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

TL;DR

This paper applies a generalized discrete Gr"{o}nwall inequality to analyze the convergence of various time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, providing optimal error estimates.

Contribution

It demonstrates how to utilize the generalized discrete Gr"{o}nwall inequality in convergence proofs for fractional subdiffusion equations, including popular fractional difference and Crank-Nicolson methods.

Findings

Proves convergence of fractional backward difference methods of orders one and two.

Establishes optimal $L^2$ error estimates in space discretization.

Shows convergence of fast time-stepping methods with simplified analysis.

Abstract

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.