Positivity Preserving High Order Finite Volume Compact-WENO Schems for Compressible Euler Equations

8/11/2019 Positivity Preserving High Order Finite Volume Compact-WENO Schems for Compressible Euler Equations

1/19

Journal of Computational Physics 274 (2014) 505523

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

A positivity-preserving high order finite volumecompact-WENO scheme for compressible Euler equations

Yan Guo a,

Tao Xiong b,,

Yufeng Shi c

a DepartmentofMathematics,ChinaUniversityofMiningandTechnology,Xuzhou,Jiangsu221116,PRChinab DepartmentofMathematics,UniversityofHouston,Houston,TX77004,USAc SchoolofElectricPowerEngineering,ChinaUniversityofMiningandTechnology,Xuzhou,Jiangsu221116,PRChina

a

r

t

i

c

l

e

i

n

f

o a

b

s

t

r

a

c

t

Article

history:

Received17January2014

Receivedinrevisedform22June2014

Accepted23June2014

Availableonline30June2014

Keywords:

Compactscheme

Finitevolume

Weightedessentiallynon-oscillatoryscheme

Positivity-preserving

CompressibleEulerequations

In thispaper,apositivity-preserving fifth-orderfinitevolumecompact-WENOscheme is

proposedforsolvingcompressibleEulerequations.Asitisknown,conservativecompact

finitevolumeschemeshavehighresolutionpropertieswhileWENO(WeightedEssentially

Non-Oscillatory) schemes are essentially non-oscillatory near flow discontinuities. We

extendtheideaofWENOschemestosomeclassicalfinitevolumecompactschemes[30],

where lowerordercompactstencilsarecombinedwithWENOnonlinearweightstoget

a higher order finite volume compact-WENO scheme. The newly developed positivity-

preserving limiter [43,42] is used to preserve positive density and internal energy for

compressible Euler equations of fluid dynamics. The HLLC (Harten, Lax, and van Leer

with Contact) approximate Riemann solver [37,4] is used to get the numerical flux at

thecellinterfaces.Numericaltestsarepresentedtodemonstratethehigh-orderaccuracy,

positivity-preserving, high-resolutionandrobustnessoftheproposedscheme.

2014

Elsevier

Inc.

All rights reserved.

1. Introduction

Computing

numerical

solutions

of

nonlinear

hyperbolic

systems

of

conservation

laws

is

an

interesting

and

challenging

work.

In

recent

years,

a

variety

of

high

resolution

schemes

which

are

high

order

accurate

for

smooth

solutions

and

non-

oscillatory

for

discontinuous

solutions

without

introducing

spurious

oscillations

have

been

proposed

for

these

problems.

WENO

schemes

[25,19,33,34,3] have

high

order

accuracy

in

smooth

region

and

keep

the

essentially

non-oscillatory

proper-

ties

for

capturing

shocks.

However,

these

classical

WENO

schemes

often

suffer

from

poor

spectral

resolution

and

excessive

numerical

dissipation.

Compact

schemes

[22] have

attracted

a

lot

of

attention

due

to

their spectral-like

resolution

properties

by

using

global

grids.Theseschemeshavethefeaturesofhigh-orderaccuracywithsmallerstencils.However,linearcompactschemesnec-

essarilyproduceGibbs-likeoscillationswhentheyaredirectlyappliedtoflowswithshockdiscontinuities,andtheamplitude

wouldnotdecreasewithmeshrefinement.Toaddressthisdifficulty,severalhybridcompactschemesareproposedtocou-

pletheENOorWENOschemes forshock-turbulence interactionproblems,e.g.,ahybridcompact-ENOschemebyAdams

andShariff[1] andahybridcompact-WENOschemebyPirozzoli[30].Anewhybridschemeasaweightedaverageofthe

compactscheme[30] andtheWENOscheme[19] wasdevelopedbyRenetal. [31].Anothercompactschemebytreating

thediscontinuityasan internalboundarywasproposedbyShen et al. [32]. Thesehybrid schemes require indicators to

* Correspondingauthor.E-mailaddresses: [email protected](Y. Guo),[email protected](T. Xiong),[email protected](Y. Shi).

http://dx.doi.org/10.1016/j.jcp.2014.06.046

0021-9991/ 2014

Elsevier

Inc.

All rights reserved.
http://dx.doi.org/10.1016/j.jcp.2014.06.046http://www.sciencedirect.com/http://www.elsevier.com/locate/jcpmailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.jcp.2014.06.046http://crossmark.crossref.org/dialog/?doi=10.1016/j.jcp.2014.06.046&domain=pdfhttp://dx.doi.org/10.1016/j.jcp.2014.06.046mailto:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/jcphttp://www.sciencedirect.com/http://dx.doi.org/10.1016/j.jcp.2014.06.046


2/19

506 Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523

detect

discontinuities

and

switch

to

a

non-compact

scheme

around

discontinuities,

spectral-like

resolution

properties

would

belost.

AclassofnonlinearcompactschemeswasproposedbyCockburnandShu[8] forshockcalculations. Itwasbasedon

the

cell-centered

compact

schemes

[22] and

combined

with

TVD

or

TVB

limiters

to

control

spurious

numerical

oscillations.

DengandMaekawa[9] andDengandZhang[10] developedaclassofnonlinearcompactschemesbasedontheENOand

WENO

ideas

respectively

by

adaptively

choosing

candidate

stencils.

Zhang

et

al. [41] proposed

increasingly

higher

order

compactschemesbasedonhigherorderWENOreconstructions[3].Insteadofinterpolatingtheconservativevariables,they

directly

interpolated

the

flux

by

using

the

LaxFriedrichs

flux

splitting

and

characteristic-wise

projections.

An

improvementof thecompactschemeconverging tosteady-statesolutionsofEulerequationswasstudies in [40].Anew linearcentral

compact

scheme

was

proposed

in

[26],

both

grid

points

and

half

grid

points

are

evolved

to

get

higher

order

accuracy

and

betterresolutions.

Jiangetal. [20] developedaclassofweightedcompactschemesbasedon thePad typeschemeofLele [22]. It isa

weighted

combination

of

two

biased

third

order

compact

stencils

and

a

central

fourth

order

compact

stencil.

