  Numerical resolution for 6-D Wigner dynamics under the Coulomb potential
faces with the combined challenges of high dimensionality, nonlocality,
oscillation and singularity. In particular, the extremely huge memory storage
of 6-D grids hinders the usage of all existing deterministic numerical scheme,
which is well-known as the curse of dimensionality. To surmount these
difficulties, we propose a massively parallel solver, termed the
CHAracteristic-Spectral-Mixed (CHASM) scheme, by fully exploiting two distinct
features of the Wigner equation: Locality of spatial advection and nonlocality
of quantum interaction. Our scheme utilizes the local cubic B-spline basis to
interpolate the local spatial advection. The key is to use a perfectly matched
boundary condition to give a closure of spline coefficients, so that
distributed pieces can recover the global one as accurately as possible owing
to the rapid decay of wavelet basis in the dual space, and communication costs
are significantly reduced. To resolve the nonlocal pseudodifferential operator
with weakly singular symbol, CHASM further adopts the truncated kernel method
to attain a highly efficient approximation. Several typical experiments
including the quantum harmonic oscillator and Hydrogen 1s state demonstrate the
accuracy and efficiency of CHASM. The non-equilibrium electron-proton couplings
are also clearly displayed and reveal the uncertainty principle and quantum
tunneling in phase space. Finally, the scalability of CHASM up to 16000 cores
is presented.
