  A class of high-order numerical algorithms for Riesz derivatives are
established through constructing new generating functions. Such new high-order
formulas can be regarded as the modification of the classical (or shifted)
Lubich's difference ones, which greatly improve the convergence orders and
stability for time-dependent problems with Riesz derivatives. In rapid
sequence, we apply the 2nd-order formula to one-dimension Riesz spatial
fractional partial differential equations to establish an unconditionally
stable finite difference scheme with convergent order $O(\tau^2+h^2)$, where
$\tau$ and $h$ are the temporal and spatial stepsizes, respectively. Finally,
some numerical experiments are performed to confirm the theoretical results and
testify the effectiveness of the derived numerical algorithms.
