A bi-fidelity method for the uncertain Vlasov-Poisson system near quasineutrality in an asymptotic-preserving particle-in-cell framework

In this paper, we study the Vlasov-Poisson system with massless electrons (VPME) near quasineutrality and with uncertainties. Based on the idea of reformulation on the Poisson equation by [P. Degond et.al., $\textit{Journal of Computational Physics}$, 229 (16), 2010, pp. 5630–5652], we first consider the deterministic problem and develop an efficient asymptotic-preserving particle-in-cell (AP-PIC) method to capture the quasineutral limit numerically, without resolving the discretizations subject to the small Debye length in plasma. The main challenge and difference compared to previous related works is that we consider the nonlinear Poisson in the VPME system which contains $e^ϕ$ (with $ϕ$ being the electric potential) and provide an explicit scheme. In the second part, we extend to study the uncertainty quantification (UQ) problem and develop an efficient bi-fidelity method for solving the VPME system with multidimensional random parameters, by choosing the Euler-Poisson equation as the low-fidelity model. Several numerical experiments are shown to demonstrate the asymptotic-preserving property of our deterministic solver and the effectiveness of our bi-fidelity method for solving the model with random uncertainties.

💡 Research Summary

This paper addresses the computational challenges posed by the Vlasov‑Poisson system with massless electrons (VPME) in the quasineutral regime, where the dimensionless Debye length ε is very small. In this regime, standard particle‑in‑cell (PIC) methods suffer from severe time‑step and mesh‑size restrictions because the electric field must resolve the Debye scale. The authors propose a two‑fold solution: (1) an asymptotic‑preserving (AP) PIC scheme that remains stable and accurate without resolving ε‑dependent scales, and (2) a bi‑fidelity uncertainty quantification (UQ) framework that dramatically reduces the cost of propagating multidimensional random inputs through the VPME model.

Reformulated Poisson equation.
Starting from the mass and momentum conservation equations for ions and electrons, the authors derive a second‑order‑in‑time reformulation of the Poisson equation:

  −∇·