High-order central difference scheme for Caputo fractional derivative

In this paper we propose a class of central difference schemes for resolving the Caputo fractional derivative. The accuracy may reach any selected integer order. More precisely, the Caputo fractional derivative operator is decomposed into symmetric and antisymmetric components. Starting from difference schemes of lower order accuracy for each component, we enhance the accuracy by a weighted average of shifted differences. The weights are calculated by matching the symbols of the scheme and the operators. We further illustrate the application of the proposed schemes to a fractional advection-diffusion equation. Together with the Crank-Nicolson algorithm, it reaches designed accuracy order, and is unconditionally stable. Numerical tests are presented to demonstrate the nice features.