Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows

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University of Nebraska - Lincoln

DigitalCommons@University of Nebraska - Lincoln

NASA Publications

National Aeronautics and Space Administration

2017

Recent developments in accuracy and stability

improvement of nonlinear filter methods for DNS

and LES of compressible flows

Helen Yee

NASA Ames Research Center, [email protected]

Björn Sjögreen

NASA Ames Research Center, [email protected]

Follow this and additional works at:

http://digitalcommons.unl.edu/nasapub

This Article is brought to you for free and open access by the National Aeronautics and Space Administration at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in NASA Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.

Yee, Helen and Sjögreen, Björn, "Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and

LES of compressible flows" (2017). NASA Publications. 250.

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JID:CAF [m5G;September4,2017;14:2]

Computers and Fluids 0 0 0 (2017) 1–18

ContentslistsavailableatScienceDirect

Computers

and

Fluids

journalhomepage:www.elsevier.com/locate/compfluid

Recent

developments

in

accuracy

and

stability

improvement

of

nonlinear

ﬁlter

methods

for

DNS

and

LES

of

compressible

ﬂows

R

H.C.

Yee

∗

,

Björn

Sjögreen

NASA Ames Research Center, United States

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 3 June 2017 Revised 8 August 2017 Accepted 14 August 2017 Available online xxx Keywords:

High order methods

High order shock-capturing methods Stability improvement of high order methods

Accuracy improvement for DNS & LES

a

b

s

t

r

a

c

t

Recent progressintheimprovement ofnumericalstability andaccuracy oftheYee andSjögreen [49] highordernonlinearfilter schemesis described.TheYee &Sjögreen adaptivenonlinearfiltermethod consistsofahighordernon-dissipativespatialbase schemeandanonlinearfilterstep.Thenonlinear filterstepconsistsofaflowsensorandthedissipativeportionofahighresolutionnonlinearhighorder shock-capturingmethodtoguidetheapplicationoftheshock-capturingdissipationwhereneeded.The nonlinearfilterideawas firstinitiatedbyYeeetal. [54]usinganartificialcompressionmethod(ACM) ofHarten[12]astheflowsensor.Thenonlinearfilterstepwasdevelopedtoreplacehighorderlinear filterssothatthesameschemecan beusedforlongtimeintegrationofdirectnumericalsimulations (DNS)andlargeeddysimulations(LES)forbothshock-freeturbulenceandturbulence-shockwaves inter-actions.Theimprovementincludesfourmajornewdevelopments:(a)Smartflowsensorsweredeveloped toreplacetheglobalACMflowsensor[21,22,50].Thesmartflowsensorprovidesthelocationsandthe estimatedstrengthofthenecessarynumericaldissipationneededattheselocationsandleavestherest oftheflowfieldfreeofshock-capturingdissipation.(b)Skew-symmetricsplittingsweredevelopedfor compressiblegasdynamicsand magnetohydrodynamics(MHD)equations[35,36]toimprovenumerical stabilityforlongtimeintegration.(c)Highorderentropystablenumericalfluxesweredevelopedasthe spatialbaseschemes forboththecompressible gasdynamicsand MHD[37,38].(d) Severaldispersion relation-preserving(DRP) centralspatialschemes wereincluded asspatialbaseschemes inthe frame-workofournonlinearfiltermethodapproach[40].Withthesenewschemeconstructionsthenonlinear filterschemesareapplicabletoawiderclassofaccurateandstableDNSandLESapplications,including forcedturbulencesimulationswherethetimeevolutionofflowsmightstartwithlowspeedshock-free turbulenceanddevelopintosupersonicspeedswithshocks.Representative testcasesforbothsmooth flowsandproblemscontainingdiscontinuitiesforcompressibleflowsareincluded.

PublishedbyElsevierLtd.

1. Introduction

The construction of spatially stable and accurate numerical methodsforlongtimeintegrationofcomplexmultiscale compress-ibleshockfreeturbulentflows,turbulentflowscontaining discon-tinuities,steepgradients, andvorticalflowsisvery differentfrom shorter time integrationof non-turbulence/non-acoustic unsteady flows andrapidly developing shock-wave interaction simulations. Standarddirectnumericalsimulations(DNS)andlargeeddy simu-lations (LES)usuallyrequirehighaccuracy schemeswithlow dis-sipativeandlowdispersiveerrorsinspaceandtime.Itiscommon

R Yee-Sjogreen 2nd paper for a special issue in honor of Prof. Eleuterio F. Toro’s

70th birthday, 2017.

