High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III)

In this paper, a series of new high-order numerical approximations to αth (0<α<1) order Caputo derivative is constructed by using rth degree interpolation approximation for the integral function, where r≥4 is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et al., 2015; Li et al., 2014), the convergence orders are O(τr+1−α), where τ is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection-diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O(τr+1−α+h2), where τ is the time stepsize, and h the space one. The stability and convergence of the proposed schemes for r=4 are also considered. Without loss of generality, we only display the numerical examples for r=4,5, which support the numerical algorithms.