AN ADVECTIVE-SPECTRAL-MIXED METHOD FOR TIME-DEPENDENT MANY-BODY WIGNER SIMULATIONS

作者：Xiong, Yunfeng; Chen, Zhenzhu; Shao, Sihong^*

来源：SIAM Journal on Scientific Computing, 2016, 38(4): B491-B520.

DOI：10.1137/15M1051373

• many-body Wigner equation
• semi-Lagrangian method
• Pauli exclusion principle
• Chebyshev spectral method
• Adams multistep scheme
• quantum transport

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 (x, t)-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in k-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.

• 出版日期2016
• 单位北京大学