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Some Grönwall inequalities for a class of discretizations of time fractional equations on nonuniform meshes

Yuanyuan Feng, Lei Li, Jian-Guo Liu, Tao Tang

TL;DR

The paper develops a resolvent-based framework for completely positive discretizations of time-fractional equations on nonuniform meshes, establishing robust continuous- and discrete-time comparison principles and discrete Grönwall inequalities that do not require step-size ratio restrictions. Using pseudo-convolution and resolvent kernels, it derives uniform-in-time bounds and decay estimates for dissipative systems and applies these results to subdiffusion and time-fractional Allen–Cahn equations. The main contributions include new comparison principles for variable-step schemes and multiple Grönwall inequalities that yield explicit Mittag–Leffler decay rates and error bounds, enhancing stability analysis under adaptive time stepping. This work provides rigorous tools for long-time stability and accurate error control in time-fractional PDEs on nonuniform grids, with direct implications for adaptive time-stepping strategies in practical simulations.

Abstract

We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have any restrictions on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen-Cahn equations for illustration.

Some Grönwall inequalities for a class of discretizations of time fractional equations on nonuniform meshes

TL;DR

The paper develops a resolvent-based framework for completely positive discretizations of time-fractional equations on nonuniform meshes, establishing robust continuous- and discrete-time comparison principles and discrete Grönwall inequalities that do not require step-size ratio restrictions. Using pseudo-convolution and resolvent kernels, it derives uniform-in-time bounds and decay estimates for dissipative systems and applies these results to subdiffusion and time-fractional Allen–Cahn equations. The main contributions include new comparison principles for variable-step schemes and multiple Grönwall inequalities that yield explicit Mittag–Leffler decay rates and error bounds, enhancing stability analysis under adaptive time stepping. This work provides rigorous tools for long-time stability and accurate error control in time-fractional PDEs on nonuniform grids, with direct implications for adaptive time-stepping strategies in practical simulations.

Abstract

We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have any restrictions on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen-Cahn equations for illustration.
Paper Structure (15 sections, 26 theorems, 152 equations)

This paper contains 15 sections, 26 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and setup
  3. The generalized Caputo derivatives and the resolvent kernels
  4. Discretization on nonuniform meshes
  5. Completely positive discretizations
  6. The comparison principles
  7. Comparison principles for time continuous case
  8. Comparison principles for numerical schemes
  9. Grönwall inequalities for completely positive schemes
  10. A Grönwall inequality for uniform bound
  11. A Grönwall inequality for decay estimates
  12. A Grönwall inequality for growing linear functions
  13. Applications to dissipative systems
  14. Example 1: subdiffusion equation
  15. Example 2: time fractional Allen-Cahn equation

Key Result

Theorem 1.1

Consider the differential form eq:diffdis that is completely positive and assume that $c_{n-j}^n$ is comparible to the average of $g_{1-\alpha}$ on the $j$th interval in some sense (see the corresponding sections for the details). If a nonnegative sequence $v_n$ satisfies that $\mathcal{D}_{\tau}^{\

Theorems & Definitions (44)

  • Theorem 1.1: Informal version of Theorem \ref{['thm:unibound']} and \ref{['thm:decayest']}
  • Theorem 1.2: Informal version of Theorem \ref{['thm:grongrow']}
  • Definition 2.1: liliu18fracliliu2018compact
  • Proposition 2.1: liliu18frac
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.1
  • ...and 34 more