A

sixth

order

centralcompact schemecanbeobtained in smooth regions.RecentlyGhoshandBaederemployed the idea in [20],and

developed

a

class

of

compact-reconstruction

finite

difference

WENO

schemes

[13].

Lower

order

biased

compact

candidate

stencils are identified at thecell interface and combinedwith theoptimalnonlinear WENO weights.The resultinghigh

order

scheme

is

upwind.

Their

scheme

was

shown

to

be

superior

spectral

accurate

and

non-oscillatory

at

discontinuities.

Inthispaper,weconsidertodesignfinitevolumehighordercompactschemesforsolvingcompressibleEulerequations.

AconservativeformulationoftheEulerequationsisgivenby

Ut+

F(U)x=

0, (1.1)

whereU andF(U) arevectorsofconservativevariablesandfluxesrespectively,whicharegivenby

U=

u1

u2

u3

=

u

E

, F(U)=

u

u2 +pu(E+p)

,

with

E=

1

2u2 + e

, e= e(,p)= p

( 1) , (1.2)

where isthedensity,p isthepressure,u istheparticlevelocity,E isthetotalenergyperunitvolume,e isthespecificinternalenergyand istheratioofspecificheat(= 1.4 foridealgas).Thesoundspeeda isdefinedas

a=

p

. (1.3)

Physically, thedensity and thepressurep shouldbothbepositive,and failure ofpreservingpositivedensityorpres-suremaycauseblow-upofthenumericalsolutions.Manyfirstorderschemeswereshowntobepositivity-preserving,such

asGodunov-typeschemes[11],fluxvectorsplittingschemes[17],LaxFriedrichsschemes [29,43],HLLCschemes[4] and

gas-kineticschemes[28,36].Somesecond-orderschemeswerealsodevelopedbasedonthesefirstorderschemes,suchas

[11,36,29,12].RecentlyZhangandShuhavedevelopedpositivity-preservingmethodsforhighorderdiscontinuousGalerkin

(DG)methods[43,44,46],finitevolumeandfinitedifferenceWENOschemes[42,45].Self-adjustingandpositivitypreserv-

ing

high

order

schemes

were

developed

by

Balsara

for

MHD

equations

[2].

Hu

et

al. have

developed

positivity-preserving

high-orderconservativeschemesbyusingafluxcut-offmethodforsolvingcompressibleEulerequations[18].Xionget.al

have

developed

a

parametrized

positivity

preserving

flux

limiters

for

finite

difference

schemes

solving

compressible

Euler

equations[39].

In

the

present

paper,

we

will

develop

a

conservative


fifth-order

finite

volume

compact-WENO(FVCW)scheme forcompressibleEulerequations.Weemploy themain idea in [13] where lowerordercompactstencils

arecombinedwiththeoptimalWENOweightstoyieldafifth-orderupwindcompactinterpolation.Asanalternativetothe

finitedifferencecompact interpolation in[13],wedesignafinitevolumecompactupwindscheme,which ismorenature

andcanbeeasilyusedonunstructuredmeshes.Wealsoemploythenewlydevelopedpositivity-preservingrescalinglim-

iter

in

[43,42] to

preserve

positive

density

and

internal

energy,

which

is

very

important

in

some

extreme

cases,

such

as

vacuumornearvacuumsolutions.TheHLLCapproximateRiemannsolver[37,4] willbeusedasthenumericalfluxatthe

element

interfaces

due

to

its

less

dissipation

and

robustness

for

solving

compressible

Euler

equations.

The

first

order

finite

volumeschemewiththeHLLCfluxisprovedtopreservepositivedensityandinternalenergy.Wewillshowthatthehigh

orderfinitevolumecompactschemewiththepositivitypreservingrescalinglimiter,canmaintainhighorderaccuracysimi-

larlyasthenon-compactfinitevolumeschemes.Numericalexperimentswillbepresentedtodemonstratethehighspectral

accuracy,highresolution,positivity-preservingandrobustnessofourproposedapproach.

The

rest

of

the

paper

is

organized

as

follows.

In

Section2,

the


finite

volume

compact-WENO

scheme

for

compressible

Euler

equations

is

presented.

Numerical

tests

for

some

benchmark

problems

of

compressible

Euler

equa-tions

are

studied

in

Section3.

Conclusions

are

made

in

Section4.


3/19

Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523 507

2. Positivity-preservingfinitevolumecompact-WENOscheme

2.1. FinitevolumeschemeforcompressibleEulerequations

Inthissection,wefirstintroducethefinitevolumescheme[23]forcompressibleEulerequations(1.1).Thecomputational

domain

[a,b] isdividedintoN cellsasfollowsa

=x 1

2


4/19


Fig. 2.1. Candidate stencils for interior points.

3

10 uj 12 + 6

10 uj+ 12 + 1

10 uj+ 32 = 1

30 uj1+19

30 uj+10

30 uj+1. (2.9)Symmetrically,wealsohave

1

10u

j 12 + 6

10u

j+ 12 + 3

10u

j+ 32 =10

30uj+

19

30uj+1+

1

30uj+2. (2.10)

These classical fifth order linear finite volume compact schemes (2.9) and (2.10) based on smaller stencils are very

accurateandkeepgoodresolutionsinsmoothregions,butunacceptablenon-physicaloscillationsaregeneratedwhenthey

aredirectlyappliedtoproblemswithdiscontinuitiesandtheamplitudewouldnotdecreaseasthegridnodesarerefined.

Inthefollowing,weadoptthemainideaof[13] toformanonlinearfinitevolumecompact-WENOscheme.Forafifth

order finite volume compact-WENO scheme, three third-order compact stencilswill beused as candidates, as shown in

Fig. 2.1.

