Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations

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 timefractional 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 multidimensional 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.