Fast sweeping methods for hyperbolic systems of conservation laws at steady state II

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Fast sweeping methods for hyperbolic systems of conservation laws at steady state II

Björn Engquist

^a,1

, Brittany D. Froese

^a,∗,2

, Yen-Hsi Richard Tsai

^a,b,3

aDepartmentofMathematicsandICES,TheUniversityofTexasatAustin,1UniversityStationC1200,Austin,TX78712,USA bRoyalInstituteofTechnologyKTH,10044Stockholm,Sweden

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received31March2014

Receivedinrevisedform25October2014 Accepted18January2015

Availableonline26January2015 Keywords:

Conservationlaws Hyperbolicequations Fastsweepingmethods Numericalanalysis

Theideaof usingfastsweeping methods forsolving stationarysystemsofconservation lawshaspreviouslybeenproposedforefficientlycomputingsolutionswithsharpshocks.

We further develop these methods to allow for a more challenging classof problems includingproblemswith sonicpoints, shocks originating inthe interior of the domain, rarefactionwaves, and two-dimensional systems. We show that fastsweeping methods canproduce higher-orderaccuracy.Computationalresultsvalidatetheclaimsofaccuracy, sharpshockcurves,andoptimalcomputationalefficiency.

©^{2015ElsevierInc.}All rights reserved.

1. Introduction

Thenumericalsolutionofsystemsofnonlinear(stationary)conservationlaws,

∇ ·

^F

(

U

) =

^a

(

U

,

x

),

x

∈ Ω

B

(

U

,

x

) =

⁰

,

x

∈ ∂Ω

⁽¹⁾

continuestobeanimportantprobleminnumericalanalysis.Amajorchallengeassociatedwiththistaskistheneedtocom- putenon-classicalsolutions[6],whichleadstotheneedtodevelopnumericalschemesthatcorrectlyresolvediscontinuities inweak(entropy)solutions.

1.1. Relatedwork

Anaturalapproachtocomputingthesestationarysolutionsistouseanexplicittime steppingorpseudotimestepping technique toevolvethesystemtosteadystate[1,2,4,16].Severalapproachesareavailableforresolvingshockfrontsinclud- ing fronttrackingschemes[10],upstream-centred schemes forconservationlaws(MUSCL)[5,27],centralschemes[17,21], essentiallynon-oscillatory(ENO)schemes[13],andweightedessentiallynon-oscillatory(WENO)schemes[20,22,23].How- ever, the computational efficiency of theseschemes is restricted by a CFL condition and the need to evolve the system fora substantialtime inordertoreachthesteadystate solution.Inorder tosubstantiallyimprovetheefficiencyofthese

*

Correspondingauthor.

E-mailaddresses:[email protected](B. Engquist),[email protected](B.D. Froese),[email protected](Y.-H.R. Tsai).

1 ThisauthorwaspartiallysupportedbyNSFDMS-1217203.

2 ThisauthorwaspartiallysupportedbyNSFDMS-1217203andanNSERCPDF.

3 ThisauthorwaspartiallysupportedbyNSFDMS-1217203,NSFDMS-1318975,andaSimonsFoundationFellowship.

http://dx.doi.org/10.1016/j.jcp.2015.01.028 0021-9991/©^{2015ElsevierInc.}All rights reserved.

videanumericalmethodfornumericallycomputingthesolutions.

In[7],weproposedanewfastsweepingapproachforsolvingstationaryconservationlaws.Thisapproachwasmotivated bythefastsweepingmethodsthathavebeenusedforsolvingHamilton–Jacobiequations[18,26,28]andinvolvescombining smooth solutionbranchesasin[14].Thesesweepingmethodsexploittheflow ofinformationalongcharacteristics,which allowssolutionstobecomputedbypassingthroughthedomaininasmallnumberofpre-determinedsweepingdirections.

Theresultingalgorithmstypicallyhavelinearcomputationalcomplexity.

Inourearlierworkweproposedabasictwo-stepsweepingmethodforsystemsofconservationlaws.

1. Smooth solution branchesare generated by means ofan update formula that is used to update the solution along differentsweepingdirections.

2. AselectionprinciplebasedontheRankine–Hugoniotandentropyconditionsforastationaryshockisusedtodetermine whichsolutionbranchisactiveateachpoint.

Weusedthisframeworktocomputethesolutiontoseveralsimpleone- andtwo-dimensionalproblems.Intwodimensions, theappropriateimplementationofthesetwostepswas informedbysomeaprioriknowledgeofthebasicstructureofthe solutions.Theresultingmethodsalsohadoptimalcomputationalcomplexityandproducedsharpshocks.

1.2. Contributionsofthisarticle

Inthisarticle,we develop thebasicidea offastsweeping methodsinorder toconstructmethods forsolvinga larger rangeofproblemsinone- andtwo-dimensions.Thesenewdevelopmentsallowustodemonstrateadditionaladvantagesof sweepingovermoretraditionalshock-capturingmethods.

In Section 2, we show how to constructhigher order sweeping methods, which preserve their globalaccuracy even inthepresenceofshocksandsonicpoints.Thisisaclearadvantageoverformally higher-ordershock-capturingmethods, whichcanberestrictedtofirst-orderaccuracywhenshockarepresent[8].

