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JOURNALOFCOMPUTATIONALPHYSICS

ARTICLENO

.

135,250–258(1997)

CP975705

ApproximateRiemannSolvers,ParameterVectors,

andDifferenceSchemes

RoyalAircraftEstablishment,Bedford,UnitedKingdom

ReceivedAugust14,1980;revisedMarch30,1981

u(x,0)ϵu

L

(xϽ0),u(x,0)ϵu

R

(xϾ0).

Severalnumericalschemesforthesolutionofhyperbolicconser-

vationlawsarebasedonexploitingtheinformationobtainedby

guedthatin

existingschemesmuchofthisinformationisdegradedandthat

isshownthatthesefeaturescanbeobtainedbyconstructinga

matrixwithacertain‘‘PropertyU.’’Matriceshavingthispropertyare

ordertoconstructthem,itisfoundhelpfultointroduce‘‘parameter

vectors’’whichnotablysimplifythestructureoftheconservation

laws.

ᮊ1981AcademicPress.

(3)

INTRODUCTION

Weconsidertheinitial-valueproblemforahyperbolic

,weseekavectoru(x,t)

suchthat

u

t

ϩF

x

ϭ0

and

u(x,0)ϭu

0

(x),(2)

(1)

whereFissomevector-valuedfunctionofu,suchthatthe

JacobianmatrixAϭѨF/Ѩuhasonlyrealeigenvalues.

Weintroducethediscreterepresentationx

i

ϭx

0

ϩi⌬x,

t

n

ϭt

0

ϩn⌬tandsupposethatu

i

n

issomeapproximation

tou(x

i

,t

n

).

Amultitudeofstrategieshavebeendevisedtoobtain

numericalresultsforthediscreteproblem,andtheirrela-

laddressinthis

papersomequestionsrelatingtothosemethodswhich

attempttoconstructthesolutionbysolvingasuccession

thattheRiemannproblem

istheinitial-valueproblemobtainedwhenthegeneraldata,

Eq.(2),isspecialisedto

ReprintedfromVolume43,Number2,October1981,pages357–372.

250

0021-9991/97$25.00

Copyright©1981byAcademicPress

Allrightsofreproductioninanyformreserved.

Thatis,westudythebreakupofasinglediscontinuity.

ThosecaseswhereF(u)islineararewellknownandessen-

tiallytrivial(see[1]forabriefdiscussion).Thosecases

whereuisscalarandFisnonlinearcanbesurprisingly

intricate,buthavebeenthoroughlyinvestigated[2].Ifu

isavectorandFisnonlinear,thentheprobleminvolves

nonlinearalgebraicequationstogetherwith,usually,logi-

calconditionswhichexpressthefactthatagivenmember

ofthewavesystemmaybepresenteitherasashockwave

wofresultsrelatingtothe

generalRiemannproblemhasbeengivenbyLax[3].In

general,themostefficientwaytosolvetheseequations

willdependonthesystemofconservationlaws(1)from

whichtheyderive;ingenuityisrequiredtoexploitspecial

specialcaseof

theunsteadyEulerequationsinonespacedimension,an

algorithmwasdevisedbyGodunov[4]andisavailablein

thebooksbyRichtmyerandMorton[5]andHolt[6].A

variantwhichconvergesmorerapidlywasworkedoutby

vanLeer[7].

TheusualwayofincorporatingtheRiemannproblem

intothenumericalsolutionistotake(u

i

n

,u

i

n

ϩ

1

),foreachi

inturn,aspairsofstatesdefiningasequenceofRiemann

problems,whicharethenthoughtofasprovidinginforma-

tionaboutthesolutionwithineachinterval(i,iϩ1).

Variousindividualmethodsarethendistinguishedbythe

fly,we

reviewthesebelow.

Godunov[4]supposedthattheinitialdatacouldbe

replacedbyapiecewiseconstantsetofstateswiththe

discontinuitiesat͕x

i

ϩ

1/2

͖.Hethenfoundtheexactsolution

tothissimplifiometimestep⌬t(less

than⌬xdividedbythegreatestwavespeedfoundinthe

Riemannsolutions)hereplacedtheexactsolutionbya

newpiecewiseconstantapproximation,whilstpreserving

first

majorextensiontothislineofapproachwasmadebyvan

Leer[7],whoapproximatedthedataandthesolutionat

APPROXIMATERIEMANNSOLVERS

251

eachsubsequenttimelevel,bypiecewiselinearsegments,

-

quiredthesolutiontoaninteractionproblemwhichwas

moregeneralthanRiemann’s,butraisedtheorderofaccu-

racyofthemethodfromonetotwo.

AparallellineofdevelopmentwasinitiatedbyGlimm

[8],whofollowedGodunovasfarastheexactsolutionto

thesimplifiedproblem,butthenobtainedthenewapproxi-

pling

producessolutions,whichareconservativeonlyonthe

average,buthastheadvantagethatnearamoving,isolated,

discontinuity,thesolutionisincrementedeitherbythefull

amountofthejump,way,initially

sharpdiscontinuitiesremainsharp,andforsometechnical

finedsampling

procedureshavebeenintroduced(,Chorin[9],

butsofartheaccuracyofthemethodapproachesunity

frombelow.

Itseemstothepresentauthorthattheexpenseofpro-

ducinganaccuratesolutiontotheRiemannproblemwould

onlybejustifiediftheabundanceofinformationwhich

istherebymadeavailablecouldbeputtosomerather

,itmustsomehowbetruethatthe

accuracywithwhichitisworthwhilesolvingtheRiemann

problemwillbelimitedbythewayweintendusingthe

mple,wemayconsidertheuseofless

andLax[10]

havedevisedanapproximateRiemannsolverparticularly

designedforincorporationintoGodunov-typeorGlimm-

roximationdeveloped

hereincouldbe(andhasbeen)usedinthesameway,but

incomparisonwiththeHartenandLaxapproximationit

suffersthetheoreticaldisadvantageofnothavinganatu-

rallyconstructedentropycondition;thispointwillbedis-

therhand,thepresent

approximationisdesignedtoprovidetheinformation

neededtoobtainhighformalaccuracy,followingthestrat-

egysetoutbyRoe[1].Thatpaperessentiallydescribeda

mechanismbywhichanyalgorithmdevelopedfornumeri-

calsolutionofthelinearadvectionequation

u

t

ϩau

x

ϭ0(4)

canbegeneralisedtothecaseofnonlinearsystems.A

largebodyofunpublishednumericalresultsforBurger’s

equation,thenonlinearshallow-waterequations,andthe

steadyandunsteadyEulerequations,demonstratesthat

allqualitativefeaturesofeachalgorithmarefaithfully

eevidencesuggests

thataccuracyalsocarriesover,atleasttothirdorder.

Anessentialstageinthemechanismistheapproximate

solutionofanonlinearRiemannproblem.

Inthispaperweconsiderapproximatesolutionswhich

areexactsolutionstoanapproximateproblem,viz.,

u

t

ϩA

˜

u

x

ϭ0,

whereA

˜

isaconstant

courseunaltered.A

˜

matrix,andthedata(u

L

,u

R

)areof

istobechosensothatitisrepresenta-

tiveoflocalconditions.

cometomindareA

˜

Candidateswhichimmediately

ϭ