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

In 1986, Dixon and McKee developed a discrete fractional Gr\"{o}nwall inequality [Z. Angew. Math. Mech., 66 (1986), pp. 535--544], which can be seen as a generalization of the classical discrete Gr\"{o}nwall inequality. However, this generalized discrete Gr\"{o}nwall inequality has 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 Gr\"{o}nwall 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 second-order fractional Crank-Nicolson type methods. We obtain the optimal $L^2$ error estimate in space discretization. The convergence of the fast time-stepping numerical methods is also proved in a simple manner.