From

(2.5),

for

the

three

candidate

stencils,

we

have

2

3

u(0)

j

1

2 +

1

3

u(0)

j+

1

2 =

1

6

(

uj

1

+5

uj),

1

3u

(1)

j 12+ 2

3u

(1)

j+ 12= 1

6(5uj+uj+1),

2

3u

(2)

j+ 12+ 1

3u

(2)

j+ 32= 1

6(uj+5uj+1). (2.11)

Giventhecellaverages{uj},anonlinearweightedcombinationof(2.11) willresultin20+ 1

3u

j 12 +0+ 2(1+ 2)

3u

j+ 12 +1

32uj+ 32

= 16

0uj1+5(0+ 1) + 2

6uj+

1+ 526

uj+1, (2.12)

where the nonlinear weights

{0,1,2

} will be specified later. Let u

j+1

2

denote the fifth order approximation of the

nodalvalue u(xj+ 12 ,tn) incell Ij .From (2.12),a fifthordercompact-WENOapproximationof u

j+ 12

basedon the stencil

{xj1,xj ,xj+1} isgivenbyu

j+ 12=u

j+ 12 . (2.13)

Insmoothregions,thefinitevolumecompact-WENOschemeyieldsafifth-orderupwindcompactscheme[30].Tocon-

struct

a

nonlinear

compact

scheme,

we

choose

a

set

of

normalized

nonlinear

weightsk [5,6] bytaking

k=z

k2l=0

zl

, zk= ck

1 +

5

k+

2, k=0, 1, 2, (2.14)

where5= |2 0| andtheclassicalsmoothindicatorsk (k=0,1,2) [33]aregivenby

0=13

12 (uj2 2uj1+uj )2+

1

4 (uj2 4uj1+ 3uj )2

,

1=13

12(uj1 2uj+uj+1)2 +

1

4(uj1uj+1)2,

2=13

12(uj 2uj+1+uj+2)2 +

1

4(3uj4uj+1+uj+2)2.

isasmallpositivenumber toavoid thedenominator tobe0, inournumerical tests,we take=1013 .Theoptimallinearweightsarec0= 210 ,c1= 510 ,c2= 310 .Theweights(2.14) aredenotedasWENO-Zweights,whichcanavoidaccuracylostatcriticalpoints[5].

For thescalarcase,a tri-diagonal system (2.12) issolved togetuj+ 12

.Letu+j+ 12

denote thefifthorderapproximation

of

the

nodal

value u(xj+ 12 ,t

n) from cell Ij+1 , following a similar procedure as above, it can be obtained by the stencil

{xj ,xj

+1,xj

+2

}.SimilartoclassicalWENOschemes,nearcriticalpoints,thecorrespondingweightapproachesto0 andthe

system

reduces

to

a

biased

bidiagonal

system.

Across

the

discontinuities,

the

fifth-order

scheme

yields

a

third-order

compactscheme

which

has

higher

resolution

than

a

third

order

non-compact

scheme.


5/19


2.3. Compact-WENOreconstructionforsystems

Inthissubsection,wewilldescribethefinitevolumecompact-WENOreconstruction forcompressibleEulerequations.

The

scalar

algorithm

(2.12) in

the

previous

subsection

will

be

applied

along

each

characteristic

field.

As

we

know,

the

conservativeEulerequations(1.1) canalsobewritteninaquasi-linearform[37]

Ut+A (U)Ux=0, (2.15)where

the

coefficient

matrix

A(U) is

the

Jacobian

matrix

of

F(U) and

can

be

written

as

A(U)=

0 1 0

12

( 3)( u2u1

)2 (3 )( u2u1

) 1u2u3

u21+ ( 1)( u2

u1)3

u3u1

32

( 1)( u2u1

)2 ( u2u1

)

.

ThetotalspecificenthalpyH isrelatedtothespecificenthalpyh,theyare

H= E+p

12

u2 + h, h= e+ p

. (2.16)

TheeigenvaluesoftheJacobianmatrixA(U) are

1=u a, 2=u, 3=u+ a, (2.17)where

a isthespeedofsound(1.3).Thecorrespondingrighteigenvectorsare

r(1) =

1

u aH ua

, r(2) =

1

u

12 u

2

, r(3) =

1

u+ aH+ ua

.

AmatrixR(U) isformedbytherighteigenvectors

R(U)= r(1), r(2), r(3). (2.18)Letting

L(U)

=R (U)1 ,then

L(U)A(U)R(U)=,here isthediagonalmatrix=diag(1,2,3).Denotingavectorl(k) tobethek-throwinL(U),then

l(1) = 12

(c2+ u/a, c1u 1/a, c1),

l(2) =(1 c2, c1u, c1),l(3) = 1

2(c2 u/a, c1u+ 1/a, c1), (2.19)

wherec1=( 1)/a2 ,c2= 12 u2c1 .Atthegridnodexj+ 12 ,denotingU

j+ 12

asthefifthorderapproximationofthenodalvaluesU(xj+ 12 ,tn) attimetn within

the

cells

Ij ,

the

scalar

finite

volume

compact-WENO

reconstruction

(2.12)is

applied

to

each

component

of

the

characteristicvariables Vj=L (URoe

j+ 12)Uj toobtain U

j+ 12,where URoe

j+ 12denotestheRoe-averageof thecell-averagevalues Uj and Uj+1

[37].

Forthesystems,acharacteristic-wisefinitevolumecompact-WENOschemeconsistsofthefollowingsteps:

1. Ateachgridnodexj+ 12

,computingtheeigenvalues(2.17) andeigenvectors(2.18)and(2.19) byusingURoej+ 12

.

2. Alongeachcharacteristicfield,computingtheweights(2.14) fromcharacteristicvariablesVj=L(URoej+ 12

)Uj .3. Applyingthescalarreconstruction(2.12) ateachcharacteristicfield

a(k)

j+ 12l(k)

j+ 12U

j 12 + b(k)

j+ 12l(k)

j+ 12U

j+ 12 + c(k)

j+ 12l(k)

j+ 12U

j+ 32 = d(k)

j+ 12l(k)

j+ 12Uj1+ e(k)

j+ 12l(k)

j+ 12Uj+ f(k)

j+ 12l(k)

j+ 12Uj+1 (2.20)

for

k

=1,

2,

3.

The

coefficients

a

(k)

j+ 12 ,

b

(k)

j+ 12 ,

c

(k)

j+ 12 ,

d

(k)

j+ 12 ,

e

(k)

j+ 12 ,

f

(k)

j+ 12 corresponding

to

the

coefficients in

(2.12),

which

can

be

obtained

from

Step

2.


