A discrete comparison principle for the time-fractional diffusion equation. (1st September 2020)

• Record Type: Journal Article
• Authors: Chen, Hu; Stynes, Martin
• Is Part Of: Computers & mathematics with applications. Volume 80:issue 5(2020)
• Page Start: 917
• Page End: 922
• Publication Date: 2020-09-01
• Subjects: Discrete comparison principle -- L1 scheme -- Weak singularity -- Convergence analysis; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2020.04.018
• Languages: English
• ISSNs: 0898-1221

Abstract

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.