Asymptotic-preserving conservative semi-Lagrangian discontinuous Galerkin schemes for the Vlasov-Poisson system in the quasi-neutral limit

TL;DR

This work tackles numerically solving the Vlasov–Poisson system in the stiff quasi-neutral limit without requiring Debye-length resolution. It develops asymptotic-preserving conservative semi-Lagrangian DG schemes (AP-CSLDG) built on a reformulated VP framework (RP^λ/RP^0), a CSLDG transport solver, and a Fourier-based Poisson solver, augmented by a positivity limiter. The authors prove AP properties, including consistency with RP^0 in the quasi-neutral limit and stability analyses, and demonstrate accuracy and robustness through extensive non-quasi-neutral and quasi-neutral numerical experiments. The approach delivers high-order accuracy in phase space, preserves mass, and remains efficient across regimes, making it suitable for multiscale plasma simulations and extensible to more complex kinetic models.

Abstract

We discretize the Vlasov-Poisson system using conservative semi-Lagrangian (CSL) discontinuous Galerkin (DG) schemes that are asymptotic preserving (AP) in the quasi-neutral limit. The proposed method (CSLDG) relies on two key ingredients: the CSLDG discretization and a reformulated Poisson equation (RPE). The use of the CSL formulation ensures local mass conservation and circumvents the Courant-Friedrichs-Lewy condition, while the DG method provides high-order accuracy for capturing fine-scale phase space structures of the distribution function. The RPE is derived by the Poisson equation coupled with moments of the Vlasov equation. The synergy between the CSLDG and RPE components makes it possible to obtain reliable numerical solutions, even when the spatial and temporal resolution might not fully resolve the Debye length. We rigorously prove that the proposed method is asymptotically stable, consistent and satisfies AP properties. Moreover, its efficiency is maintained across non-quasi-neutral and quasi-neutral regimes. These properties of our approach are essential for accurate and robust numerical simulation of complex electrostatic plasmas. Several numerical experiments verify the accuracy, stability and efficiency of the proposed CSLDG schemes.

Figures (13)

• Figure 3.1: Schematic illustration of the CSLDG scheme in 1D.
• Figure 4.1: Illustration of the AP properties.
• Figure 5.1: Two stream instability I: $P^1$ (left) and $P^2$ (right) polynomial spaces, time evolution of the logarithm of electrostatic energy log$(\varepsilon_p)$ with different CFL numbers.
• Figure 5.2: Two stream instability I: $P^1$ (left) and $P^2$ (right) polynomial spaces, time evolution of the relative deviations for mass (top) and $L^2$ norm (bottom) with different CFL numbers.
• Figure 5.3: Two stream instability I: $P^1$ (left) and $P^2$ (right) polynomial spaces, time evolution of the relative deviations for entropy (top) and energy (bottom) with different CFL numbers.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Remark 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2: Conservation of the particle load