In Section 3,we consider theproblemof sweeping andmatching techniquesin two dimensions. Inour earlierwork, we proposed a technique forsweeping andmatching whenthe exact solutionis composed ofsmooth solutionbranches separated by a shock curve. We now develop these techniques to allow usto solve problems with a more complicated characteristicstructure.In oneofthe problemswe study,ashockbeginsin themiddleofthedomain ratherthanatthe boundary.Asecondexampleinvolvesrarefactionwaves.Thesemethodsproducesharpshockcurvesandedges,indistinction tothesmearedoutresultsproducedbycapturingmethods.

Finally,in Section 4,we focus onthe solutionof aclass of two-dimensionalsystems. Inthissetting, we cannot view sweepingandmatchingasdistinctstepssinceonlypartialboundarydatamaybeavailableinsomeregions.Tosolvethese systems,wecombinetheusualselectionprinciple(Rankine–Hugoniotconditions)withaconditionthatsolutionsareregular oneithersideoftheshock.Inthissection,weshowhowtoconstructanefficientmethodthatsimultaneouslyconstructsa smoothsolutionstateandacleanshockcurve.

2. Accuracy

The developmentofhigher-order schemesforsolving systemsofconservationlawshasbeenan importantarea ofre- search, withthe essentially non-oscillatory(ENO) [13,24] andweighted essentially non-oscillatory (WENO) [20] schemes becoming very popular approaches. When solutions includeshocks, thesemethods willnaturally producean O(¹) error neartheshock.Dependingonthestructureofcharacteristicsinthesystem,thiserrorcan propagateandcausetheglobal solutionaccuracytobeonlyfirst-ordereveninregionswherethesolutionissmooth[8].

Thefastsweepingmethodsinvolvecomputingsmoothsolutionbranches,whicharejoinedtogetherbydirectlyimposing theRankine–Hugoniotconditionsatashock.Themethodsarealsoveryflexibleinthesensethatwecansolvethenecessary ODEsorPDEsusinganyconsistent,stablescheme.Thisallowsustoeasilycomputebothsmoothsolutionbranchesandthe shock location with higher accuracy. In particular, we can compute higher-order solutions in settings where traditional shock-capturingmethodsarelimitedtofirst-orderaccuracy.

2.1. Solutionswithshocks

Inthissectionwedescribeahigher-orderleft-to-rightsweepingmethod;theoppositesweepingdirectioncanbehandled inthesameway.

Inone-dimensions,systemstaketheform

⎧ ⎨

⎩

f

(

U

)

x

=

^a

(

U

,

x

),

xL

<

x

<

xR

U

(

x

) =

^UL

,

x

=

^xL

B_R

U

(

x

)

=

⁰

,

x

=

^xR

.

(2)

WeintroducetheoperatorPxLxU_L todenotethesolutionU atthepointx.Thisisobtainedbypropagatingtheleftboundary condition UL fromxL toxviathesolutionoftheODE

V_x

=

^a

f⁻¹

(

V

),

x

,

x

>

x_L

V

(

x

) =

f

(

UL

),

x

=

xL

.

⁽³⁾

WealsointroducethejumpoperatorΦU₋,whichgivesanentropy-satisfyingsolutionoftheRankine–Hugoniotconditions

f

(Φ

U₋

) =

f

(

U₋

).

(4)

When the solutionconsistsoftwo smooth statesseparatedby a single shock, thesweeping methodrequires solving the followingequationfortheunknownshocklocationx_S:

B_R

P^hx_Sx_R

Φ

Px^h_Lx_SU_L

=

⁰

.

(5)

HereP^{h referstoadiscreteversionofthe}propagationoperatorandhdenotesthestepsizeonthegrid.

Theorem 1 in [7]ensuresthat ifthe methodusedto solvethe necessaryODEs isconsistent, stable, andhas accuracy O(^hk),thenthesolutioncomputedbythesweepingmethodwillalsohaveaccuracyO(^hk)inL¹.Asthefirstexampleinthis paper,we demonstratethat higher-orderODEsolverscan easilybeincorporatedintothemethod.Here weprovideanex- amplewheretraditionalshock-capturingmethodsareonlyfirst-orderaccurate,whilethefastsweepingmethodsuccessfully achieveshigheraccuracy.

2.1.1. Nozzleproblem(shock)

Weconsideranozzleproblemthatwasdiscussedin[8].Inthatwork,theauthorssolvethisproblemusingafourth-order accurateENOmethodandasecond-orderaccurateTVDscheme.TheyconsidertheL²errorinaregiontotheleft(0<x<

4.5)andright(5.5<x<10)oftheshock.Intheleftregion,solutionsobtaintheexpectedaccuracy.However,totheright oftheshock,theaccuracyisonlyfirst-order;thisisaconsequenceofthecharacteristicstructureattheshock.

Thesystemwewanttosolveis

⎛

⎝ ρ

A

ρ

u A E A

⎞

⎠

t

+

⎛

⎝ ρ

u A

( ρ

u²

+

p

)

A

u A

(

E

+

^p

)

⎞

⎠

x

=

⎛

⎝

0 p A

(

x

)

0

⎞

⎠ ,

(6)

whichwewanttosolvetosteadystateonthedomainx∈ [⁰,10]^{.Herethenozzleareais} A

(

x

) =

¹

.

398

+

⁰

.

347 tanh

(

0

.

8x

−

⁴

),

thepressureis p

= ( γ ₋

1

)

E

−

¹

2

ρ

u²

= ρ

R T

,

andwetake

γ

=¹.4 and R=⁸.3144.Weenforcetheboundaryconditions

ρ =

⁰

.

502

,

u

=

¹

.

299

,

p

=

⁰

.

