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

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1512-1539, August 2025.
Abstract. In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order [math] in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several new pointwise-in-time Sobolev regularity estimates that are useful for numerical analysis. Then we develop a time-stepping scheme to solve quasilinear subdiffusion, based on convolution quadrature generated by the second-order backward differentiation formula with a correction at the first step. Further, we establish that the convergence order of the scheme is [math] without imposing any additional assumption on the regularity of the solution, which is high-order in the sense that its convergence rate is higher than the first-order convergence of the vanilla scheme. 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 are presented to show the sharpness of the error estimate.