Numerical analysis of linear and nonlinear time-fractional subdiffusion equations
Yubo Yang, Fanhai Zeng
TL;DR
This work addresses numerical analysis of linear and nonlinear time-fractional subdiffusion equations driven by the Caputo derivative ${_{0}^{C}{\mathcal{D}}^{\beta}_t u}$ with $0<\beta<1$. It develops a discrete fractional Grönwall inequality for convolution quadrature generated by the generalized Newton–Gregory formula (order up to two) and leverages a temporal–spatial error splitting framework to establish unconditional stability and convergence of Galerkin spectral methods, including a semi-implicit scheme for nonlinear problems. The main contributions are the discrete inequality, unconditional convergence results for both linear and nonlinear problems, and supporting numerical experiments confirming the predicted temporal and spatial convergence behavior, with discussion of how solution regularity affects rates. The results enhance the reliability of high-order time-stepping and spectral discretizations for subdiffusion, with potential impact on simulations of anomalous diffusion in heterogeneous media and related applications.
Abstract
In this paper, a new type of the discrete fractional Gr{ö}nwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear time-fractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Gr{ö}nwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.