Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion

TL;DR

This paper analyzes the regularity of solutions to a quasilinear subdiffusion model with fractional time derivatives and introduces a high-order convolution quadrature scheme that achieves near-optimal convergence without extra regularity assumptions.

Contribution

It provides new regularity estimates for quasilinear subdiffusion and develops a high-order time stepping scheme with proven convergence order under minimal regularity assumptions.

Findings

Regularity estimates for solutions using smoothing properties.

A second-order backward differentiation convolution quadrature scheme.

Numerical experiments confirming the sharpness of the error estimate.

Abstract

In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order $\alpha \in (0,1)$ in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several pointwise-in-time regularity estimates that are useful for numerical analysis. Then we develop a high-order time stepping scheme for solving quasilinear subdiffusion, based on convolution quadrature generated by second-order backward differentiation formula with correction at the first step. Further, we establish that the convergence order of the scheme is $O(\tau^{1+\alpha-\epsilon})$ without imposing any additional assumption on the regularity of the solution. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions show the sharpness of the error estimate.