Abstract: A series of semi-weighted implicit finite difference schemes for solving one-dimensional fractional advection-dispersion equations with variable coefficients on a finite domain are considered in this paper. The schemes are proved  unconditionally stable and second-order accuracy in spatial grid size for the problem with order of fractional derivative belonging to $[(\sqrt{17}-1)/2,2].$ Numerical examples are provided to verify the theoretical analysis.

Key words: semi-weighted implicit finite difference schemes, fractional advection-dispersion equations, unconditionally stable