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1、JournalofDifferentialEquations162,140173(2000)doi:10.1006jdeq.1999.3653,availableonlineathttp:www.idealibrary.comonRelaxationLimitforPiecewiseSmoothSolutionstoSystemsofConservationLawsWen-QingXuCourantInstitute,NewYorkUniversity,251Mercerstreet,NewYork,NY,10012E-mail:wenqingcims.nyu.eduRece

2、ivedSeptember15,1998Inthispaperwestudytheasymptoticequivalenceofageneralsystemof1-DconservationlawsandthecorrespondingrelaxationmodelproposedbyS.JinandZ.Xin(1995,Comm.PureAppl.Math.48,235277)inthelimitofsmallrelaxationrate.Itisshownthatiftherelaxationsystemsatisfiesthesubcharacteristiccondi-tio

3、nandthesolutionofthehyperbolicconservationlawsispiecewisesmoothwithafinitenumberofnoninteractingshockssatisfyingtheentropycondition,thenthereexistsolutionsoftherelaxationsystemsthatconvergetothesolutionoftheoriginalconservationlaws(``equilibrium''system)atarateoforder=astherateofrelaxation=goest

4、ozero.Theproofusesamatchedasymptoticanalysisandanenergyestimaterelatedtothenonlinearstabilitytheoryforviscousshockprofiles.2000AcademicPress1.INTRODUCTIONIn[3],JinandXinproposedtoapproximateageneralsystemofhyper-bolicconservationlawsntu+xf(u)=0,u#R(1.1)usingthefollowingrelaxationmodeltu+xv=

5、0,nu,v#R(1.2)1tv+axu=&(v&f(u))=where=>0istherateofrelaxationwhichiscomparabletotheconceptofmeanfreepathinkinetictheory(Boltzmannequation,BGKmodel)andaisapositiveconstantsatisfyingthesubcharacteristiccondition2a>*,*=max

6、*i(u)

7、,1in,(1.3)where*1(u)<}}}<*n(u)aretheeigenvaluesoftheJacobianmatrixf

8、$(u).1400022-03960035.00Copyright2000byAcademicPressAllrightsofreproductioninanyformreserved.ZERORELAXATIONLIMIT141Oneofthemainadvantagesofthesystem(1.2)isitslocalrelaxationstructureandlinearityinconvection,thusonecansolvethissystemquiteeasilybyunderresolvedstablenumericaldiscretizationsusing

9、neitherRiemannsolversspatiallynornonlinearsystemsofalgebraicequationssolverstemporally.Notethattheformulationoftherelaxationmodel(1.2)naturallyextendstomultidimensionalcase;see[3].Inthispaperwerestrictourselvestothesimpleron