3809 atx=^{0 and}

ρ ₌

0

.

7519 atx=^10.

We can integrate the system to obtain an implicit expression for the exact solution, which can be approximated to machine precision. Thisgives usan exactsolution touse tocompute the errorin ourapproximations. We note that the exactsolutioncontainsashockatx≈5 (Fig. 1).

Weperformthesweeping(integration)usingtwodifferentmethods:thetrapezoidruleandafourth-orderRunge–Kutta method.WeplottheerrorinthedensityinFig. 2.WealsoprovidetheL²solutionerrorintheintervals[⁰,4.5]^and[⁵.5,10] (totheleftandrightoftheshock);seeTable 1.Indistinctiontoshock-capturingmethods,we findthat thefastsweeping methodspreservethehigher-orderaccuracyonbothsidesoftheshock.

Fig. 1.Density at steady state for nozzle problem.

Fig. 2.Error in density for the solution of the nozzle problem by sweeping with (a) the trapezoid rule and (b) RK4.

2.2. Solutionswithturningpoints

Amore challenging situationoccurs whenthe solutioncontains aturning points;that is, whenone (or more) ofthe eigenvalues of theflux gradient (∇^{f) changes signsmoothly. Here we show how to develop} higher-orderfast sweeping methodsforsolvingone-dimensionalsystemsthatcontainturningpoints.

Wedescribe indetailthesolutionofa problemwitha singleturning point,wherean eigenvaluesmoothlytransitions from negative to positive. The same technique can be used to solve problems withmultiple turning points, as outlined in[7].

Notethat this approachis onlyneededwhen an eigenvaluetransitions fromnegative to positive throughthe turning point. In the situation wherean eigenvalue transitionscontinuously frompositive to negative, the turning point can be interpretedasadegenerateshock;thejumpoperatorwillhaveacontinuous solutionthatsatisfiestheLaxentropycondi-

Table 1

L²errorindensitytotheleftandrightoftheshockforthesolutionofthenozzleproblembysweeping.

Scheme N 0<x<4.5 5.5<x<10 Shock location

Error inL² Rate Error inL² Rate Error Rate

Trap 50 6.663×^{10−5 1}.438×^{10−4 1}.412×¹⁰⁻³

Trap 100 1.149×^{10−5 2.5 0}.242×^{10−4 2.6 0}.428×^{10−3 1.7}

Trap 200 0.201×10⁻⁵ 2.5 0.042×10⁻⁴ 2.5 0.116×10⁻³ 1.9

Trap 400 0.035×10⁻⁵ 2.5 0.007×10⁻⁴ 2.5 0.030×10⁻³ 1.9

RK4 50 1.829×10⁻⁸ 7.934×10⁻⁸ 2.017×10⁻⁶

RK4 100 0.076×^{10−8 4.6 0}.276×^{10−8 4.8 0}.113×^{10−6 4.2}

RK4 200 0.003×^{10−8 4.5 0}.012×^{10−8 4.5 0}.007×^{10−6 4.1}

tions[19].InthiscasewecansimplyusetheRankine–Hugoniotconditionsandentropyconditionstodeterminetheturning pointlocation.

RecallthatataturningpointxT,whereaneigenvalueλivanishes,thesolutionUT mustsatisfyacompatibilitycondition

P⁻¹a

(

U_T

,

x_T

)

i

=

⁰

.

(7)

HerethematrixP isdefinedviaadecompositionoftheJacobianoftheflux,

P⁻¹

∇

^f

(

UT

)

P

= Λ,

(8)

whereΛisadiagonalmatrixcontainingtheeigenvaluesof∇^{f.Forawell-posedproblem,thepresenceofaturningpoint} willbeaccompaniedby“incomplete”boundaryconditions.Thatis,insteadofprovidingvaluesforallsolutioncomponents, thegivenboundaryconditionontheleftwillbeacurvethroughstatespacethatisparameterisedbyanunknown

α

,

U

(

xL

) =

^UαL

.

Thentheunknownboundaryconditionisdeterminedbycompatibilityconditionattheturningpoint.Thatis,wesearchfor theunknown

α

thatsolvestheequation

P⁻¹a

PxLxTU^α_L

,

x_T

i

=

⁰

.

(9)

Inthesesituations,wecan stilluseaconventional ODEsolver tosolvethesystemofODES fromx_L uptoa gridpoint xj<xT thatisnearthesonicpoint.

Next,weneedtosolvethefollowingsystemfortheunknownsolutionUT andtheturningpointlocationxT=^xj+^h.

U_T

=

P_x^h_j,xT^Uj

λ

i

(

UT

) =

⁰

.

⁽¹⁰⁾

In theaboveequation,we requirethat λi isthelargestnegative eigenvaluesincetheeigenvalueissmoothly transitioning fromnegativetopositive.

In [7], we used a forward Euler step to approximate both the solution at the turning point andthe location of the turningpoint.However,thisapproachdoesnotgeneralisenaturallytohigher-ordermethodsbecauseofthe(possible)need to accurately invert∇^{f at or neara point where an eigenvalue vanishes. Instead, we willrely on the regularity of the} solutionneartheturningpoint.

WerecallthatwehavethevaluesofbothU andUx=(∇^f)⁻¹aatthegridpointxk,k^{j.Wewillusethesevaluesto} compute U_j+1byextrapolation.Forexample,ifweletx=x_j−^xj−1 and0^hx,wecanusetheformula

U

(

x_j

+

^h

) =

1

+

^h

x

U

(

x_j

) +

¹ 2h

1

+

^h

x

U_x

(

x_j

)

−

^h

xU

(

xj−¹

) −

¹ 2h

1

+

^h

x

Ux

(

xj−¹

) +

O

x⁴

.

