A discrete comparison principle for the time-fractional diffusion equation. (1st September 2020) - Chen, Hu
Abstract: We prove a discrete comparison principle (equivalent to a discrete maximum principle) for the L1 discretisation of the Caputo time derivative and the standard 3-point discretisation of the spatial derivative in the time-fractional initial–boundary value problem D t α u − p Δ u + c ( x, t ) u = f, where p > 0 but no assumption is made on the sign of c . (Previously, any discrete comparison principle relied on the assumption that c ≥ 0 .) The result reveals why one needs a certain condition on the temporal mesh that has been assumed by several authors when analysing numerical methods for this problem. Then this comparison principle is used to give a new error analysis for the case c ≥ 0 by means of a barrier function. All the analysis can be extended to certain other discretisations of the fractional derivative.