positivity preserving high order finite volume compact-weno schems for compressible euler equations
TRANSCRIPT
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8/11/2019 Positivity Preserving High Order Finite Volume Compact-WENO Schems for Compressible Euler Equations
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Journal of Computational Physics 274 (2014) 505523
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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
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e
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f
o a
b
s
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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 -
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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
positivity-preserving
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
positivity-preserving
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.
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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
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508 Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523
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.
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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.
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510 Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523
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
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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)
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512 Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523
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
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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)
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514 Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523
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=
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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
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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
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Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523 517
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
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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
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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
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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
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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
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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
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Y. Guo et al. / Journal of Computational Physics 274 (2014) 505523 523
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).
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