6/19


4. Rewriting

Eq.(2.20)to

be

Aj+ 12 Uj 12 +B j+ 12 Uj+ 12 +Cj+ 12 Uj+ 32 =D j+ 12 Uj1+Ej+ 12 Uj+Fj+ 12 Uj+1 (2.21)

where

Aj+ 12 =

a(1)

j

+12

l(1)

j

+12

a(2)

j+ 12l(2)

j+ 12

a(3)

j+ 12l(3)

j+ 12

, B j+ 12 =

b(1)

j

+12

l(1)

j

+12

b(2)

j+ 12l(2)

j+ 12

b(3)

j+ 12l(3)

j+ 12

, Cj+ 12 =

c(1)

j

+12

l(1)

j

+12

c(2)

j+ 12l(2)

j+ 12

c(3)

j+ 12l(3)

j+ 12

,

Dj+ 12 =

d(1)

j+ 12l(1)

j+ 12

d(2)

j+ 12l(2)

j+ 12

d(3)

j+ 12l(3)

j+ 12

, Ej+ 12 =

e(1)

j+ 12l(1)

j+ 12

e(2)

j+ 12l(2)

j+ 12

e(3)

j+ 12l(3)

j+ 12

, Fj+ 12 =

f(1)

j+ 12l(1)

j+ 12

f(2)

j+ 12l(2)

j+ 12

f(3)

j+ 12l(3)

j+ 12

.

Noticingthatl(k)

j+ 12fork

=1,2,3 areallvectors,a3

3 blocktri-diagonalsystem(2.21) issolvedbyusingthechasing

method[15]toobtainUj+ 12 .

From(2.21),afifthordercompact-WENOapproximationofUj+ 12

basedonthestencil{xj1,xj ,xj+1} isgivenby

Uj+ 12

=Uj+ 12 . (2.22)

LettingU+j+ 12

denotethefifthorderapproximationofthenodalvalueU(xj+ 12 ,tn) fromcell Ij+1 ,followingasimilarproce-

dureasabove,itcanbeobtainedbythestencil{xj ,xj+1,xj+2}.

2.4. Positivity-preservingandHLLCapproximateRiemannsolver

ForcompressibleEulerequations,theRiemannsolutionsconsistofacontactwaveandtwoacousticwaves,eithermay

be

a

shock

or

a

rarefaction

wave.

In

[14],

Godunov

presented

a

first-order

upwind

scheme

which

could

capture

shock

waveswithoutintroducingnonphysicalspuriousoscillations.TheimportantpartoftheGodunov-typemethodistheexact

orapproximatesolutionsoftheRiemannproblem.ExactsolutionstotheRiemannproblemisdifficultortooexpensiveto

be

obtained.

Approximate

Riemann

solvers

are

often

used

to

build

Godunov-type

numerical

schemes.

The

HLLC

approximate

Riemannsolver[37,4] hasbeenprovedtobeverysimple,reliableandrobust.In[4],Battenetal.proposedanappropriate

choiceoftheacousticwavespeedsrequiredbyHLLCandprovedthattheresultingnumericalmethodresolvesisolatedshock

and

contact

waves

exactly,

and

is

positively

conservative

which

will

be

reviewed

in

the

following.

For

the

HLLC

flux,

two

averaged

statesU

l,Ur betweenthetwoacousticwavesSL ,SR areconsidered,whicharesepa-

ratedbythecontactwavewhosespeedisdenotedbySM.TheapproximateRiemannsolutionwithtwostatesUl andUr is

definedas

UHLLC =

Ul, ifSL > 0,U

l, ifSL 0 0,

Fl=Fl+SL (UlUl ), ifSL 0


7/19


To

determineU

l,

the

following

assumption

has

been

made

[4]

SM=ul=ur=u. (2.25)whichgivesthecontactwavevelocity

SM=rur(SR ur) lul(SL ul) +plpr

l(SR

ur)

l(SL

ul)

. (2.26)

and

l= l SLulSLSM ,

p= l(ulSL )(ulSM) +pl,

lu

l= (SLul )l ul+(ppl )

SLSM ,

El= (SLul)Elplul+p SM

SLSM .

(2.27)

Therightstarstatecanbeobtainedsymmetrically.

Tomaketheschemepreservingpositivity,theacousticwavespeedsarecomputedfrom

SL= min

ulal,ua

, SR=min

ur+ar,u+a

, (2.28)

where

u= ul+urR1+R ,a=

( 1)[H 12u2],

H= (Hl+HrR )1+R ,

R=

rl

.

(2.29)

Defining

the

set

of

physically

realistic

states

as

those

with

positive

densities

and

internal

energies

by

G=

U=

u

E

, >0, e=

E u

2

2 >0

, (2.30)

thenG isaconvexset[4].

We

now

consider

a

first

order

finite

volume

scheme

Un+1j

=Unj

F

Unj ,Unj+1

FUnj1,Unj , (2.31)where F(,) isaHLLCfluxand= t

h .Forapositivelyconservativescheme(2.31),if Unj ,j=1, ,N,iscontainedinG ,

then Un+1j ,j=1, ,N,willalsolie insideG .ThiswouldbeguaranteedbyprovingtheintermediatestatesUl G ifwehave

UlG ,andprovingUr G ifwehaveUrG ,becauseG isaconvexset(fordetailssee[4]).Inthefollowing,wewillshowtheleftstarstateU

lG ,whilesimilarargumentsholdfortherightstarstateUr G .

Thatis,whenUlG ,whichisequivalentto

l >0, El1

2 lu

2

l >0, (2.32)

wewillhave

l >0, (2.33)

and

El1

2l u

l

2>0. (2.34)

From(2.27),wecanget

l = lSL ul

SLSM. (2.35)

SM

in

(2.26) is

an

averaged

velocity,

from

(2.28) we

have

SL < SM, SL < ul, (2.36)


