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Approximate solutions for the Vlasov--Poisson system with boundary layers

Chang-Yeol Jung, Bongsuk Kwon, Masahiro Suzuki, Masahiro Takayama

Abstract

We construct the approximate solutions to the Vlasov--Poisson system in a half-space, which arises in the study of the quasi-neutral limit problem in the presence of a sharp boundary layer, referred as to the plasma sheath in the context of plasma physics. The quasi-neutrality is an important characteristic of plasmas and its scale is characterized by a small parameter, called the Debye length. We present the approximate equations obtained by a formal expansion in the parameter and study the properties of the approximate solutions. Moreover, we present numerical experiments demonstrating that the approximate solutions converge to those of the Vlasov--Poisson system as the parameter goes to zero.

Approximate solutions for the Vlasov--Poisson system with boundary layers

Abstract

We construct the approximate solutions to the Vlasov--Poisson system in a half-space, which arises in the study of the quasi-neutral limit problem in the presence of a sharp boundary layer, referred as to the plasma sheath in the context of plasma physics. The quasi-neutrality is an important characteristic of plasmas and its scale is characterized by a small parameter, called the Debye length. We present the approximate equations obtained by a formal expansion in the parameter and study the properties of the approximate solutions. Moreover, we present numerical experiments demonstrating that the approximate solutions converge to those of the Vlasov--Poisson system as the parameter goes to zero.
Paper Structure (13 sections, 4 theorems, 126 equations, 6 figures)

This paper contains 13 sections, 4 theorems, 126 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Approximate solutions by asymptotic expansion
  3. Equations of outer solutions
  4. Equations of inner solutions
  5. The explicit expressions of finite expansions
  6. Homogeneous initial data
  7. Estimates of the inner solutions
  8. Zeroth order inner solutions
  9. $j$th order inner solutions
  10. Estimates of the errors $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$
  11. Numerical experiments
  12. Some discussion
  13. Numerical scheme

Key Result

Lemma 2.1

Let $g$ be an analytic function. There holds formally that where $\mathcal{J}^{j-1}_g(u)$ is independent of $u^{j}$ and determined only by $u^0,u^1,\ldots,u^{j-1}$.

Figures (6)

  • Figure 1: The plots of $f$, $F^0$, $\phi$, and $\Phi^0$ at $t=0.1$ for various $\varepsilon$'s.
  • Figure 2: Log-log plots of the errors $f-f^0-F^0$ and $\phi-\phi^0-\Phi^0$ at $t=0.1$, $(x,\xi) \in [0,1]\times[-4,0]$ in (a) $L^2((0,1)\times(-4,0))$, $L^2((0,1))$, and in (b) $L^\infty((0,1)\times(-4,0))$, $L^\infty((0,1))$.
  • Figure 3: The plots of $f$, $f^0$, and $F^0$ for $\varepsilon = 10^{-2}$ with the initial and boundary conditions in Case (I).
  • Figure 5: The plots of $f$, $f^0$, and $F^0$ for $\varepsilon = 10^{-2}$ with the initial and boundary conditions in Case (II).
  • Figure 7: The plots of $\int (f,f^0,F^0) d\xi$ and $\int \xi (f,f^0,F^0) d\xi$ at $t=0.1$ for $\varepsilon = 10^{-2}$ with the initial and boundary conditions in Cases (I) and (II).
  • ...and 1 more figures

Theorems & Definitions (7)

  • Lemma 2.1: JKS21
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof