Subdiffusion with Time-Dependent Coefficients: Improved Regularity and Second-Order Time Stepping

TL;DR

This paper develops a second-order time discretization method for subdiffusion equations with time-dependent coefficients, establishing regularity results and proving convergence of the method even with nonsmooth data.

Contribution

It introduces a second-order convolution quadrature scheme with correction for subdiffusion equations with time-dependent coefficients, improving accuracy and regularity analysis.

Findings

Second-order convergence achieved with proper correction.

Regularity estimates for subdiffusion equations with time-dependent coefficients.

Numerical experiments confirm theoretical convergence rates.

Abstract

This article concerns second-order time discretization of subdiffusion equations with time-dependent diffusion coefficients. High-order differentiability and regularity estimates are established for subdiffusion equations with time-dependent coefficients. Using these regularity results and a perturbation argument of freezing the diffusion coefficient, we prove that the convolution quadrature generated by the second-order backward differentiation formula, with proper correction at the first time step, can achieve second-order convergence for both nonsmooth initial data and incompatible source term. Numerical experiments are consistent with the theoretical results.