A Sampling-Based Adaptive-Rank Approach to the Wigner–Poisson System

Abstract.

We develop a mass-conserving, adaptive-rank solver for the 1D1V Wigner–Poisson system. Our work is motivated by applications to the study of the stopping power of [Formula: see text] particles at the National Ignition Facility. In this regime, electrons are in a warm dense state, requiring more than a standard kinetic model. They are hot enough to neglect Pauli exclusion, yet quantum enough to require accounting for uncertainty. The Wigner–Poisson system captures these effects but presents challenges due to its nonlocal nature. Based on a second-order Strang splitting method, we first design a full-rank solver with a structure-preserving Fourier update that ensures the intermediate solutions remain real-valued (up to machine precision), improving upon previous methods. Simulations demonstrate that the solutions exhibit a low-rank structure for moderate to high dimensionless Planck constants ([Formula: see text]). This observed low-rank structure motivates the development of an adaptive-rank solver, built on a semi-Lagrangian adaptive-rank scheme for advection and an adaptive-rank, structure-preserving Fourier update for the Wigner integral terms, with a rigorous proof of structure-preserving property provided. Our solver achieves [Formula: see text] complexity in both storage and computation time, while preserving mass and maintaining momentum accuracy up to the truncation error. The adaptive-rank simulations are visually indistinguishable from the full-rank simulations in capturing solution structures. These results highlight the potential of adaptive-rank methods for high-dimensional Wigner–Poisson simulations, paving the way toward fully kinetic studies of stopping power in warm dense plasmas.

Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/sgong11/WignerPoissonAdaptiveRank and in the supplementary materials ( WignerPoissonAdaptiveRank-main.zip [55.4KB]). [Formula: see text]