8/19


and

l >0 iseasilyobtained.Usingrelations(2.27) and(2.36),(2.34)canberewrittenas

(ulSL )El+p lulp SM+((SL ul)lulpl+p)2

2l(SL ul)>0, (2.37)

whichisequivalentto

1

2 l(SM ul)2

p lSM

ul

ulSL + pl

1 >0. (2.38)To

guarantee

this

inequality

for

any

value

ofSM ul ,thediscriminantoftheabovequadraticfunctionofSM ul shouldbe

negative,

which

gives

the

following

condition

p2l

(ulSL )2 22l el


9/19


Noticing

thatU+

j 12= Qj (x1j ) andUj+ 12 =Qj (

xMj ),j,thescheme(2.42) canberearrangedasfollows

Un+1j

=M

=1 Qj

xj FUj+ 12 , U+j+ 12FU+

j 12, U

j+ 12

+FU+j 12

, Uj+ 12

FUj 12

, U+j 12

=M1=2 Qj

xj +1U+j 12

1

F

U+j 12 , Uj+ 12FUj 12 , U+j 12

+M

Uj+ 12

M

F

Uj+ 12

, U+j+ 12

FU+j 12

, Uj+ 12

=M1=2

Qjxj +1 H1+MHM,

where

H1=U+j 12

1

F

U+j 12

, Uj+ 12

FUj 12

, U+j 12

,

HM=Uj+ 12

M

F

Uj+ 12 , U+j+ 12

FU+j 12 , Uj+ 12 .The

above

two

equations

are

both

of

the

form

(2.31),

therefore H1 and HM areinthesetG duetoU

j+ 1

2

G ,j andtheCFLcondition(2.44) withtheHLLCflux(2.24) andtheacousticwavespeeds(2.28).Now Un+1j G isprovedsince it isaconvexcombinationofH1 ,HM andQj (xj) for2 M 1,whichareallinG .

Similartotheapproachin[42,43],thepositivity-preservinglimiterforthepresentschemeintheone-dimensionalspace

willbeconstructed.Theeasy-implementationalgorithmofWENOschemesin[42]willbeadopted:

1. Setupasmallpositiveparameter=minj{1013,nj }.2. Compute

the

limiter

1= min nj

nj min

, 1

, (2.46)

wheremin={

j+ 12

,+j 1

2

,j(x1j )} and

jx1j

= nj1+

j 12M

j+ 121 21

. (2.47)

3. Modifythedensitybyletting

j(x)=1

j (x) nj+nj . (2.48)

Getj

+12

and+j

12

from

j+ 12

=1

j+ 12

nj+nj ,

+j 12

=1

+j 12

nj+nj .

Denote

W1j=Uj+ 12 , W2j=U+j 12 ,

W3j=Un

j1U+

j 12MU

j+ 121 21

.

4. Get2= min=1,2,3 t frommodifyingtheinternalenergy:

For=1,2,3: if

e(Wj )< ,solvethefollowingquadraticequationsfort asin[43]

e

1 t

Unj+ t Wj= (2.49)


10/19


Table 3.1

NumericalerrorsandordersforExample 1.

N L1 error L1 order L error L order L2 error L2 order

10 7.802E04 6.506E04 5.874E0420 1.493E05 5.71 1.716E05 5.24 1.263E05 5.5440 3.260E07 5.52 2.942E07 5.87 2.625E07 5.5980 9.107E09 5.16 9.117E09 5.01 7.162E09 5.20

160 2.695E

10 5.08 2.903E

10 4.97 2.113E

10 5.08

320 8.169E12 5.04 9.202E12 4.98 6.413E12 5.04

Ife(Wj ) ,lett=1.Denote

Uj+ 12

=2

Uj+ 12

Unj+Unj , U+j 12 =2

U+

j 12Unj

+Unj .5. Thescheme(2.42) withthepositivity-preservinglimiterwouldbe

Un+1j

=Unj (F

Uj+ 12

,U+j+ 12

FUj 12

,U+j 12

. (2.50)

Remark2.Toprove that the limiterwillnotdestroythehighorderaccuracyofdensity forsmoothsolutions, forafifth

orderscheme,weneedtoshow

j

(x)

j

(x)=

O (x5) in(2.48).Inthepresentcompactscheme,althoughj+ 12

and+j 12

are

obtained

globally,

which

are

different

from

those

in

[42,43],

the

constructed

polynomialj (x) from(2.43) canbeseen

locally.Thus,theproofofpreservinghighorderaccuracyofdensityissimilartothatin[42,43].Similarargumentsholdfor

the

internal

energy.

So

the

scheme

(2.50) is

conservative,

high

order

accurate

and

positivity

preserving.

2.5. Temporaldiscretization

Strong

stability

preserving

(SSP)

high

order

RungeKutta

time

discretization

[16] will

be

used

to

improve

the

temporal

accuracyforthescheme(2.50).Thethird-orderSSPRungeKuttamethodis

U(1) =Un + tLUn,U(2) = 3

4Un + 1

4U(1) + 1

4t LU

(1)

,Un+1 = 1

3Un + 2

3U(2) +2

3t L

U(2)

, (2.51)

whereL(U) isthespatialoperator.Similarto[43],forSSPhighordertimediscretizations,thelimiterwillbeusedateach

stageoneachtimestep.

3. Numericalexamples

Inthissection,wewillinvestigatethenumericalperformanceofthepresentpositivity-preservingfifth-orderfinitevol-

ume

compact-WENO

(FVCW)

scheme.

The

fifth-order

WENO

scheme

[6] will

be

denoted

as

WENO-Z

and

the

original

fifthorderWENOschemeofJiangandShu[19] isdenotedasWENO-JS.WewillcomparetheFVCWschemetoWENO-JS

andWENO-Zschemes.Forallthenumericaltests,thethird-orderSSPRungeKuttamethod(2.51) isusedundertheCFL

condition(2.44)unlessotherwisespecified.ThenumericalsolutionsarecomputedwithN gridnodesanduptotimet.

Example1.Advectionofdensityperturbation.Theinitialconditionsfordensity,velocityandpressurearespecified,respec-

tively,

as

(x, 0)=1 + 0.2sin(x), u(x, 0)=1, p(x, 0)= 1.Theexactsolutionofdensityis(x,t)=1+ 0.2sin( (x t)).

Thecomputational domain is [0,2] and theboundary condition is periodic. The L1 , L2 and L errors and orders att=2 for the present finite volume compact-WENO scheme are shown in Table 3.1. Here the time step is taken to bet= 1|u|+a h5/3 .Wecanclearlyobservefifth-orderaccuracyforthisproblem.

Inthisexamplewithsmoothexactsolutions,wealsocomparethecomputationalcostbetweentheFVCWschemeand

the WENO-JS scheme. As we know, the FVCW scheme has high resolutions, however, a 33 block tri-diagonal system(2.21) needs to be solved at each grid nodexj+ 12 and at each stage of each time step. This might be computationallyexpensive.

