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.

翻译：本文研究了一类拟线性次扩散模型，该模型包含阶数为 $\alpha \in (0,1)$ 的时间分数阶导数及非线性扩散系数。首先，利用线性次扩散解算子的光滑性及摄动论证，我们证明了若干对数值分析有用的逐点时间正则性估计。随后，我们基于第一步带校正的二阶向后差分公式生成的卷积求积，发展了一种求解拟线性次扩散的高阶时间步进格式。进一步地，我们证明了在不强加任何额外解正则性假设的前提下，该格式的收敛阶为 $O(\tau^{1+\alpha-\epsilon})$。分析依赖于非线性摄动余项的精细 Sobolev 正则性及离散解算子的光滑性质。在二维空间中的若干数值实验验证了误差估计的尖锐性。