically compatible fractional Grönwall inequality that bridges the discrete and continuous regimes as the mesh refines. The results unlock rigorous pointwise-in-time error and stability analysis for non-uniform time stepping, particularly the L1 scheme on graded meshes, with explicit error rates dependent on the grading parameter and fractional order. Numerical verifications on growing linear ODEs and reaction-diffusion problems corroborate the theory and demonstrate practical implications for grid design and error control in time-fractional simulations. The paper also provides concise DCS estimates and a necessity result for the Caputo discrete coefficient, informing the stability considerations for higher-order fractional schemes.

Abstract

In this work, we will give proper estimates for the discrete convolution complementary (DCC) kernels, which leads to the asymptotically compatible fractional Grönwall inequality. The consequence can be applied in the analysis of the stability and pointwise-in-time error of difference-type schemes on a non-uniform mesh. The pointwise error is explicitly bound when a non-uniform time grid is given by a specific scale function e.g. graded mesh, can be given directly. Numerical experiments towards the conclusion of this work validate the error analysis.
Paper Structure (11 sections, 10 theorems, 65 equations, 2 figures, 4 tables)

This paper contains 11 sections, 10 theorems, 65 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Asymptotically Compatible Fractional Grönwall Inequality
  3. The asymptotic analysis of the DCC
  4. The Asymptotically compatible fractional Grönwall inequality
  5. Error analysis
  6. Numerical Verification
  7. The growing linear ODE
  8. The reaction-diffusion equation
  9. Conclusions
  10. The estimate of DCS
  11. A proof of necessity

Key Result

lemma thmcounterlemma

For all $1 \le k+1 \le j \le n$,

Figures (2)

  • Figure 1: The diagram of the $p_{n-k}^{(n)}$ and $\tilde{p}_{n-k}^{(n)}$ with fixed $n=32$
  • Figure 2: The graphs of $p_{n-k}^{(n)}$ and $\tilde{p}_{n-k}^{(n)}$ with fixed $n=1024$.

Theorems & Definitions (22)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma: The asymptotic analysis of DCC
  • proof
  • remark thmcounterremark
  • remark thmcounterremark
  • theorem 1
  • proof
  • corollary thmcountercorollary
  • lemma thmcounterlemma: $L^2$-stability
  • ...and 12 more