However,

we

will

demonstrate

by

this

example

that

the

FVCW

scheme

would

still

be

more

efficient.

Two

kinds

of

reconstructions

for

systems

are

considered.

One

is

based

on

a

characteristic

variable

reconstruction,

the

other

is

directlyreconstructing

on

the

conservative

variables.

We

take

relatively

coarser

grids

and

choose

the

time

step

to

satisfy|u|+a=


11/19


Table 3.2

NumericalerrorsandcomputationalcostforWENO-JSandFVCWschemesforExample 1.Conservativevariablereconstruction.

FVCW WENO-JS

N L1 error L error L2 error CPU cost (s) N L1 error L error L2 error CPU cost (s)7 3.780E03 2.796E03 2.939E03 1.56E002 15 2.236E03 1.899E03 1.792E03 3.13E02

14 7.819E05 8.125E05 6.366E05 3.12E02 30 7.510E05 7.187E05 6.295E05 0.1128 2.065E06 1.537E06 1.579E06 0.14 60 2.352E06 2.353E06 1.919E06 0.4756 5.879E08 4.699E08 4.511E08 0.58 120 7.336E08 7.082E08 5.878E08 1.88

112 1.945E09 1.482E09 1.506E09 2.22 240 2.280E09 2.022E09 1.824E09 7.55224 8.882E11 6.897E11 6.926E11 8.86 480 6.977E11 5.909E11 5.541E11 29.95

Table 3.3

NumericalerrorsandcomputationalcostforWENO-JSandFVCWschemesforExample 1.Characteristicvariablereconstruction.

FVCW WENO-JS

N L1 error L error L2 error CPU cost (s) N L1 error L error L2 error CPU cost (s)7 3.780E03 2.796E03 2.939E03 3.13E02 15 2.236E03 1.899E03 1.792E03 4.69E02

14 7.819E05 8.125E05 6.366E05 0.11 30 7.509E05 7.183E05 6.293E05 0.1228 2.065E06 1.537E06 1.579E06 0.47 60 2.351E06 2.346E06 1.917E06 0.6956 5.879E08 4.699E08 4.511E08 1.86 120 7.318E08 6.969E08 5.859E08 2.72

112 1.945E09 1.482E09 1.506E09 7.34 240 2.259E09 1.943E09 1.802E09 10.84224 8.882E11 6.898E11 6.926E11 29.22 480 6.800E11 5.616E11 5.377E11 43.27

Fig. 3.1. ComparisonofCPUcostversusL1 errorsfortheWENO-JSandFVCWschemes.Left:conservativevariablereconstructioninTable 3.2;Right:

characteristicvariablereconstructioninTable 3.3.

0.16,sothatthespatialerrorwouldalwaysdominate. InTable 3.2,weshowthecomputationalcostbetweentheFVCW

scheme

and

the

WENO-JS

scheme

for

the

conservative

variable

reconstruction

case.

For

this

case,

without

characteristic

decomposition,

only

tri-diagonal

(not

block

tri-diagonal)

systems

need

to

be

solved

along

each

component,

less

CPU

cost

wouldbeneeded.WecanseeatacomparableL1 errorlevel,thecomputationalcostfortheFVCWschemeismuchlessthan

theWENO-JSschemeespeciallywhentheerrorissmall,whichcanalsobeseenfromFig. 3.1 (left),wherethecomparison

of

the

CPU

cost

versus

the L1 errors isdisplayed.Similarly inTable 3.3 andFig. 3.1 (right) forthecharacteristicvariable

case,

we

can

also

observe

less

computational

cost

for

the

FVCW

scheme

when

it

has

comparable

error

to

the

WENO-JS

scheme.Similardiscussionscanbefoundin[13].WenotethattheFVCWschemewithconservativevariablereconstruction

ismoreefficientthanthecharacteristicvariablereconstructionforsmoothsolutions.Howeverfordiscontinuoussolutions,

the

characteristic

variable

reconstruction

would

perform

better

to

control

spurious

numerical

oscillations.

In

this

paper,

for

thefollowingexamples,wewillmainlyadoptthecharacteristicvariablereconstruction.

Example2.Thisexampleistheone-dimensionalLaxshocktubeproblem[21]withthefollowingRiemanninitialconditions

(, u,p) =

(0.445, 0.698, 3.528), 5 x


12/19


Fig. 3.2. The density (left) and pressure (right) profiles of the Lax problem (3.1)at t=1.4.

WENO-JSschemewithN=100 andCPUcost0.55s,botharebetterthantheWENO-JSschemewithN=60.Itshowsthecompactschemehasbetterresolutionsthanthenon-compactscheme.Atthesameresolution,thecompactschemecantake

much

coarser

grids

while

with

comparable

computational

cost

as

the

non-compact

scheme.

Example3.Thisexampleistheone-dimensionalSodshocktubeproblem[35]withthefollowingRiemanninitialconditions

(, u,p) =

(0.125, 0, 1), 5 x


13/19


Fig. 3.3. The comparison of density for the Lax problem(3.1) with the WENO-JS scheme and the FVCW scheme at t=1.4.

Fig. 3.4. The density profiles of the Sod problem (3.2)at t=2.0.

Example4.Inthisexample,theonedimensionalMach3shock-turbulencewaveinteractionproblem[34]istestedwiththe

followinginitialconditions

(, u,p) =

(3.857143, 2.629369, 10.33333), 5x


14/19


Fig. 3.5. Shock-turbulence interaction(3.3)with N= 200 at t=1.8.

Fig. 3.6. Shock-turbulence interaction(3.3)with N

=400 at t

=1.8.

Fig. 3.7. Blastwave interaction problem (3.4) with N= 200 at t=0.038.

oscillatorybehavioracross theshockwave.Thenumerical solution isgreatly improvedwith N=400 and thenumericalresults

are

shown

in

Fig. 3.6.

Example5. The one dimensional blastwave interaction problem of Woodward and Collela [38] has the following initial

conditions

(, u,p) =(1, 0, 1000), 0

x


15/19


Fig. 3.8. Blastwave interaction problem(3.4) with N= 400 at t=0.038.

