  As a phase space language for quantum mechanics, the Wigner function approach
bears a close analogy to classical mechanics and has been drawing growing
attention, especially in simulating quantum many-body systems. However,
deterministic numerical solutions have been almost exclusively confined to
one-dimensional one-body systems and few results are reported even for
one-dimensional two-body problems. This paper serves as the first attempt to
solve the time-dependent many-body Wigner equation through a grid-based
advective-spectral-mixed method. The main feature of the method is to resolve
the linear advection in $(\bm{x},t)$-space by an explicit three-step
characteristic scheme coupled with the piecewise cubic spline interpolation,
while the Chebyshev spectral element method in $\bm k$-space is adopted for
accurate calculation of the nonlocal pseudo-differential term. Not only the
time step of the resulting method is not restricted by the usual CFL condition
and thus a large time step is allowed, but also the mass conservation can be
maintained. In particular, for the system consisting of identical particles,
the advective-spectral-mixed method can also rigorously preserve physical
symmetry relations. The performance is validated through several typical
numerical experiments, like the Gaussian barrier scattering, electron-electron
interaction and a Helium-like system, where the third-order accuracy against
both grid spacing and time stepping is observed.
