An analysis of the Grunwald-Letnikov scheme for initial-value problems with weakly singular solutions

A convergence analysis is given for the GrunwaldLetnikov discretisation of a RiemannLiouville fractional initial-value problem on a uniform mesh t(m) = m tau with m = 0, 1, . . . , M. For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t = 0. Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the GrunwaldLetnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the GrunwaldLetnikov scheme is 0 (tau t(m)(alpha-1)) at each mesh point t(m); hence the scheme is globally only 0 (tau(alpha)) accurate, but it is 0 (tau) accurate for mesh points t(m) that are bounded away from t = 0. Numerical results for a test example show that these theoretical results are sharp. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.