LinearandNonlinearWaves

A.MignonePhysicsDepartmentTurinUniversity

I.THESCALARADVECTIONEQUATION

TheAdvectionEquation:Theory•  Firstorderpartialdifferentialequation(PDE)in(x,t):

•  HyperbolicPDE:informationpropagatesacrossdomainatfinitespeedàmethodofcharacteristics

•  Characteristiccurvessatisfy:

•  Alongeachcharacteristics:àThesolutionisconstantalongcharacteristiccurves.

U(x-at,0)

U(x,t)

TheAdvectionEquation:Theory•  forconstanta:thecharacteristicsarestraightparallellinesandthe

solutiontothePDEisauniformshiftoftheinitialprofile:

•  Thesolutionshiftstotheright(fora>0)ortotheleft(a<0):

II.LINEARSYSTEMSOFHYPERBOLICCONSERVATIONLAWS

SystemofEquations:Theory•  Weturnourattentiontothesystemofequations(PDE)

whereisthevectorofunknowns.Aisam × mconstantmatrix.

•  Forexample,form=3,onehas

SystemofEquations:Theory•  ThesystemishyperbolicifAhasrealeigenvalues,λ1≤…≤λmand

acompletesetoflinearlyindependentrightandlefteigenvectorsrkandlk(rj ⋅lk =δjk)suchthat

•  ForconveniencewedefinethematricesΛ=diag(λk),and

•  SothatA⋅R=R⋅Λ,L⋅A=Λ⋅L,L⋅R=R⋅L=I,L⋅A⋅R=Λ.

SystemofEquations:Theory•  Thelinearsystemcanbereducedtoasetofdecoupledlinear

advectionequations.•  MultiplytheoriginalsystemofPDE’sbyLontheleft:

•  Definethecharacteristicvariablesw=L⋅qsothat

•  SinceΛisdiagonal,theseequationsarenotcoupledanymore.

SystemofEquations:Theory•  Inthisform,thesystemdecouplesintomindependentadvection

equationsforthecharacteristicvariables:

where(k=1,2,…,m)isacharacteristicvariable.

•  Whenm=3onehas,forinstance:

SystemofEquations:Theory•  Themadvectionequationscanbesolvedindependentlybyapplyingthe

standardsolutiontechniquesdevelopedforthescalarequation.

•  Inparticular,onecanwritetheexactanalyticalsolutionforthek-thcharacteristicfieldas

i.e.,theinitialprofileofwkshiftswithuniformvelocityλk,andistheinitialprofile.•  Thecharacteristicsarethusconstantalongthecurvesdx/dt=λk

SystemofEquations:ExactSolution•  Oncethesolutionincharacteristicspaceisknown,wecansolvethe

originalsystemviatheinversetransformation

•  Thecharacteristicvariablesarethusthecoefficientsoftherighteigenvectorexpansionofq.

•  Thesolutiontothelinearsystemreducestoalinearcombinationofmlinearwavestravelingwithvelocitiesλk.

•  Expressingeverythingintermsoftheoriginalvariablesq,

III.NONLINEARSCALARHYPERBOLICPDE:BURGER’SEQUATION

NonlinearAdvectionEquation•  Weturnourattentiontothescalarconservationlaw

•  Wheref(u)is,ingeneral,anonlinearfunctionofu.

•  Togainsomeinsightsontheroleplayedbynonlineareffects,westartbyconsideringtheinviscidBurger’sequation:

NonlinearAdvectionEquation•  WecanwriteBurger’sequationalsoas

•  Inthisform,Burger’sequationresemblesthelinearadvectionequation,exceptthatthevelocityisnolongerconstantbutitisequaltothesolutionitself.

•  Thecharacteristiccurveforthisequationis

•  àuisconstantalongthecurvedx/dt=u(x,t)àcharacteristicsareagainstraightlines:valuesofuassociatedwithsomefluidelementdonotchangeasthatelementmoves.

NonlinearAdvectionEquation•  From

onecanpredictthat,highervaluesofuwillpropagatefasterthanlowervalues:thisleadstoawavesteepening,sinceupstreamvalueswilladvancesfasterthandownstreamvalues.

NonlinearAdvectionEquation•  Indeed,att=1thewaveprofilewilllooklike:

•  thewavesteepens…

NonlinearAdvectionEquation•  Ifwewaitmore,weshouldgetsomethinglikethis:

•  Amulti-valuefunctions?!àClearlyNOTphysical!

???

BurgerEquation:ShockWaves•  Thecorrectphysicalsolutionistoplaceadiscontinuitythere:ashockwave.

•  Sincethesolutionisnolongersmooth,thedifferentialformisnotvalidanymoreandweneedtoconsidertheintegralform.

Shock position

BurgerEquation:ShockWaves•  Thisishowthesolutionshouldlooklike:

•  SuchsolutionstothePDEarecalledweaksolutions.