Fig. 3.9. The results of the low density and low internal energy problem (3.5)with N=400 at t=0.1.

andreflectiveboundaryconditions.Thefinal time ist=0.038.The initialpressuregradientsgeneratetwodensityshockwaves

which

collide

and

interact

at

later

time.

The

solution

of

this

problem

contains

rarefactions,

interaction

of

shock

waves

andthecollisionofstrongshockwaves.Theexactsolutionofthistestproblemisareferencesolutioncomputedbythe

WENO-JSschemewith3200 gridpoints.ThedensityobtainedwithWENO-JS,WENO-ZandthepresentFVCWschemesat

t=0.038 with200 cellsareshowninFig. 3.7.ThezoomedregionsofthedensityprofileFig. 3.7(b)showthatthepresentFVCWschemegivesbetterresolutionthan theother twoschemes.Thenumericalsolution isalsogreatly improvedwith

N=400 andthenumericalresultsareshowninFig. 3.8.

Example

6.

In

this

test,

we

consider

a

one-dimensional

low

density

and

low

internal

energy

Riemann

problem

with

thefollowing

initial

conditions


16/19


Fig. 3.10. The results of the strong shock wave problem (3.6)with N=200 at t=2.5 106 .

(, u,p) = (1, 2, 0.4), 0x


17/19


Fig. 3.11. One-dimensionalproblemsinvolvingvacuumornearvacuum,h=0.005:(left)doublerarefactionproblem(3.7) att=0.6;(right)planarSedovblast-waveproblem(3.8)att

=0.001.

positivevaluesof2.120E04 and2.201E04 respectively.Forthisproblemwithvacuumornear-vacuumsolutions,someoscillations

can

also

be

observed

which

might

be

due

to

the

same

reason

as

described

in

Example 6.

Example9.Thisone-dimensionaltestproblemistheplanarSedovblast-waveproblemwiththefollowinginitialconditions

(, u,p) =

(1, 0, 4 1013), 0


18/19


Fig. 3.12. The results of the Leblanc problem(3.9)at t= 6.0. N= 400 (left), N= 1000 (right).

Example

10.

LeBlanc

shock

tube

problem.

In

this

extreme

shock

tube

problem,

the

computational

domain

is

[0,

9] filled

withaperfectgaswith=5/3.Theinitialconditionsarewithhighratioofjumpsfortheinternalenergyanddensity.Thejumpfortheinternalenergyis106 andthejumpforthedensityis103 .Theinitialconditionsaregivenby

(, u, e)=

(1, 0, 0.1), 0x


19/19


Acknowledgements

The

work

was

partly

supported

by

the

Fundamental

Research

Funds

for

the

Central

Universities

(2010QNA39,

2010LKSX02).

The

third

author

acknowledges

the

funding

support

of

this

research

by

the

Fundamental

Research

Funds

fortheCentralUniversities(2012QNB07).

References

[1] N.

Adams,

K.

Shariff,

A

high-resolution

hybrid

compact-ENO

scheme

for

shock-turbulence

interaction

problems,

J.

Comput.

Phys.

127

(1996)

2751.[2] D.S.Balsara,Self-adjusting,positivitypreservinghighorderschemesforhydrodynamicsandmagnetohydrodynamics,J.Comput.Phys.231(2012)

75047517.

[3] D.S.Balsara,C.-W.Shu,Monotonicitypreservingweightedessentiallynon-oscillatoryschemeswithincreasinglyhighorderofaccuracy,J.Comput.Phys.

160(2000)405452.[4] P.Batten,N.Clarke,C.Lambert,D.Causon,OnthechoiceofwavespeedsfortheHLLCRiemannsolver,SIAMJ.Sci.Comput.18(1997)15531570.

[5] R.Borges,M.Carmona,B.Costa,W.Don,Animprovedweightedessentiallynon-oscillatoryschemeforhyperbolicconservationlaws,J.Comput.Phys.

227(2008)31913211.[6] M.Castro,B.Costa,W.Don,Highorderweightedessentiallynon-oscillatoryWENO-Zschemesforhyperbolicconservationlaws,J.Comput.Phys.230

(2011)17661792.

[7] J.Cheng,C.-W.Shu,Positivity-preservingLagrangianschemeformulti-materialcompressibleflow,J.Comput.Phys.257(2014)143168.

[8] B.Cockburn,C.-W.Shu,Nonlinearlystablecompactschemesforshockcalculations,SIAMJ.Numer.Anal.31(1994)607627.[9] X.Deng,H.Maekawa,Compacthigh-orderaccuratenonlinearschemes,J.Comput.Phys.130(1997)7791.

[10] X.Deng,H.Zhang,Developinghigh-orderweightedcompactnonlinearschemes,J.Comput.Phys.165(2000)2244.[11] B.Einfeldt,C.-D.Munz,P.L.Roe,B.Sjgreen,OnGodunov-typemethodsnearlowdensities,J.Comput.Phys.92(1991)273295.

[12] J.Estivalezes,P.Villedieu,High-orderpositivity-preservingkineticschemesforthecompressibleEuler equations,SIAMJ.Numer.Anal.33(1996)

20502067.[13] D.Ghosh,J.Baeder,CompactreconstructionschemeswithweightedENOlimitingforhyperbolicconservationlaws,SIAMJ.Sci.Comput.34(2012)

16781706.

[14] S.K.Godunov,Adifference methodforthenumericalcalculationofdiscontinuoussolutionsofhydrodynamicequations,Mat.Sb.89(1959)271306.

[15] Y.Gong,Y.X.Long,LUmethodforsolvingblocktridiagonalmatrixequationsanditsapplications,J.DalianUniv.Technol.37(1997)406409.[16] S.Gottlieb,D.Ketcheson,C.-W.Shu,Highorderstrongstabilitypreservingtimediscretizations, J.Sci.Comput.38(2009)251289.

[17] J.Gressier,P.Villedieu,J.-M.Moschetta,Positivityoffluxvectorsplittingschemes,J.Comput.Phys.155(1999)199220.[18] X.Y.Hu,N.A.Adams,C.-W.Shu,Positivity-preservingmethodforhigh-orderconservativeschemessolvingcompressibleEuler equations,J.Comput.

Phys.242(2013)169180.

