An explicit multistep method for the Wigner problem

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.