Convergence in Positive Time for a Finite Difference Method Applied to a Fractional Convection-Diffusion Problem

Abstract

A standard finite difference method on a uniform mesh is used to solve a time-fractional convection-diffusion initial-boundary value problem. Such problems typically exhibit a mild singularity at the initial time t = 0 {t=0} . It is proved that the rate of convergence of the maximum nodal error on any subdomain that is bounded away from t = 0 {t=0} is higher than the rate obtained when the maximum nodal error is measured over the entire space-time domain. Numerical results are provided to illustrate the theoretical error bounds.