BurgerEquation:ShockWaves•  Let’strytounderstandwhathappensbylookingatthe

characteristics.•  Considertwostatesinitiallyseparatedbyajumpataninterface:

•  Here,thecharacteristicvelocitiesontheleftaregreaterthanthoseontheright.

uL

uR

u(x)

x

BurgerEquation:ShockWaves•  Thecharacteristicwillintersect,creatingashockwave:

•  Theshockspeedissuchthatλ(uL)>S>λ(uR).Thisiscalledtheentropycondition.

t

x

t

x

NonlinearAdvectionEquation•  TheshockspeedScanbefoundusingtheRankine-Hugoniotjump

conditions,obtainedfromtheintegralformoftheequation:

•  ForBurger’sequationf(u)=u2/2,onefindstheshockspeedas

BurgerEquation:RarefactionWaves•  Let’sconsidertheoppositesituation:

•  Here,thecharacteristicvelocitiesontheleftaresmallerthanthoseontheright.

uL

uR u(x)

x

BurgerEquation:RarefactionWaves•  Nowthecharacteristicswilldiverge:

•  Puttingashockwavebetweenthetwostateswouldbeincorrect,sinceitwouldviolatetheentropycondition.Instead,thepropersolutionisararefactionwave.

t

x

t

x

tail head

BurgerEquation:RarefactionWaves

•  Theheadoftherarefactionmovesatthespeedλ(uR),whereasthetailmovesatthespeedλ(uL).

•  Thegeneralconditionforararefactionwaveisλ(uL)<λ(uR)

•  BothrarefactionsandshocksarepresentinthesolutionstotheEulerequation.Bothwavesarenonlinear.

•  Ararefactionwaveisanonlinearwavethatsmoothlyconnectstheleftandtherightstate.Itisanexpansionwave.

•  Thesolutioncanonlybeself-similarandtakesontherangeofvaluesbetweenuLanduR.

BurgerEquation:RiemannSolver•  Theseresultscanbeusedtowritethegeneralsolutiontothe

RiemannproblemforBurger’sequation:

–  IfuL>uRthesolutionisadiscontinuity(shockwave).Inthiscase

–  IfuL<uRthesolutionisararefactionwave.Inthiscase

NonlinearAdvectionEquation•  Solutionslooklike

•  forararefactionandashock,respectively.

IV.NONLINEARSYSTEMSOFCONSERVATIONLAW

NonlinearSystems•  Muchofwhatisknownaboutthenumericalsolutionofhyperbolic

systemsofnonlinearequationscomesfromtheresultsobtainedinthelinearcaseorsimplenonlinearscalarequations.

•  Thekeyideaistoexploittheconservativeformandassumethesystemcanbelocally“frozen”ateachgridinterface.

•  However,thisstillrequiresthesolutionoftheRiemannproblem,whichbecomesincreasinglydifficultforcomplicatedsetofhyperbolicP.D.E.

EulerEquations•  Systemofconservationlawsdescribingconservationofmass,

momentumandenergy:

•  TotalenergydensityEisthesumofthermal+Kineticterms:•  ClosurerequiresanEquationofState(EoS).Foranidealgasonehas

EulerEquations:CharacteristicStructure•  Theequationsofgasdynamicscanalsobewrittenin“quasi-linear”

orprimitiveform.In1D:

whereV=[ρ,vx,p]isavectorofprimitivevariable,cs=(γp/ρ)1/2istheadiabaticspeedofsound.

•  Itiscalled“quasi-linear”since,differentlyfromthelinearcasewherewehadA=const,hereA=A(V).

EulerEquations:CharacteristicStructure•  Thequasi-linearformcanbeusedtofindtheeigenvector

decompositionofthematrixA:

•  Associatedtotheeigenvalues:

•  Thesearethecharacteristicspeedsofthesystem,i.e.,thespeedsatwhichinformationpropagates.Theytellusalotaboutthestructureofthesolution.

EulerEquations:RiemannProblem•  Bylookingattheexpressionsfortherighteigenvectors,

•  weseethatacrosswaves1and3,allvariablesjump.Thesearenonlinearwaves,eithershocksorrarefactionswaves.

•  Acrosswave2,onlydensityjumps.Velocityandpressureareconstant.Thisdefinesthecontactdiscontinuity.

•  Thecharacteristiccurveassociatedwiththislinearwaveisdx/dt=u,anditisastraightline.Sincevxisconstantacrossthiswave,theflowisneitherconvergingordiverging.

EulerEquations:RiemannProblem•  ThesolutiontotheRiemannproblemlookslike

•  Theouterwavescanbeeithershocksorrarefactions.•  Themiddlewaveisalwaysacontactdiscontinuity.•  Intotalonehas4unknowns:,sinceonlydensityjumps

acrossthecontactdiscontinuity.

x

t (contact) (shock or rarefaction)

(shock or rarefaction)

EulerEquations:RiemannProblem•  Dependingontheinitialdiscontinuity,atotalof4patternscan

emergefromthesolution:

x

t C S R

x

t C S R

x

t C R

x

t C S S R