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High order Asymptotic Preserving penalized numerical schemes for the Euler-Poisson system in the quasi-neutral limit

Nicolas Crouseilles, Giacomo Dimarco, Saurav Samantaray

Abstract

In this work, we focus on the development of high-order Implicit-Explicit (IMEX) finite volume numerical methods for plasmas in quasineutral regimes. At large temporal and spatial scales, plasmas tend to be quasineutral, meaning that the local net charge density is nearly zero. However, at small time and spatial scales, measured by the the Debye length, quasineutrality breaks down. In such regimes, standard numerical methods face severe stability constraints, rendering them practically unusable. To address this issue, we introduce and analyze a class of penalized IMEX Runge-Kutta methods for the Euler-Poisson (EP) system, specifically designed to handle the quasineutral limit. These schemes are uniformly stable with respect to the Debye length and degenerate into high-order methods as the quasineutral limit is approached. Several numerical tests confirm that the proposed methods exhibit the desired properties.

High order Asymptotic Preserving penalized numerical schemes for the Euler-Poisson system in the quasi-neutral limit

Abstract

In this work, we focus on the development of high-order Implicit-Explicit (IMEX) finite volume numerical methods for plasmas in quasineutral regimes. At large temporal and spatial scales, plasmas tend to be quasineutral, meaning that the local net charge density is nearly zero. However, at small time and spatial scales, measured by the the Debye length, quasineutrality breaks down. In such regimes, standard numerical methods face severe stability constraints, rendering them practically unusable. To address this issue, we introduce and analyze a class of penalized IMEX Runge-Kutta methods for the Euler-Poisson (EP) system, specifically designed to handle the quasineutral limit. These schemes are uniformly stable with respect to the Debye length and degenerate into high-order methods as the quasineutral limit is approached. Several numerical tests confirm that the proposed methods exhibit the desired properties.
Paper Structure (27 sections, 4 theorems, 78 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 27 sections, 4 theorems, 78 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. One-fluid Euler-Poisson System and its quasi-neutral limit
  3. Time Discretisation
  4. First order time discretisation
  5. IMEX-RK and indirect approach for DAEs of index 1
  6. IMEX-RK Time Discretisation
  7. Indirect Approach for RK Methods for DAEs of Index 1
  8. High Order Time Semi-discretisation
  9. Asymptotic Preserving property
  10. Preserving the constraints
  11. 1. First iteration ($n=0$)
  12. First stage $(i=1)$
  13. Induction on the stages of the first iteration
  14. 2. Second iteration ($n=1$)
  15. First stage $(i=1)$
  16. ...and 12 more sections

Key Result

Proposition 1

Let us assume that the initial conditions i.e. are well-prepared in the sense of Definition def:wp_data and $\lambda^2\phi^0 = \rho^0-1$. Then, the time semi-discrete scheme updaterho-updateq initialised with the above initial conditions enjoys the following properties when $\lambda\to 0$ Moreover, under the same conditions on the initial conditions, the scheme updaterho-updateq degenerates when

Figures (11)

  • Figure 1: Double Butcher tableau of an IMEX-RK scheme.
  • Figure 2: Double Butcher tableaux of type-A Additive IMEX schemes. Top: DP1-A(2, 4, 2)and Bottom: DP2-A(2, 4, 2).
  • Figure 3: Double Butcher tableaux of type-CK Additive IMEX schemes.ARS (2,2,2). Here, $\gamma=1-\sqrt{2}/2$, $\sigma = 1 / 2 \gamma$ and $\delta=1 - \sigma$.
  • Figure 4: Case 1, resolved mesh ($N=10^4$) with Penalised type-A IMEX and SI-IMEX Schemes. $\lambda = 10^{-4}$. Left: $x_1\to |\rho(t, x_1)-1|$, Center: $x_1\to |\nabla \cdot u(t, x_1)|$ , Right: $x_1\to |\phi(t, x_1)|$, at time $t = 10^{-1}$, $K = 16$.
  • Figure 5: Case 1, under-resolved mesh $N = 10^2$ with Penalised type-A IMEX and SI-IMEX Schemes. $\lambda = 10^{-4}$. Left: $x_1\to |\rho(t, x_1)-1|$, Center: $x_1\to |\nabla \cdot u(t, x_1)|$, Right: $x_1\to |\phi(t, x_1)|$, at time $t = 10^{-1}$, $K = 16$.
  • ...and 6 more figures

Theorems & Definitions (18)

  • Definition 1
  • Proposition 1
  • proof
  • Remark 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Proposition 2
  • ...and 8 more