∗ _{Corresponding author.}

E-mail addresses: [email protected] (H.C. Yee), [email protected] (B. Sjögreen).

tohavenumericallyinduced highfrequencyoscillations (spurious numericalartifacts)duetolongtimeintegrationofnon-dissipative orlow-dissipativefinitediscretizations. Agoodnumericalmethod forDNS and LES should be able to minimize these spurious os-cillations while maintainingstability andaccuracy during an en-tirelong-timeevolution.Thispaperonlyaddressesthespatial dis-cretizationbythemethod-of-linesapproach.Controllinglow dissi-pativeandlowdispersivetemporalerrorsisimportantbutoutside the scope of this investigation. Highly accurate appropriate tem-poral discretizations and, when appropriate, smalltime steps are assumedtobeusedinconjunctionwiththecurrentdevelopment. Numerical stability and accuracy considerations are an intri-catebalancing actforturbulenceflowswithdiscontinuities. More stable numerical methods usually contain more numerical dissi-pation than their higher accuracy method counterparts. Improv-ing nonlinear stability without smearing physical turbulent fluc-tuations for long time integrations are conflicting requirements

http://dx.doi.org/10.1016/j.compﬂuid.2017.08.028

0045-7930/Published by Elsevier Ltd.

Pleasecitethisarticleas:H.C.Yee,B.Sjögreen,Recentdevelopmentsinaccuracyandstabilityimprovementofnonlinearﬁltermethods

This document is a U.S. government work and is not subject to copyright in the United States.

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Fig. 1. Smooth initial data of the linear advection problem.

fornumerical methods development. Since the turn ofthis cen-tury, manyoptimized compact andnon-compact WENOschemes have been developed to address some of the pacing difficulties. See,e.g.,[10,16,27].ThesenumericalmethodsareveryhighinCPU operation counts andmost often still suffer from numerical sta-bility/accuracyforlong time integration.Other optimized numer-ical methods for combating these conflicting requirements com-bine the non-dissipative or low dissipative, and low dispersive spatialschemes with high order high resolution shock-capturing schemes.Theblendingofthesetwotypesofschemesrequires ex-tremecaretoensurenumericalconservationandstabilityat inter-facelocations[30].Othermoreefficientnumericalmethodswhich avoidthe interfacing problem are the [21,22,33,33,49,54] nonlin-earfilterschemes. Numerical stabilitycanbe improved by skew-symmetricsplittingofthe inviscid flux derivatives[35,36,53]and byhighorderstableentropyconservativenumericalfluxes[37,38]. Anothersourceofaccuracyimprovementisthedispersion relation-preserving(DRP) schemes forcomputational aeroacoustics (CAA)

[40].

Nonlinearfilter schemes: TheYee andSjögreen[50] adaptive nonlinear filter method consists of a high order non-dissipative spatial base scheme and a nonlinear filter step. The nonlinear filter step consists of a flow sensor and the dissipative portion ofa highresolution highorder shock-capturingmethod to guide theapplication of the shock-capturingdissipation where needed. The nonlinear filteridea wasfirst initiated by Yee etal. [54] us-ingan artificialcompressionmethod(ACM)ofHarten [12]asthe flow sensor. The nonlinear filter step was developed to replace highorderlinearfilterssothat thesameschemecanbe usedfor long time integration of direct numerical simulations (DNS) and large eddy simulations (LES) for both shock-free turbulence and turbulence-shockwave interactions. Smartflow sensors were de-velopedatalaterstagebythesameinvestigatorsandcollaborators in[21,22,33,33,49]. The smart flow sensor provides the locations andtheestimatedstrengthofthenecessarynumericaldissipation neededattheselocationsandleavestherestoftheflowfieldfree ofshock-capturingdissipations.Itisnotedthatthenonlinearfilter approachofYeeandSjögreen[50]requiresoneRiemannsolverper timesteppergridpointforeachspatialdirection.Itisindependent ofthetimediscretization.However,hybridschemes(switching be-tweenhighorder non-dissipativemethods andhighorder shock-capturingmethods)wouldrequirefourRiemannsolvers per time steppergridpointforeachspatialdirectionif,e.g.,afourth-stage Runge-Kuttatimediscretizationisused.Unlikethehybridmethod, ourhighlyparallelizableadaptive nonlinearfilterschemes donot

rely on switching between schemes to avoid the related numer-ical instability and conservation consideration at switching loca-tions.Thesenonlinearﬁlterschemeswithadaptivenumerical dis-sipation control inhigh ordershock-capturingschemes andtheir hybridcousins haveshownexcellent performanceforcertain tur-bulenttestcases.Formorepractical3DtestcasesofDNSandLES ofcompressibleshock-freeturbulence,lowspeedturbulencewith shocklets, andsupersonicturbulence fornon-periodic boundaries incurvilineargeometries,some improvementinnumerical stabil-ityisneededwithoutresortingtoaddednumericaldissipationthat caninterferewiththeaccuracyofnumericalsimulations.

