2016/01/01 by Yunfeng Xiong, Zhenzhu Chen, Sihong Shao · 32 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Chemical Physics Studies #Advection #Electromagnetic Simulation and Numerical Methods #Mathematical analysis #Mathematics #Partial differential equation #Physics #Piecewise #Quantum #Quantum mechanics #Quantum, superfluid, helium dynamics #Wigner distribution function
paper · doi:10.1137/15m1051373
published in SIAM Journal on Scientific Computing 38(4), B491-B520 (Society for Industrial and Applied Mathematics)
openalex publication_date 2016/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/30
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 (\boldsymbolx,t)-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in \boldsymbolk-space is adopted for accurate calculation of the nonlocal pseudo-differential term. Not only is the time step of the resulting method 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.