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

TL;DR

The paper develops an asymptotic-preserving particle-in-cell method for the Vlasov-Poisson system with massless electrons near quasineutrality, addressing the nonlinear Poisson term $e^{\phi}$ and ensuring uniform accuracy without resolving the Debye length. It then couples this high-fidelity AP-PIC solver with a low-fidelity Euler-Poisson model in a bi-fidelity stochastic collocation framework to efficiently quantify uncertainty from multidimensional random parameters. Numerical experiments demonstrate both the AP property in the deterministic setting and the effectiveness of the bi-fidelity approach across several uncertainty scenarios, including KL-expansion-based inputs. The work offers a practical pathway to accurate, scalable simulations of kinetic plasmas with uncertain parameters, with potential extensions to higher dimensions and higher-order time discretizations.

Abstract

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.

Figures (13)

• Figure 1: Flowchart of our AP-PIC method for solving the VPME system \ref{['RF_VPME1']}--\ref{['RF_VPME2']}.
• Figure 2: (Two schemes for solving the reformulated Poisson, $\varepsilon = 0.01$). Density $n$ (left) and velocity $u$ (right) at $t=2$ by an implicit scheme \ref{['Full-scheme']} with Newton's iteration (red) and an explicit scheme based on penalty \ref{['eq:RF_Poisson_explicit']} (blue).
• Figure 3: (AP property for the RF-VPME in Subsection \ref{['sec:AP']}). The $l_{2}$ errors $\mathcal{E}_{l_2}$ of density $n$, velocity $u$ and potential $\phi$ for $\varepsilon = 1$ at $t = 0.2$. Here, the $x$-axis is $\log(\Delta x)$, and the $y$-axis is $\log(\mathcal{E}_{l_2})$. The mesh sizes in space are $N_x = 10, 20, 40$ and $50$.
• Figure 4: (AP property for the RF-VPME in Subsection \ref{['sec:AP']}). The $l_{2}$ errors $\mathcal{E}_{l_2}$ of density $n$, velocity $u$ and potential $\phi$ for $\varepsilon = 0.1, 0.01$ and $10^{-4}$ at $t = 0.2$. Here, the $x$-axis is $\log(\Delta x)$, and the $y$-axis is $\log(\mathcal{E}_{l_2})$. The mesh sizes in space are $N_x = 25, 50, 100$ and $200$.
• Figure 5: (Long time evolution of RF-VPME in Subsection \ref{['sec:AP']}). The relative errors $\mathcal{E}_{r}$ of density $n$, velocity $u$ and potential $\phi$ with $\varepsilon = 0.1, 0.01$ and $10^{-4}$ between RF-VPME and Quasi-VP models, from $t = 0$ to $t = 15$. Here, the $x$-axis is time $t$, and the $y$-axis is $\log(\mathcal{E}_{r})$. The mesh size in space is $N_x=100$.