Skew-symmetric splitting of the inviscid ﬂux derivative:

Starting in the early 1980s skew-symmetric splitting of certain components of the inviscid flux derivatives in conjunction with central schemes was shown to help withnumerical stabilityfor longtimeintegration.Forcertainsplittingstheycanprovidea sta-ble energy norm estimate for the Euler equations with smooth flows. For other skew-symmetric formulations they can provide a discrete momentum conservation or a discrete kinetic energy preservation property. SeeArakawa [1],Blaisdell etal.[2], Ducros etal.[8],Kotov etal.[21, 22],SjögreenandYee[34],Yeeand Sjö-green[49,50],Yeeetal.[53]forsomediscussionsandperformance ofthe combinedapproachfor DNSandLESapplications.A semi-conservative skew-symmetric splitting (entropy splitting) of Yee etal.[53]inconjunctionwiththenonlinearfilterapproachto im-prove numerical stability without added ad hoc numerical dissi-pation wasconductedin2000.Ithasbeenutilizedextensivelyin DNS of shock-free turbulence. See [32] and their later work for their wideapplications.Fortheirskew-symmetricsplitting exten-siontotheidealmagnetohydrodynamics(MHD),seeSjögreenand Yee[35,36],Yeeetal.[53].Notethatsome oftheskew-symmetric splittings for the gas dynamics flux derivatives are not applica-bleand/orcannotbestraightforwardlyextendedtotheidealMHD

[53].Their degreeof stabilityimprovement isalso dependent on theMHDgoverningequationformulation.

High orderentropyconservativeschemes: Entropy conserva-tive schemes [7,42,48] are another class of methods that might havebetterstabilitypropertiesthanstraightforwardpurecentered discretizations and compact spatial schemes. Here, entropy con-servative schemes refer to conservative schemes satisfying a dis-creteentropyequation.Inviewofthefactthatmethodsproposed in [7,42,48] are low order and their linear numerical dissipation approachesforshock-capturingrequirefurtherimprovement, Sjö-greenandYeecombinedsomeoftheseideastoconstructaformof thehighorderconservativeentropynumericalfluxes.Startingwith thehighorderentropyconservativedevelopmentofSjögreenand Yee[34]forgasdynamicsinsmoothflows,constructionofefficient highorder conservativenumerical fluxesforproblemscontaining discontinuitiesandfortheidealMHDarereportedinSjögreenand Yee[37,38].Notethattheextensionofhighorderentropy conser-vative numericalfluxesthat were developedfor gasdynamics to theMHDisnot straightforwardduetothenon-strictlyhyperbolic natureoftheconservativeidealMHDequations.See[37,38].

DRP schemes: DRP schemes (optimized low dispersion

schemes)forCAAarealsoaclassofmethodsthatmighthave bet-ter accuracythan purecentered schemes.UnliketypicalDNSand LESnumericalconsiderations,themagnitudeofacousticsolutions issimilartonumericalnoisebutisdifferentfromnumerically in-ducedhighfrequencyoscillations duetolong time integrationof non-dissipative or low dissipative ﬁnite discretizations. Here, the term“DRP” schemeshasbeenusedloosely,accordingtotherecent deﬁnitionofDRPmethodsbyTam[44],toincludegeneralschemes thatperformvariousoptimizationstoreducenumericaldispersion errors for different applications. Most CAA-related DRP methods employed techniquesto minimize dispersion errorto resolve lin-earacousticwavesoverlongdistanceswithoutcompromisingthe

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Fig. 2. Gaussian pulse: C08 (top left) and optimized schemes without any linear or nonlinear ﬁlter, DRP4S7 (top right), DRP4S9 (bottom right), and STO9 (bottom left). Solutions at t = 3 of the linear advection problem. Computed solution plotted in blue color, exact solution shown in black color. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

realphysicalbehavior ofthe waveformpropagationoftheinitial boundary valueproblem(IBVP).A largepercentageofDRP meth-odsutilizeleastsquares,L1-norm,L2-norm,L∞-norm,andother

in-tegralmetrics tominimize thenumericalwavenumber errorover prescribedintervalsinordertoobtainthegridstencilcoeﬃcients. The resulting DRPschemes usually havewider grid stencils than theirstandardcentralschemesofthesameorderofaccuracy.Low dispersivetemporaldiscretizationandspecialtreatmentsforIBVPs ofthedifferentCAAapplicationsarealsoneeded.SeeTam[44,45], Brambley [6], Haras andTaasan [11], andLinders and Nordström

[24], Linders etal. [25] for formulations and overviews. Some of the DRP schemes might perform poorly for decaying or growing oscillations.SeeBrambley[6]forastudy.Fordiscontinuousinitial data andlongtime wave propagationsofsmooth acoustic waves, variousspaceandtimeDRPlinearfiltersareneeded. Foracoustic wavesinteractingwithshocksandturbulenceinducednoise,DRP schemeswithlinearfiltersaloneusually arenotcapableof simu-latingsuchflows.

