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 $\alpha$ particles at the National Ignition Facility (NIF). 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 ($H \ge 0.1$). This observed low rank structure motivates the development of an adaptive-rank solver, built on a Semi-Lagrangian adaptive-rank (SLAR) 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 $O(N)$ 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.