A BGK approximation to scalar conservation laws with discontinuous flux

Florent Berthelin

Description

A BGK approximation to scalar conservation laws with discontinuous flux F. Berthelin?and J. Vovelle† July 3, 2009 Abstract We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem. Keywords: scalar conservation laws – discontinuous flux – BGK model – relax- ation limit Mathematics Subject Classification: 35L65 – 35F10 – 35D05 1 Introduction In this paper we consider the equation ∂tf ? + ∂x(k(x)a(?)f ?) = ?u? ? f? ? , t > 0, x ? R, ? ? R, (1) with the initial condition f?|t=0 = f0, in Rx ? R?. (2) Here k is given by k = kL1I(?∞,0) + kR1I(0,+∞), where 1IB is the characteristic function of a set B, ? 7? a(?) is a continuous function on R such that ?u ? [0, 1], ∫ u 0 a(?)d? ≥ 0, ∫ 1 0 a(?)d? = 0, (3) and, in (1), ?u? , the so-called equilibrium function associated to f? is defined by u?

• kr kl


• bgk approximation


• called equilibrium


• scalar


• let f0 ?


• conservation laws


• flux function

Sujets

• Jordan
• Scalar
• Conservation law

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

Extrait

A BGK approximation to scalar conservation laws with discontinuous flux F. Berthelin ∗ and J. Vovelle † July 3, 2009

Abstract We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem. Keywords : scalar conservation laws – discontinuous flux – BGK model – relax- ation limit Mathematics Subject Classification : 35L65 – 35F10 – 35D05 1 Introduction In this paper we consider the equation ∂ t f ε + ∂ x ( k ( x ) a ( ξ ) f ε ) = χ u ε ε − f ε , t > 0 , x ∈ R , ξ ∈ R , (1) with the initial condition f ε | t =0 = f 0 , in R x × R ξ . (2) Here k is given by k = k L 1I ( −∞ , 0) + k R 1I (0 , + ∞ ) , where 1I B is the characteristic function of a set B , ξ 7→ a ( ξ ) is a continuous function on R such that ∀ u ∈ [0 Z u ( ξ ) dξ ≥ 0 , Z 0 1 a ( ξ ) dξ = 0 , (3) , 1] , a 0 and, in (1), χ u ε , the so-called equilibrium function associated to f ε is defined by u ε ( t Z R ( t, x, ξ ) d , x ) = f ε ξ, χ α ( ξ )=1I ]0 ,α [ ( ξ ) − 1I ] α, 0[ ( ξ ) , ∗ LaboratoireJ.A.Dieudonn´e,UMR6621CNRS,Universite´deNice,ParcValrose,06108 Nice cedex 2, France, Email: [e-mail] † InstitutCamilleJordan,UMR5218CNRS,Universit´eClaudeBernardLyon1,43boule- vard du 11 novembre 1918, 69622 Villeurbanne Cedex, France. Email: [e-mail]- lyon1.fr

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