(11)

Similarly, wecan usemoreorfeweroftheneighbouringpoints inordertoobtain anapproximation thathasthedesired orderofaccuracy.Noticethatthisformulacanbeinterpretedasanon-traditionallinearmultistepmethod.

Wewillusethisformulafortwopurposes:

•^{Itisusedtocompute U}j+1withouttheneedforinvertingthefluxfunction.

•^{Itisusedtodeterminethelocationoftheturningpoint.Thatis,wefindthevalueofhforwhich}λ(U(xj+^h))=^0.The turningpointisthengivenbyx_T=^xj+^h.

ThisgivesusanapproachforpropagatingasolutionfromtheleftboundaryxL allthewayuptotheturningpoints xT. Usingthismethod,wecansolve(9)forthe“missing”boundaryconditions.Fromhere,wecancontinuetouseaconventional ODEsolvertopropagatethesolutionfromxj+¹ totherightendofthedomain.

Fig. 3.Solution to the nozzle problem: (a) pressure and (b) eigenvalues.

2.2.1. Nozzleproblem(sonicpointandshock)

Weprovidesomecomputationalresultsforthenozzleproblemthatwecomputedin[7]usingforwardEuler.

⎛

⎝ ρ

A

ρ

u A E A

⎞

⎠

t

+

⎛

⎝ ρ

u A

( ρ

u²

+

p

)

A

u A

(

E

+

^p

)

⎞

⎠

x

=

⎛

⎝

_{p A}⁰

(

x

)

0

⎞

⎠ ,

(12)

whichwewanttosolvetosteadystateonthedomainx∈ [⁰,3]^. Herethepressureis

p

= ( γ ₋

1

)

E

−

¹ 2

ρ

u²

= ρ

R T andthesoundspeedis

c

= γ

p

/ ρ .

TheeigenvaluesoftheJacobianareλ1=^u−^c,λ2=^u,andλ3=^u+^c.

Wetakethecross-sectionalareatobe A

(

x

) =

¹

+

²

.

2

(

x

−

¹

.

5

)

²

,

thegasconstant

γ

=¹.4,andR=⁸.3144.Weconsidertheboundaryconditions p_L

=

¹

,

p_R

=

⁰

.

6784

,

T_L

=

³⁰⁰

.

Boundary data for the remaining solution components is computed via compatibility conditions as detailed in [7, Sec- tion 3.5.1].

Thesolution (pressure andeigenvalues)are inFig. 3.We compute thesolutionusingthe trapezoidruleanda fourth- order Runge–Kutta method. The error is presented in Fig. 4 and Table 2. Despite the challenge of this problem, we successfullycomputethesolutiontothedesiredaccuracy.

3. Shockcapturingversusmatching

In[7],wealsooutlineda basicsweepingapproachfortwo-dimensionalproblems,whichwas testedon severalsimple examples.Theapproachconsistedoftwobasicsteps:

1. Computesmoothsolutionbranchesbysweepingintheboundaryconditions.

2. UsetheRankine–Hugoniotconditionstogenerateashockcurvethatseparatestwosmoothstates.

Insomesituations,itispossibletogenerateasolution(includingshocks)bysweepinginonedirectiononly.Thisleads to asimplemethod forsolvingstationary systemsthat canhavea fairly complicatedcharacteristicstructure.However, it alsonegatesoneofthebenefitsoffastsweepingmethods,whichistheabilitytoproducesharpshockcurvesbymatching smoothstates.

Now we show how thebasic sweeping methodcan be usedto constructsolutions that havemore complicatedchar- acteristic structures.Thisessentially involvesbreakingtheproblemintosmallsub-domains, whereasimplefastsweeping methodcanbeapplied.

Fig. 4.Error in pressure for the solution of the nozzle problem by sweeping with (a) the trapezoid rule and (b) RK4.

Table 2

L²errorinpressureforthesolutionbysweepingofthenozzleproblemwithasonicpointandshock.

Scheme N Error inL² Rate

Trap 50 2.292×10⁻⁴

Trap 100 0.287×10⁻⁴ 2.9

Trap 200 0.039×10⁻⁴ 2.7

RK4 50 3.318×¹⁰⁻⁵

RK4 100 0.164×^{10−5 4.3}

RK4 200 0.010×^{10−5 4.0}

3.1. Shockcurvegeneratedininteriorofdomain

Oneexamplewheresweepingisdesirablebutnon-trivialistheexample

u² 2

x

+

^uy

=

^{0 (13)}

subjecttotheboundaryconditions

u

(

0

,

y

) =

1

.

5

,

u

(

1

,

y

) = −

0

.

5

,

u

(

x

,

0

) =

1

.

5

−

2x

.

Inthisexample, f(u)=¹₂^u2andg(u)=^u.

Thebasicsweepingandmatchingtechniqueinvolvesgrowingashockcurvethatbeginsataboundarypoint,whereasin thisexampletheshockbeginsinthe interiorofthedomain.Alternatively,wecansimplysweep fromthebottomusinga conservativescheme.However,thiswillproduceasmeared-outshock(Fig. 6(a)).

Nowwedescribeanalternativeprocedure,wherethesweepingstepisonlyusedtogeneratesmoothsolutionbranches.

