CORRECTION OF HIGH-ORDER BDF CONVOLUTION QUADRATURE FOR FRACTIONAL EVOLUTION EQUATIONS

We develop proper correction formulas at the starting k - 1 steps to restore the desired k th-order convergence rate of the k-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired k th-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case alpha is an element of (0, 1) and sketch the proof for the di ff usion-wave case alpha is an element of(1, 2). Extensive numerical examples are provided to illustrate the e ff ectiveness of the proposed scheme.