Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete stability estimates for the pressureless Euler-Poisson-Boltzmann equations in the Quasi-Neutral limit

Mehdi Badsi, Nicolas Crouseilles

TL;DR

This work develops and analyzes a fully implicit finite-volume scheme for the 1D pressureless Euler-Poisson-Boltzmann system on the torus, incorporating a discretization of the forcing term that yields unconditional discrete energy decay. It introduces a discrete modulated-energy framework to establish nonlinear stability around constant states in the quasi-neutral limit and proves the existence of the scheme via a Brouwer-fixed-point argument, along with discrete renormalized continuity and energy estimates. The authors provide detailed numerical experiments, including well- and ill-prepared data and a convergence study as the Debye length $\\varepsilon$ tends to zero, demonstrating stability, energy dissipation, and convergence toward the limiting isothermal Euler regime under suitable conditions. The results offer a robust, entropy-stable discretization approach for EPB-type plasmas in the quasi-neutral limit and deliver practical tools for assessing stability and convergence of numerical schemes in plasma modeling. Overall, the paper makes a significant contribution by bridging rigorous discrete stability theory with practical, implicit schemes for nonlinear plasma dynamics in the quasi-neutral setting.

Abstract

We propose and study a fully implicit finite volume scheme for the pressureless Euler-Poisson-Boltzmann equations on the one dimensional torus. Especially, we design a consistent and dissipative discretization of the force term which yields an unconditional energy decay. In addition, we establish a discrete analogue of the modulated energy estimate around constant states with a small velocity. Numerical experiments are carried to illustrate our theoretical results and to assess the accuracy of our scheme. A test case of the literature is also illustrated.

Discrete stability estimates for the pressureless Euler-Poisson-Boltzmann equations in the Quasi-Neutral limit

TL;DR

This work develops and analyzes a fully implicit finite-volume scheme for the 1D pressureless Euler-Poisson-Boltzmann system on the torus, incorporating a discretization of the forcing term that yields unconditional discrete energy decay. It introduces a discrete modulated-energy framework to establish nonlinear stability around constant states in the quasi-neutral limit and proves the existence of the scheme via a Brouwer-fixed-point argument, along with discrete renormalized continuity and energy estimates. The authors provide detailed numerical experiments, including well- and ill-prepared data and a convergence study as the Debye length tends to zero, demonstrating stability, energy dissipation, and convergence toward the limiting isothermal Euler regime under suitable conditions. The results offer a robust, entropy-stable discretization approach for EPB-type plasmas in the quasi-neutral limit and deliver practical tools for assessing stability and convergence of numerical schemes in plasma modeling. Overall, the paper makes a significant contribution by bridging rigorous discrete stability theory with practical, implicit schemes for nonlinear plasma dynamics in the quasi-neutral setting.

Abstract

We propose and study a fully implicit finite volume scheme for the pressureless Euler-Poisson-Boltzmann equations on the one dimensional torus. Especially, we design a consistent and dissipative discretization of the force term which yields an unconditional energy decay. In addition, we establish a discrete analogue of the modulated energy estimate around constant states with a small velocity. Numerical experiments are carried to illustrate our theoretical results and to assess the accuracy of our scheme. A test case of the literature is also illustrated.

Paper Structure

This paper contains 21 sections, 14 theorems, 172 equations, 6 figures, 4 tables.

Table of Contents

  1. keywords:
  2. MSCcodes:
  3. Introduction
  4. Conservation properties
  5. The modulated energy estimate
  6. Discretization
  7. The time implicit scheme
  8. A priori estimates and existence
  9. Definition of a Mapping.
  10. Stability of $B_{M}.$
  11. Discrete modulated energy estimate
  12. Numerical experiment
  13. Description of the non linear solver
  14. Non linear stability around constant states
  15. Convergence as $\varepsilon \rightarrow 0$ with a fixed mesh-size
  16. ...and 6 more sections

Key Result

Proposition 1

(Conservations) Let $(\rho_{\varepsilon},u_{\varepsilon},\phi_{\varepsilon})$ be a strong solution to EPB on $[0,T]$ with $\rho_{\varepsilon} > 0.$ Then we have for $t \in [0,T]:$ where $\mathcal{H}$ is the total energy given by where

Figures (6)

  • Figure 1: Left: time evolution of the modulated energy on $[0,T]$ with $\varepsilon = 10^{-1}$ for three values of $\Delta x$. Right: density $\rho(T=0.2,\cdot)$ with $\varepsilon = 10^{-1}$ for three values of $\Delta x$.
  • Figure 2: Plots of the initial density $\rho(0, \cdot)$ and final density $\rho(20\Delta t, \cdot)$ for $\varepsilon \in \lbrace 0.1, 0.05, 0.025\rbrace$ on the fine mesh: $\Delta x=10^{-3}$ and $\Delta t=\frac{1}{2}\Delta x$.
  • Figure 3: Time evolution of $t\in [0, 5]\mapsto \|\rho(t, \cdot)-\bar{\rho}\|_{L^2}/\varepsilon^2$ for $\varepsilon \in \lbrace 0.01, 0.05, 0.1\rbrace$. $\Delta x=1/200$ and $\Delta t=\frac{1}{2}\Delta x$
  • Figure 4: Plots of the initial density $\rho(0, x)$ and final density $\rho(20\Delta t, x)$ for $\varepsilon \in \lbrace 0.1, 0.05, 0.025\rbrace$ on a fine mesh: $\Delta x=10^{-3}$ and $\Delta t=\frac{1}{2}\Delta x$.
  • Figure 5: Five-branch test: comparison of the scheme for $\varepsilon=10^{-2}, 10^{-4}$ and the asymptotic scheme $\varepsilon=0$. Left: density $\rho(t=0.5, \cdot)$. Right: velocity $u(t=0.5, \cdot)$. $\Delta x=2\pi/400$ and $\Delta t=\frac{1}{2}\Delta x$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 18 more