A Structure-preserving Adaptive-Rank Approach to the High-Dimensional Wigner-Poisson System
This work addresses the computational bottleneck of high-dimensional quantum kinetic simulations by enabling efficient low-rank approximations while preserving essential physical symmetries and conservation laws.
The authors developed a structure-preserving adaptive-rank solver for the high-dimensional Wigner-Poisson system that achieves roundoff-level conservation of mass, momentum, and energy, and approximately linear scaling with grid points, enabling long-time 3D3V simulations without full tensor assembly.
The Wigner-Poisson system is a deterministic phase-space model for quantum kinetic electron dynamics, but high-dimensional simulations are limited by the full 3D3V phase space and the nonlocal Wigner potential. We develop a structure-preserving, sampling-based adaptive-rank solver in hierarchical Tucker format for finite-$H$ regimes in which Wigner-Poisson solutions exhibit exploitable low-rank structure. The central difficulty is that adaptive compression can destroy the Fourier-Hermitian tensor symmetry required for a real inverse velocity transform and can break discrete global conservation laws. We address these issues with a Fourier-Hermitian-symmetry-aware sampling and mapping procedure and a global moment correction enforcing mass, momentum, and self-consistent total energy. Numerical tests for two-stream instability and strong Landau damping in 2D2V and 3D3V show roundoff-level conservation, preservation of the real-valued inverse transform, and approximately linear scaling with respect to the number of grid points per coordinate over the tested rank range. The results demonstrate that long-time 3D3V Wigner-Poisson simulations can be performed without assembling the full phase-space tensor.