Convergence Outside the Initial Layer for a Numerical Method for the Time-Fractional Heat Equation

Abstract

In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order $\delta$ where $0<\delta <1$. It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time $t=0$; this reduces the global order of convergence of the finite difference method to $O(h^2+\tau ^\delta )$, where h and $\tau$ are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from $t=0$. This motivates us to investigate if the finite difference method is more accurate away from $t=0$. Numerical experiments with various non-smooth and incompatible initial conditions show that, away from $t=0$, one obtains $O(h^2+\tau )$ convergence.