An advective-spectral-mixed method for time-dependent many-body Wigner simulations

TL;DR

This paper introduces a novel grid-based advective-spectral-mixed numerical method for solving the time-dependent many-body Wigner equation, enabling accurate, large time step simulations while preserving physical properties.

Contribution

It is the first to develop a grid-based advective-spectral-mixed method specifically for many-body Wigner equations, combining characteristic schemes with spectral methods for improved accuracy and efficiency.

Findings

Achieved third-order accuracy in numerical experiments

Validated method on Gaussian barrier, electron interactions, and Helium-like systems

Method preserves physical symmetry and mass conservation

Abstract

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.