  It is well known that using high-order numerical algorithms to solve
fractional differential equations leads to almost the same computational cost
with low-order ones but the accuracy (or convergence order) is greatly
improved, due to the nonlocal properties of fractional operators. Therefore,
developing some high-order numerical approximation formulas for fractional
derivatives play a more important role in numerically solving fractional
differential equations. This paper focuses on constructing (generalized)
high-order fractional-compact numerical approximation formulas for Riesz
derivatives. Then we apply the developed formulas to the one- and two-dimension
Riesz spatial fractional reaction-dispersion equations. The stability and
convergence of the derived numerical algorithms are strictly studied by using
the energy analysis method. Finally, numerical simulations are given to
demonstrate the efficiency and convergence orders of the presented numerical
algorithms.