[19] G.-S.Jiang,C.-W.Shu,EfficientimplementationofweightedENOschemes,J.Comput.Phys.126(1996)202228.[20] L.Jiang,H.Shan,C.Liu,Weightedcompactschemeforshockcapturing,Int.J.Comput.FluidDyn.15(2001)147155.

[21] P.D.Lax,Weaksolutionsofnonlinearhyperbolicequationsandtheirnumericalcomputation,Commun.PureAppl.Math.7(1954)159193.

[22] S.Lele,Compactfinitedifferenceschemeswithspectral-likeresolution,J.Comput.Phys.103(1992)1642.[23] R.J.LeVeque,FiniteVolumeMethodsforHyperbolicProblems,vol. 31,CambridgeUniversityPress,2002.

[24] W.Liu,J.Cheng,C.-W.Shu,HighorderconservativeLagrangianschemeswithLaxWendroff type timediscretizationforthecompressibleEuler

equations,J.Comput.Phys.228(2009)88728891.

[25] X.

Liu,

S.

Osher,

T.

Chan,

Weighted

essentially

non-oscillatory

schemes,

J.

Comput.

Phys.

115

(1994)

200212.[26] X.Liu,S.Zhang,H.Zhang,C.-W.Shu,Anewclassofcentralcompactschemeswithspectral-likeresolutionI:Linearschemes,J.Comput.Phys.248

(2013)235256.

[27] R.Loubre,M.J.Shashkov,Asubcellremappingmethodonstaggeredpolygonalgridsforarbitrary-LagrangianEulerianmethods,J.Comput.Phys.209

(2005)105138.[28] B.Perthame,Boltzmanntypeschemesforgasdynamicsandtheentropyproperty,SIAMJ.Numer.Anal.27(1990)14051421.

[29] B.Perthame,C.-W.Shu,OnpositivitypreservingfinitevolumeschemesforEulerequations,Numer.Math.73(1996)119130.[30] S.Pirozzoli,Conservativehybridcompact-WENOschemesforshock-turbulenceinteraction,J.Comput.Phys.178(2002)81117.

[31] Y.Ren,M.Liu,H.Zhang,Acharacteristic-wise hybridcompact-WENOschemeforsolvinghyperbolicconservationlaws,J.Comput.Phys.192(2003)

365386.[32] Y.Shen,G.Yang,Z.Gao,High-resolutionfinitecompactdifferenceschemesforhyperbolicconservationlaws,J.Comput.Phys.216(2006)114137.

[33] C.Shu,Essentiallynon-oscillatoryandweightedessentiallynon-oscillatoryschemesforhyperbolicconservationlaws,in:AdvancedNumericalApprox-

imationofNonlinearHyperbolicEquations,1998,pp. 325432.

[34] C.-W.Shu,S.Osher,Efficientimplementationofessentiallynon-oscillatoryshock-capturingschemes,II,J.Comput.Phys.83(1989)3278.[35] G.A.Sod,Asurveyofseveralfinitedifferencemethodsforsystemsofnonlinearhyperbolicconservationlaws,J.Comput.Phys.27(1978)131.

[36] T.Tao,K.Xu,Gas-kineticschemesforthecompressibleEulerequations:positivity-preservinganalysis,Z.Angew.Math.Phys.50(1999)258281.

[37] E.F.

Toro,

Riemann

Solvers

and

Numerical

Methods

for

Fluid

Dynamics:

A Practical

Introduction,

Springer,

2009.[38] P.Woodward,P.Colella,Thenumericalsimulationoftwo-dimensionalfluidflowwithstrongshocks,J.Comput.Phys.54(1984)115173.[39] T.Xiong,J.-M.Qiu,Z.Xu,ParametrizedpositivitypreservingfluxlimitersforthehighorderfinitedifferenceWENOschemesolvingcompressibleEuler

equations,http://arxiv.org/abs/1403.0594 ,2013,submitted.

[40] S.Zhang,X.Deng,M.Mao,C.Shu,ImprovementofconvergencetosteadystatesolutionsofEulerequationswithweightedcompactnonlinearschemes,

ActaMath.Appl.Sin.29(2013)449464.

[41] S.Zhang,S.Jiang,C.Shu,Developmentofnonlinearweightedcompactschemeswithincreasinglyhigherorderaccuracy,J.Comput.Phys.227(2008)

72947321.[42] X.Zhang,C.Shu,Maximum-principle-satisfyingandpositivity-preservinghigh-orderschemesforconservationlaws:surveyandnewdevelopments,

Proc.R.Soc.A,Math.Phys.Eng.Sci.467(2011)27522776.

[43] X.Zhang,C.-W.Shu,Onpositivity-preservinghighorderdiscontinuousGalerkinschemesforcompressibleEuler equationsonrectangularmeshes,

J. Comput.Phys.229(2010)89188934.[44] X.Zhang,C.-W.Shu,Positivity-preservinghighorderdiscontinuousGalerkinschemesforcompressibleEulerequationswithsourceterms,J.Comput.

Phys.230(2011)12381248.

[45] X.Zhang,C.-W.Shu,Positivity-preservinghighorderfinitedifferenceWENOschemesforcompressibleEulerequations,J.Comput.Phys.231(2012)

22452258.

[46] X.

Zhang,

Y.

Xia,

C.-W.

Shu,

Maximum-principle-satisfying

and


high

order

discontinuous

Galerkin

schemes

for

conservation

laws

on

triangular

meshes,

J.

Sci.

Comput.

50

(2012)

2962.
http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6164616D733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C736172613230313273656C66s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62616C73617261323030306D6F6E6F746F6E6963697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib62617474656E3139393763686F696365s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib626F7267657332303038696D70726F766564s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib63617374726F3230313168696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib4368656E6732303134706F7369746976697479s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib636F636B6275726E313939346E6F6E6C696E6561726C79s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6731393937636F6D70616374s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib64656E6732303030646576656C6F70696E67s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib65696E66656C647431393931676F64756E6F76s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevier.com/S0021-9991(14)00453-7/bib6573746976616C657A65733139393668696768s1http://refhub.elsevi

Positivity Preserving High Order Finite Volume Compact-WENO Schems for Compressible Euler Equations

Documents

Transcript of Positivity Preserving High Order Finite Volume Compact-WENO Schems for Compressible Euler Equations