Inparticular,thismeansthatwedonotneedtouseaconservativescheme.

Fig. 5.Solution of(13). (a) Bottom, (b) left, and (c) right solution branches. Matching done in the (d) bottom and (e) top parts of the domain.

Firstweneedtosolvetheconservationlawintheregionbelowtheshock.

1. Webeginbyidentifyingthepointx_∗ alongthebottomboundarywherethecharacteristicsbecomevertical;thisisthe pointwhere f(u(x^∗,0))=^0.

2. Wegeneratea bottomsolutionbranch u_B.Thiscanbe accomplishedusinganyscheme,includinganon-conservative scheme,sincewewillonlymakeuseofasmoothportionofthesolution.SeeFig. 5(a).Noticethatwehavechosento useanon-conservativescheme,sothattheresultingshockcurvedoesnotsatisfytheRankine–Hugoniotconditions.

3. NextwegenerateleftandrightsolutionbranchesuL anduR bysweepingfromthesides.SeeFigs. 5(b)–5(c).

4. Wecomputeanewsolutionbranchbymatchingu_B andu_L intheregionx<x_∗.Thismatchingwillstartatthecorner (0,0)andcontinueuntilx=^x∗,wherethecharacteristicsfromtheleftboundarycollidewiththeverticalcharacteristics from(x_∗,0).Weuse(x_∗,y_∗)todenotetheendpointofthiscomputedcurve.WesimilarlymatchthisresultwithuR in theregionx>x∗^{, y}<y_∗.SeeFig. 5(d).

Theresultofthisprocedureisasolution^u˜ ^{thatsatisfiesthe}conservationlawintheregion y<y_∗ belowtheshock.

Inorder tocompute the top portionofthe solution,we need onlysolve thesame PDEin thedomain [⁰,1]× [^y∗,1] subjecttotheboundaryconditions

u

(

0

,

y

) =

¹

.

5

,

u

(

1

,

y

) = −

⁰

.

5

,

u

(

x

,

y_∗

) = ˜

^u

(

x

,

y_∗

).

Thisproblemnowinvolvesashockoriginatingfromtheboundary(at(x_∗,y_∗)),whichcanbesolvedusingastraightforward applicationofthesweepingandmatchingtechniques.SeeFig. 5(e).

Fig. 6.Solutionof(13)computedbyshock-capturingusing(a) Lax–Friedrichsor(b) Lax–Wendroff.Solutioncomputedbysweepingusing(c) Lax–Friedrichs or(d) Lax–Wendroff.Allsolutionsarepresentedusingthesamecolouraxes,whichareadjustedtoallowfortheovershootsintheLax–Wendroffscheme.

Table 3

ComputationtimesonanN×Ngridforthesolutionof(13)bysweeping.

N 32 64 128 256 512

CPU Time (s) 0.19 0.56 1.99 7.40 30.03

Solutions constructedby thisprocedure areshownin Fig. 6.Toillustrate theadvantages ofthesweeping method,we use Lax–Friedrichs orLax–Wendroff schemes to perform the sweeping steps. For comparison, we also present solutions computing via shock-capturingwith Lax–Friedrichs orLax–Wendroff. As desired, thesweeping methodproduces a clean shockcurve,withnoneofthesmearingoroscillationsresultingfromthecapturingmethods.

ComputationtimesaredisplayedinTable 3andvalidatetheclaimthatthemethodhaslinearcomputationalcost.

Thisexampledemonstratesseveraladvantagesofthefastsweepingapproach.

•^{Sharp shockcapturing: thisresultcontains noneof the smearingor} oscillationsthat accompany a traditional shock- capturingmethod.

•Flexibility:anyconsistent,stableschemecanbeusedtocomputesmooth solutionbranches.Inthisexample,a simple non-conservativeschemewasused,butthisdidnotaffectthecorrectnessofthecomputedshockcurve.

•Challenging shock structures can be handled by breaking the problem down into simpler problems in appropriate sub-domains.

•^{Themethodisvery}computationallyefficient.

3.2. Rarefaction

Asecond examplewheresweepingisdesirableisinproblemsthatinvolverarefaction.Weconsiderthesameequation asintheprevioussection,

u² 2

x

+

^uy

=

⁰

,

(14)

inthedomain[−¹,1]× [⁰,1]^{.Weenforcetheboundarycondition} u

(

x

,

0

) =

−

1

,

x

<

0 1

/

2

,

x

>

0

.

Inthissetting,thediscontinuityattheboundarypoint(x_∗,0)=(0,0)willresultinararefactionwaveratherthanashock.

Fig. 7.Solutionof(14).(a) Left,(b) right,and(c) topsolutionbranches.Solutioncomputedusing(d) sweepingandmatchingor(e) ashockcapturing scheme.

Asinthepreviousexample,wecould solvethisby asinglesweep fromthebottomboundaryasinFig. 7(e).However, whilethisexampledoesnotincludeashockcurve,thesolutionisnotdifferentiable.Capturingschemesdonotproducethe sharpedgethatisdesirable.

Weemploythefollowingsweepingprocedure.

1. Compute a left branch by sweeping fromthe bottom, usingthe given boundary conditions atx<0,y=^{0,and ex-} tending thesesmoothly along theremainder ofthebottom boundary.Similarly, computea rightsolutionbranch.See Figs. 7(a)–7(b).

2. Noticethatthesetwosolutionbranchescannotbematchedacrossanentropysatisfyingshockcurve.Instead,theymust beconnectedthroughararefactionwave.

