  In this paper, the coupled fractional Ginzburg-Landau equations are first
time investigated numerically. A linearized implicit finite difference scheme
is proposed. The scheme involves three time levels, is unconditionally stable
and second-order accurate in both time and space variables. The unique
solvability, the unconditional stability and optimal pointwise error estimates
are obtained by using the energy method and mathematical induction. Moreover,
the proposed second-order method can be easily extended into the fourth-order
method by using an average finite difference operator for spatial fractional
derivatives and Richardson extrapolation for time variable. Finally, numerical
results are presented to confirm the theoretical results.
