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

TL;DR

This paper introduces an asymptotic-preserving particle-in-cell method for the Vlasov-Poisson system near quasineutrality, and develops a bi-fidelity approach for uncertainty quantification with random parameters.

Contribution

It presents a novel explicit scheme for the nonlinear Poisson problem in the VPME system and extends to a bi-fidelity UQ method using the Euler-Poisson as a low-fidelity model.

Findings

The deterministic solver accurately captures the quasineutral limit.

The bi-fidelity method effectively handles uncertainties with reduced computational cost.

Numerical experiments confirm the method's efficiency and accuracy.

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 $e^{\phi}$ (with $\phi$ 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.