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An Asymptotic Preserving Scheme for the Euler-Poisson-Boltzmann System in the Quasineutral Limit

K. R. Arun, R. Ghorai

TL;DR

The paper addresses efficient, stable numerical simulation of the EPB system across dispersive and quasineutral regimes by designing an asymptotic preserving, energy-stable semi-implicit finite-volume scheme on a MAC grid. It introduces stabilisation terms in the fluxes and source, proves discrete energy balance and existence of solutions, and establishes weak consistency and AP convergence to the ICE limit as $\varepsilon\to0$. The method is validated through a suite of numerical tests, demonstrating robustness in resolving plasma sheaths and preserving correct quasineutral behavior without prohibitive time-step restrictions. The work offers a practically impactful tool for simulating low-temperature plasma dynamics with multiscale features, balancing accuracy, stability, and computational efficiency.

Abstract

In this paper, we study an asymptotic preserving (AP), energy stable and positivity preserving semi-implicit finite volume scheme for the Euler-Poisson-Boltzmann (EPB) system in the quasineutral limit. The key to energy stability is the addition of appropriate stabilisation terms into the convective fluxes of mass and momenta, and the source term. The space-time fully-discrete scheme admits the positivity of the mass density, and is consistent with the weak formulation of the EPB system upon mesh refinement. In the quasineutral limit, the numerical scheme yields a consistent, semi-implicit discretisation of the isothermal compressible Euler system, thus leading to the AP property. Several benchmark numerical case studies are performed to confirm the robustness and efficacy of the proposed scheme in the dispersive as well as the quasineutral regimes. The numerical results also corroborates scheme's ability to very well resolve plasma sheaths and the related dynamics, which indicates its potential to applications involving low-temperature plasma problems.

An Asymptotic Preserving Scheme for the Euler-Poisson-Boltzmann System in the Quasineutral Limit

TL;DR

The paper addresses efficient, stable numerical simulation of the EPB system across dispersive and quasineutral regimes by designing an asymptotic preserving, energy-stable semi-implicit finite-volume scheme on a MAC grid. It introduces stabilisation terms in the fluxes and source, proves discrete energy balance and existence of solutions, and establishes weak consistency and AP convergence to the ICE limit as . The method is validated through a suite of numerical tests, demonstrating robustness in resolving plasma sheaths and preserving correct quasineutral behavior without prohibitive time-step restrictions. The work offers a practically impactful tool for simulating low-temperature plasma dynamics with multiscale features, balancing accuracy, stability, and computational efficiency.

Abstract

In this paper, we study an asymptotic preserving (AP), energy stable and positivity preserving semi-implicit finite volume scheme for the Euler-Poisson-Boltzmann (EPB) system in the quasineutral limit. The key to energy stability is the addition of appropriate stabilisation terms into the convective fluxes of mass and momenta, and the source term. The space-time fully-discrete scheme admits the positivity of the mass density, and is consistent with the weak formulation of the EPB system upon mesh refinement. In the quasineutral limit, the numerical scheme yields a consistent, semi-implicit discretisation of the isothermal compressible Euler system, thus leading to the AP property. Several benchmark numerical case studies are performed to confirm the robustness and efficacy of the proposed scheme in the dispersive as well as the quasineutral regimes. The numerical results also corroborates scheme's ability to very well resolve plasma sheaths and the related dynamics, which indicates its potential to applications involving low-temperature plasma problems.
Paper Structure (24 sections, 21 theorems, 94 equations, 12 figures)

This paper contains 24 sections, 21 theorems, 94 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Euler-Poisson-Boltzmann System and the Quasineutral Limit
  3. Apriori Energy Estimates
  4. Quasineutral limit
  5. Stabilisation
  6. Space Discretisation and Discrete Differential Operators
  7. Mesh and Unknowns
  8. Discrete Convection Fluxes and Differential Operators
  9. Energy Stable Semi-implicit Scheme
  10. Discrete Identities
  11. Existence of a Numerical Solution
  12. Weak Consistency of the Scheme
  13. Quasineutral Limit of the Scheme
  14. Numerical Results
  15. Five Branch Problem
  16. ...and 9 more sections

Key Result

Proposition 2.2

The classical solutions of eq:cons_mas-eq:poisson-boltz satisfy the following identities:

Figures (12)

  • Figure 1: Ion density, velocity, potential plots for the five-branch test case at $T=1$ for $\varepsilon=1$.
  • Figure 2: Ion density, velocity, potential plots for the five-branch test case at $T=1$ for $\varepsilon=10^{-2}$.
  • Figure 3: Density, velocity and potential profiles for the shock tube test case at $T=0.2$ for $\varepsilon=10^{-2}$.
  • Figure 4: The density, velocity and potential profiles for the shock tube test case for different times corresponding to $\varepsilon=10^{-4}$.
  • Figure 5: Zoom of the density, velocity and potential profiles for the shock tube test case at $T=0.1$ corresponding to $\varepsilon=10^{-4}$.
  • ...and 7 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 45 more