  In this paper, two kinds of high-order compact finite difference schemes for
second-order derivative are developed. Then a second-order numerical scheme for
Riemann-Liouvile derivative is established based on fractional center
difference operator. We apply these methods to fractional anomalous
subdiffusion equation to construct two kinds of novel numerical schemes. The
solvability, stability and convergence analysis of these difference schemes are
studied by Fourier method in details. The convergence orders of these numerical
schemes are $\mathcal {O}(\tau^2+h^6)$ and $\mathcal {O}(\tau^2+h^8)$,
respectively. Finally, numerical experiments are displayed which are in line
with the theoretical analysis.
