Optimal Convergence Rates in Time-Fractional Discretisations: the L1, L1 and Alikhanov Schemes

Consider the discretisation of the initial-value problem D alpha u(t) = f (t) for 0 < t < T with u(0) = u0, where D alpha u(t) is a Caputo derivative of order alpha is an element of (0, 1), using the L1 scheme on a graded mesh with N points. It is well known that one can prove the maximum nodal error in the computed solution is at most (R) (N-min{r alpha,2-alpha}), where r >= 1 is the mesh grading parameter (r = 1 generates a uniform mesh). Numerical experiments indicate that this error bound is sharp, but no proof of its sharpness has been given. In the present paper the sharpness of this bound is proved, and the sharpness of the analogous nodal error bounds for the L1 and Alikhanov schemes will also be proved, using modifications of the L1 analysis.