3. Computeatopbranchbysweepingfromthetopboundary,solvingthePDEintheform

−

^f

(

u

)

_x

−

^g

(

u

)

_y

=

⁰

.

Todeterminethecorrectboundaryvaluesat(x,1),werecall thatthecharacteristicemanatingfromthispointshould intersectthebottomboundaryatthepoint(x_∗,0)wherethegivenboundaryconditionsarediscontinuous.Additionally, theslopeofthischaracteristiclinewillbegivenby g(u)/f(u).Consequently,thecorrectboundaryvalueateachpoint (x,1)canbedeterminedbysolving

f

(

u

) =

^g

(

u

)(

x

−

^x∗

).

SeeFig. 7(c).

Table 4

ComputationtimesonanN×^{Ngridforthesolutionof(14)bysweeping.}

N 32 64 128 256 512

CPU Time (s) 0.24 0.73 2.83 10.98 42.48

4. Matchthetop andleft solutionbranchesstartingfromthepoint(x_L,1)onthetopboundarywherethe twosolution branches agree. In thissetting, the matching curve is constructed so that the resulting solution will be continuous.

Similarly,matchthisresultwiththerightsolutionbranch.

See Fig. 7(d) for a picture of the solution computed by this procedure using a Lax–Friedrichs approximation of the flux. Thisresultismuch sharperthan thesolutioninFig. 7(e),whichwas computedusingLax–Friedrichs capturing.This techniqueisalsocomputationallyefficient,asevidencedinTable 4.

Remark1. Thefirst step ofthisprocess required usto producea smooth extension of theboundary conditionsinto the region x>0.Inthisexample,aconstantextension wasnatural.Whentheboundarydataisnotconstant,manyextensions are possible.Topreventdramaticchangesintheorientationofthecharacteristics,itisdesirabletominimisethevariation intheboundarydata.Onepossibleextensionofmoregeneralboundarydatau(x,0)intotheregionx>0 canbeobtained bycomputing

˜

u

(

x

,

0

) =

⎧ ⎨

⎩

u

(

x

,

0

)

x

<

0

u

( ,

0

)

x

> >

0 u

(

0

,

0

) +

^x

(

u

( ,

0

) −

u

(

0

,

0

))

0

<

x

< .

Thisresultcanbeconvolvedwithasmoothingkernelinordertoobtainasmoothextension.

4. Two-dimensionalsystems

Constructingsweepingmethodsfortwo-dimensionalsystemsinvolvesadditionalchallengesthatarenotpresentinscalar problems.Onemainchallenge istheissueof“incomplete”boundaryconditions,whichisalsopresentinone-dimensional systems.

4.1. Shockreflection

Asettingwherethisoccursisthefollowingshockreflectionproblem,whichinvolvessolvingthestationaryEulerequa- tions

⎛

⎜ ⎜

⎝ ρ

u

ρ

u²

+

^p

ρ

uv u

(

E

+

^p

)

⎞

⎟ ⎟

⎠

x

+

⎛

⎜ ⎜

⎝ ρ

v

ρ

uv

ρ

v²

+

p v

(

E

+

^p

)

⎞

⎟ ⎟

⎠

y

=

^{0 (15)}

inthedomain

0

^x

⁴

,

0

^y

¹

with p=(

γ

−¹)(E−¹₂

ρ

(u²+^v2))and

γ

=¹.4.

Following[3,12],weenforcetheboundaryconditions

( ρ ,

u

,

v

,

p

) =

(

1

.

69997

,

2

.

61934

,−

0

.

50632

,

1

.

528191

)

y

=

1

(

1

,

2

.

9

,

0

,

1

/ γ )

x

=

⁰

.

Areflectioncondition(i.e.v=^{0)isimposedat y}=^{0 andnoboundaryconditionsaregivenatx}=^4.

In[7,Theorem 3],weshowedthatthisproblemcouldbesolvedusingasingleleft-to-rightsweepbysolvingaparaxial formoftheequations.Inthatsetting,theshockisresolvedusingashockcapturingscheme.AsinthediscussioninSection3, thiswillproduceasmearedoutshock.SeeFig. 10.

Theinitialstepsofafastsweepingmethodcanbecarriedoutonthisproblem.

1. Generateasmoothleftstatebysweepingintheboundaryconditionatx=^0.

2. Generateasmoothtopstatebysweepingintheboundaryconditionaty=^1.

3. Matchtheleftandtopstatesbyconstructingashockcurvethatstartsatthepoint(0,1)wheretheboundaryconditions arediscontinuous.

Fig. 8.Density for the 2D Euler shock reflection problem computed by matching top and left solution branches.

The resultofthisprocedureis picturedinFig. 8.Theshockseparatingthe topandleft brancheseventually intersects thebottomboundary,asexpected.Totherightofthisintersectionpointx_∗,thegivenbottomboundarycondition(v=^0)is notsatisfied.Thisindicatesthatanothershockcurvewillneedtoextendfromthisboundary,separatingthetopstatefrom a thirdright state.However, determining thisrightstate isnon-trivialsince onlyone boundary condition isgivenatthe bottom,andnonearegivenontherightsideofthedomain.

Wedonotice,though,thattheinitialrightstate U_right andnormalncanbedeterminedatthepoint(x_∗,0)wherethe shockoriginates. This isbecausethe reflectionboundary condition givesusone componentofthe rightstate: v_right=^0.

