  An explicit multistep scheme is proposed for solving the initial-value Wigner
problem. In this scheme, the integrated form of the Wigner equation is
approximated by extrapolation or interpolation polynomials on backwards
characteristics, and the pseudo-differential operator is tackled by the
spectral collocation method. Since it exploits the exact Lagrangian advection,
the time stepping of the multistep scheme is not restricted by the CFL-type
condition. It is also demonstrated that the calculations of the Wigner
potential can be carried out by two successive FFTs, thereby reducing the
computational complexity dramatically. Numerical examples illustrating its
accuracy are presented.
