  We propose a locally one dimensional (LOD) finite difference method for
multidimensional Riesz fractional diffusion equation with variable coefficients
on a finite domain. The numerical method is second-order convergent in both
space and time directions, and its unconditional stability is strictly proved.
Comparing with the popular first-order finite difference method for fractional
operator, the form of obtained matrix algebraic equation is changed from
$(I-A)u^{k+1}=u^k+b^{k+1}$ to $(I-{\widetilde A})u^{k+1}=(I+{\widetilde
B})u^k+{\tilde b}^{k+1/2}$; the three matrices $A$, ${\widetilde A}$ and
${\widetilde B}$ are all Toeplitz-like, i.e., they have completely same
structure and the computational count for matrix vector multiplication is
$\mathcal{O}(N {log} N)$; and the computational costs for solving the two
matrix algebraic equations are almost the same. The LOD-multigrid method is
used to solve the resulting matrix algebraic equation, and the computational
count is $\mathcal{O}(N {log} N)$ and the required storage is $\mathcal{O}(N)$,
where $N$ is the number of grid points. Finally, the extensive numerical
experiments are performed to show the powerfulness of the second-order scheme
and the LOD-multigrid method.
