In 1986, Dixon and McKee (Z Angew Math Mech 66:535–544, 1986) developed a discrete fractional Gronwall inequality, which can be seen as a generalization of the classical discrete Gronwall inequality. However, this generalized discrete Gronwall inequality and its variant (Al-Maskari and Karaa in SIAM J Numer Anal 57:1524–1544, 2019) have not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Gronwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the fractional Crank-Nicolson type methods. We obtain the optimal \(L^2\) error estimate in space discretization for multi-dimensional problems. The convergence of the fast time-stepping numerical methods is also proved in a simple manner. The present work unifies the convergence analysis of several existing time-stepping schemes. Numerical examples are provided to verify the effectiveness of the present method.

1986 年，Dixon 和 McKee (Z Angew Math Mech 66:535–544, 1986) 开发了离散分数 Gronwall 不等式，可以看作是经典离散 Gronwall 不等式的推广。然而，这种广义离散 Gronwall 不等式及其变体（SIAM J Numer Anal 57:1524–1544, 2019 中的 Al-Maskari 和 Karaa）尚未广泛应用于时间分数演化的时间步长方法的数值分析中方程。本文的主要目的是展示如何应用广义离散 Gronwall 不等式来证明一类时间步长数值方法的收敛性，用于时间分数非线性子扩散方程，包括流行的分数后向差分类型的一阶和二、分数Crank-Nicolson型方法。我们得到最优的多维问题空间离散化中的\(L^2\)误差估计。快速时间步数值方法的收敛性也以简单的方式得到证明。目前的工作统一了几种现有时间步长方案的收敛性分析。提供了数值例子来验证本方法的有效性。