Thusthejumpconditions

f

Utop

(

x

)

−

^f

U_right

(

x

) ,

g

Utop

(

x

)

−

^g

U_right

(

x

)

·

ⁿ

=

⁰

yieldfourequationsforthenormalnandthethreeremainingcomponentsofU_right.

Theideathat weproposeistofocusonthesearchfortheunknownshockcurve y=φ (x)thatoriginatesfromx_∗ and separatesthetopandrightstates.RecallthatwecaneasilycomputethematchedtopstateU_top(x)throughoutthedomain.

Ifwe are given thiscurve then we can employ the followingprocedure to constructthe solution on thefar side ofthe shock.

1. Applythejumpconditions

f

U_top

(

x

)

−

^f

U_right

(

x

) ,

g

U_top

(

x

)

−

^g

U_right

(

x

)

·

ⁿ

=

⁰

atpointsontheshockcurve,rejectingthecontinuoussolutionU_top(x)=^Uright(x). 2. Solvethesystem

⎧ ⎨

⎩

f

(

U

)

_x

= −

^g

(

U

)

_y

+

^a

(

U

,

x

,

y

),

x

>

x_∗

,

0

<

y

< φ (

x

)

U

(

x

) =

^Uright

(

x

),

x

>

x_∗

,

y

= φ (

x

)

v

(

x

) =

⁰

,

x

>

x_∗

,

y

=

⁰

.

We can alsoapply the above procedure using an arbitraryshockcurve y=φ (x).However, in thiscase, the reflected solutioncomingfromtheboundary y=^{0 willbe}incompatiblewiththeresultofapplyingthejumpconditionsacrossthe shock.Theresultwillbetheformationofasharplayeratsomepointinthedomain,whichcontradictsthehypothesisthat the datashould producea single reflected shock. SeeFigs. 9(a)–9(b)forexamples oflayers that form neartwo different candidateshockcurves.Thusregularityofthecomputedrightsolutionbranchisthecondition thatdeterminesthecorrect shockcurve.

Knowingtheorientation oftheshockcurve near(x_∗,0),we turnourattentionto extendingthiscurveandrightstate U_right.Theapproachforgeneratingtheshockcurveandresultingrightstateisthentofindthe(discrete)curve(x_i,φi)such thatasmoothnessconditionissatisfied,

S

Φ

Utop

(

xi

);

ni

=

0

,

x_∗

<

xi

<

xR

.

(16) In the computations below, the smoothness indicator is a discrete estimate ofthe second derivative in the y-direction, whichwe set equalto zero.Forhigher-orderapproximations, ahigher-order derivative couldbe used.This approachcan be motivatedbyfinitedifference approximations,which arederived bytruncating Taylorseries,essentiallyassuming that higherderivativesarezero.Alternatively,wecouldchoosetominimisesomeindicatorofsolutionsmoothness.

Inproblemswherealleigenvaluesof∇^{f arepositive,itisnotnecessarytoconstructtheentirecurve y}=φ (x)atonce.

Instead,itcanbe generatedonestepatatime usinga singleleft-to-rightsweep.Consequently,construction oftheshock

Fig. 9.Density for the 2D Euler shock reflection problem computed using the sweeping approach with incorrect shock curves.

curve requiresthesolutionofasequenceofnonlinearscalarequationsforeachφi ratherthenthesolutionofanonlinear systemforallvaluesofφ.

Theapproachisasfollows.

1. Tosweeptherightsolutionbranchandshockcurvefromx_i−1tox_ifirstguessatthenewlocationoftheshockcurveφi. 2. ThenormaltothecurvecanbeestimatedandthejumpconditionsappliedtogetthevalueofU_right at(x_i,φi). 3. DoasinglesweepingsteptocomputeU_right fromx_i−1 tox_i forall0<y_j< φi.Enforcethereflectionconditionatthe

bottomandthecorrectvaluesofU_right attheshock.

4. If thevalue (xi,φi)extends theshockcurve with anincorrect orientation,it willnot be possible toconnect the top state Utop toasinglesmooth rightstateU_right via thisshockcurve.Instead,theincorrectshockcurve willtriggerthe formation ofadditionalintermediate states,resulting inabreakdown intheregularityof Uright that originatesatthe point(xi−¹,φi−¹).

5. Computea smoothness indicator at (x_i,φi) and usethis value to adjustthe value ofφi via Newton’smethod. Ifan undesirable intermediate statehasoriginatedat(x_i−1,φi−1),discrete approximationsofderivativesinthe y-direction willbelargenear(x_i,φi).Thusu_{y y} emergesasanaturalindicatorofsmoothness.

The solutioncomputedusingthisfastsweeping approach ispicturedin Fig. 10(b).Itcontains a sharpshockcurve,as desired. We emphasise againthat computing thissolutionwas done efficiently, requiringonly three sweeps throughthe data.ComputationtimesarepresentedinTable 5andvalidateourclaimsofoptimalcomputationalcomplexity.

4.2. Obliqueshock(piecewiseconstant)

We consider a second example of a two-dimensional system and use the sameprocedure described in the previous section. Inthisobliqueshockproblem,auniformflowimpinges onan impermeablewall.Following[9],weposethetwo- dimensionalstationaryEulerequationsinthedomain.

Fig. 10.Densityforthe2DEulershockreflectionproblemcomputedusing(a) ashock-capturingmethodtosolveaparaxialformoftheequationsand (b) sweeping.

