A Reduced-Order Model for the High-Field Vlasov-Poisson-Fokker-Planck System Based on a Micro-Macro Decomposition

Abstract

We develop a projection-based reduced-order model for the high-field Vlasov-Poisson-Fokker-Planck system. Based on a micro-macro decomposition, the proposed Micro-ROM applies proper orthogonal decomposition (POD)-Galerkin reduction only to the microscopic fluctuation while retaining the macroscopic density on the full spatial grid. The field-dependent Maxwellian and collision operator introduce additional computational costs that cannot be eliminated by Galerkin projection alone. To remove the dependence on the velocity-grid resolution during online simulations, we approximate the projected Maxwellian terms using precomputed Chebyshev expansions of functions of the local electric field and exploit a separable discretization of the collision operator to assemble the reduced collision matrix exactly from precomputed blocks. A first-order implicit-explicit scheme is employed to evolve the reduced coefficients and the density. The resulting method conserves discrete mass and formally recovers a discretization of the limiting high-field model as $\varepsilon\rightarrow 0$. Its online computational cost per time step is independent of the velocity-grid resolution. Numerical experiments demonstrate the accuracy and efficiency of the proposed method across a range of parameters, including the high-field regime.