Entropy based moment closure for kinetic equations: Riemann problem and invariant regions

Description

Entropy-based moment closure for kinetic equations: Riemann problem and invariant regions Jean-Franc¸ois Coulombel and Thierry Goudon CNRS & Universite Lille 1, Laboratoire Paul Painleve, UMR CNRS 8524 Cite scientifique, 59655 VILLENEUVE D'ASCQ Cedex, France and Team SIMPAF, INRIA Futurs E-mails: , November 8, 2005 Abstract We study a nonlinear hyperbolic system of balance laws that arises from an entropy- based moment closure of a kinetic equation. We show that the corresponding homogeneous Riemann problem can be solved without smallness assumption, and we exhibit invariant regions. AMS subject classification: 82C40, 35L60 35L67 1 Introduction This paper is devoted to the analysis of the following PDEs system ? ? ? ∂t?+ ∂xJ = 0 , ?2 ∂tJ + ∂x ( ?? (?J ? )) = ?J , (1) where the unknown are the density ?, and the current J , while ? is a positive scaling parameter. The function ? that appears in (1) is defined in the following way: ? : (?1,+1) ?? ]0,+∞[ u 7?? u2 + G? ( G?1(u) ) = F ?? F ( G?1(u) ) , (2) where we have let ?? ? R , F(?) := sinh(?)? , G(?) := coth(

• defined up


• solution does


• ?u ?


• r?1


• system


• ?? ?


• corresponding homogeneous


• based moment closure


• corresponding eigenvectors

Sujets

• Goudon
• Painlevé
• System

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 33
• Langue: English

Extrait

Entropy-based moment closure for kinetic equations:
Riemann problem and invariant regions

Jean-Fran¸coisCoulombeland ThierryGoudon

CNRS & Universit´ Lille 1, Laboratoire Paul Painlev´, UMR CNRS 8524
Cit´ scientifique, 59655 VILLENEUVE D’ASCQ Cedex, France
and Team SIMPAF, INRIA Futurs
E-mails:[e-mail], [e-mail]

November8,2005

Abstract
We study a nonlinear hyperbolic system of balance laws that arises from an
entropybased moment closure of a kinetic equation.We show that the corresponding homogeneous
Riemann problem can be solved without smallness assumption, and we exhibit invariant
regions.
AMS subject classification:82C40, 35L60 35L67

1

Introduction

This paper is devoted to the analysis of the following PDEs system

∂tρ+∂xJ= 0,


εJ(1)
2
ε ∂tJ+∂xρ ψ=−J ,
ρ
where the unknown are the densityρ, and the currentJ, whileεis a positive scaling parameter.
The functionψthat appears in (1) is defined in the following way:
ψ: (−1,+1)−→]0,+∞[
′′
F(2)
2′ −1−1
u7−→u+G G(u) =G(u),
F
where we have let
′
sinh(β) 1F(β)
∀β∈R,F(β) :=,G(β) := coth(β)−=.(3)
β βF(β)
∞ −1
We note thatGis aCdiffeomorphism fromRonto (−1,1), so the use of the inverseGis
legitimate. Forfuture purposes, it is convenient to remark that
Z
+1
βv
F(β) =e dµ(v),
−1
where, here and below,dµstands for the normalized Lebesgue measure on (−1,+1). Itis also
worth noting thatF, andψare even functions, whileGis an odd function.We will show below
thatψis strictly convex.The following relations will be often used throughout the paper:
1
′ ′
F(0) = 1,G(0) = 0, ψ(0) =G(0) =, ψ(0) = 0.
3

1