A weighted and shifted difference formula is constructed
based on the Lubich operators, which gives a forth-order
and unconditionally stable difference scheme for the Cauchy problem
of space fractional diffusion equations.
The novelty of the proposed method here is
that only four weighted parameters are required,
compared to eight parameters used in the previous work,
to achieve the fourth-order accuracy and to ensure the stability
at the same time.
To verify the efficiency of the proposed scheme,
several numerical experiments for both one-dimensional
and two-dimensional fractional diffusion problems are provided.
In this work,
the convergence rate for the time derivative is $O(\tau^2)$
by using the Crank-Nicolson scheme.
To improve this convergence, one can apply an idea
we adopted in a later work on tempered fractional diffusion equations.
See [S5] in the publication list.