Table 5

ComputationtimesonanN×^{Ngridfortheshockreflectionproblem.}

N 32 64 128 256 512

CPU Time (s) 6.4 24.4 85.7 301.6 1096.8

(

x

,

y

) ∈ [

⁰

,

1

] × [

⁰

,

1

/

2

]

y

(

x

−

¹

/

2

)

tan

δ .

Ahorizontalflowisenforcedattheleftboundary,

( ρ ,

u

,

v

,

p

) = (

1

,

3

,

0

,

1

/ γ ),

x

=

⁰

andthebottomboundaryisanimpermeablewall, v

=

0

,

x

<

1

/

2

,

y

=

⁰

utan

δ,

x

>

1

/

2

,

y

= (

x

−

¹

/

2

)

tan

δ.

Whentheuniformhorizontalflowimpingesonthewedgeatthebottomofthedomain,ashockwillform.

Wesolvethisproblemsettingδ=^{15◦ astheanglethebottomwedgemakeswiththe}horizontal. Weexpecttheshock tomakeanangleofapproximately32.2^◦ withthehorizontal,andtheflowbeyondtheshockshouldhaveaMachnumber ofapproximately2.255[9].

Solutionofthisproblemviafastsweepingrequiresatwo-stepprocess:

1. Generateasmoothleftstatebysweepingintheboundaryconditionsatx=^0.

2. UsethemethoddescribedinSection 4.1toconstructa shockcurve andsmooth rightstate emanatingfromthepoint (1/2,0)wherethebottomboundaryconditionisdiscontinuous.

Fig. 11.Mach number computed using sweeping for the oblique shock problem.

Table 6

ComputationtimesonanN×Ngridfortheobliqueshockproblemwithpiecewiseconstantdata.

N 32 64 128 256 512

CPU Time (s) 8.1 30.1 109.2 331.9 1334.3

Table 7

ComputationtimesonanN×^{Ngridfortheobliqueshockproblemwith}non-constantdata.

N 32 64 128 256 512

CPU Time (s) 7.5 23.8 81.5 299.0 1147.2

The computedMach numberisdisplayed inFig. 11.Inparticular,therightstatehasaMachnumberofapproximately 2.2549, whilethecomputedshockmakesan angleof32.240^◦ withthehorizontal.Thisisinagreementwiththeexpected solution.Wealsoobserveoptimalcomputationalcomplexity;seeTable 6.

4.3. Obliqueshock(nonconstant)

Becausetheexamplespresentedintheprevioussubsectioninvolvedpiecewiseconstantdata,theresultingshockcurves were linear.Tobetter demonstratetheabilitiesofthe fastsweepingmethod,we repeattheoblique shockwave problem, thistimeimposingnon-constantdataattheleftboundary.Theflowatx=^{0 isnowgivenby}

( ρ ,

u

,

v

,

p

) =

1

,

3

,

10y

1

2

−

y

,

1

/ γ −

0

.

3 sin

(

4

π

y

)

,

x

=

0

.

ThecomputedMachnumber,whichnowincludesvariablestatesandanonlinearshockcurve,isdisplayedinFig. 12(a).

Asinthepreviousexample,thissolutionwascomputedusingaLax–Friedrichsapproximation,butproducesmuchsharper resultsthanasolutioncomputedbysimplyevolvingaLax–Friedrichsschemetosteadystate(Fig. 12(b)).Thecomputational costofthesweepingmethodisnogreaterthanitwasforthesimplerpiecewiseconstantcase(Table 7).

Noexactsolutionisavailableforthisproblem.Forreference,weincludeaplotoftheMachnumberobtainedbyevolving aLax–Friedrichsschemetosteadystateonamorerefined(801×^801)grid(Fig. 12(c)).

5. Conclusions

Inthisarticle,wehavefurtherdevelopedafastsweeping approachforcomputingsteadystatesolutions tosystemsof conservationlaws,whichwasfirstintroducedin[7].Wedemonstratedthepossibilityofconstructingmethodsthatachieve higher accuracythanis possiblewithtraditionalshock-capturingmethods.We alsoextended theuseoftwo-dimensional sweeping methodsto problemsthathavea morecomplicatedshockstructure,includingshocks thatbeginintheinterior ofthedomainandrarefaction.Finally,wedevelopedfastsweeping methodsfora classoftwo-dimensionalsystems.Inall cases,ourmethodproduced resultswithcorrect, sharpshockcurves.Computationalexperiments alsovalidatedthe claim thatthemethodshaveoptimalcomputationalcomplexity.

Future challengesincludetheextension ofthesetechniques tohigherdimensions. In problemswithmorechallenging structures, itmaynotbestraightforwardtogenerateanexplicitformulationoftheshocksurface. Analternativeapproach wouldbetorepresenttheshockimplicitly;forexample,usingalevelsetformulation.

Fig. 12.Machnumberfortheobliqueshockproblemwithnonconstantdata.Solutionobtainedby (a) sweepingor(b) evolvingaLax–Friedrichsschemeto steadystateona200×^{200 grid.}(c) ReferencesolutioncomputedbyevolvingaLax–Friedrichsschemetosteadystateonan 801×^{801 grid.}

Acknowledgements

Yen-HsiRichard Tsaithanks the NationalCenterfor Theoretical Sciences,Taipei,Taiwan forhosting hisvisit to Taipei, wheretheresearchforthisarticlewasinitiated.HealsothanksI-LiangChernforstimulatingconversationonrelatedtopics.

References

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195 (2)(2004)474–507.

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