According to Tam [44], optimized compact schemes are also DRP schemes. For over 20 years high order compact spatial dis-cretizations in conjunctionwithlinear highorder compactﬁlters have been methods of choice for many DNS and LES of incom-pressibleandlowspeedcompressibleturbulent/acousticﬂowsdue to their advantageofrequiring avery low numberofgrid points

per wavelength and flexibility in geometry handling. However, most optimized compact schemes were not designed for long-timeintegration andadditional constraintsareneeded. SeeHaras andTaásan [11] for the construction of compact finitedifference schemes for long time integration. In addition, the advantage of compactschemesseemstorequireadditionalinvestigationand re-search forcompressibleturbulent flows containing moderateand strongshockwaves.Onepopularmethodisbyemployinga blend-ingofhighordercompactspatialschemeswithhighorder shock-capturingschemes.Anothermoreefficientapproachforturbulence with discontinuities is the nonlinear filter approach of Sjögreen andYee[33],YeeandSjögreen[49],Yeeetal.[54].Theyemployed thehighordercompactschemeastheir spatialbasescheme.The YeeandSjögreenstudies[51]indicatedthatforshock-wave turbu-lence interactions the accuracy performance of compactschemes issimilarto thecentral schemeofthesameorderunderthe Yee andSjögreennonlinearfilterapproach.

Objectives:Here recentprogressinhighorder,nonlinearfilter numericalmethoddevelopmentforDNSandLESapplicationsis re-viewed.Theimprovementincludesfourmajornewdevelopments: (a)Smartflowsensorswere developedtoreplacetheglobalACM flow sensor [21,22,50]. The smart flow sensor provides the loca-tions andthe estimatedstrength ofthe necessarynumerical dis-sipationneededattheselocationsandleavestherestofthe flow

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Fig. 3. Gussian pulse: Standard nonlinear filter scheme C08+WENO7fi (top left). Optimized nonlinear filter schemes, DRP4S7+WENO7fi (top right), DRP4S9+WENO7fi (middle left), STO9+WENO7fi (middle right), and standard shock-capturing scheme WENO7 (bottom left). Solutions at time t = 3 of the linear advection problem. Computed solution plotted in blue color, exact solution shown in black. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Square pulse: C08 (top left). Optimized schemes without linear or nonlinear ﬁlter, DRP4S7 (top right), DRP4S9 (bottom left), and STO9 (bottom right). Solutions at t = 3 of the linear advection problem with square pulse initial data. Computed solution plotted in blue color, exact solution shown in black. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

fieldfreeofshock-capturingdissipation.(b)Skew-symmetric split-tingsweredevelopedforthecompressiblegasdynamicsandMHD equations[35,36]toimprovenumericalstabilityforlongtime inte-gration.(c)Highorderentropystablenumericalfluxeswere devel-opedasthespatialbaseschemesforboththecompressiblegas dy-namicsandMHDequations[37,38].(d)Severaldispersion relation-preserving (DRP) central spatial schemes were included as spa-tialbaseschemesintheframeworkofournonlinearfilterscheme methodapproach[40].

This paperonly considersseveral DRPcentral spatialschemes as the base scheme in the framework of the Yee and Sjögreen

[50] low dissipative nonlinear filter method approach. DRP time discretizations are notconsidered. Fortime discretizationwe uti-lize the low dissipative fourth-order Runge–Kutta method with small time steps for the investigationto minimize dispersion er-ror dueto timediscretization.Theinvestigationis focusedonthe possible gain inaccuracy by highorder entropynumerical fluxes and DRP schemes as the base scheme over the standard central schemes of the same grid stencil for generalDNS and LES com-pressibleflowcomputations.Asmentionedbefore,CAAfocuseson dispersionerrorforlongtimelinearwavepropagationratherthan the formal order of accuracy of the scheme. The resulting DRP schemes usually havewider grid stencils andan increase in CPU

operationscount compared to their standard central schemes of thesameorderofaccuracy.Fordiscontinuousinitialdataandlong time wave propagations of smooth acoustic waves various space andtimeDRPlinearfiltersareneeded.Foracousticwaves interact-ingwithshocksandturbulenceinducednoise,DRPschemeswith linearfiltersaloneusuallyarenotcapableofsimulatingsuchflows. Due to thisfact, here, the Yee and Sjögreennonlinear filter step withshock-capturingandlongtimeintegrationpropertiesreplaces thespatialDRPlinearfilter.

With these new scheme constructions the nonlinear ﬁlter schemes are applicable to a wider class of accurate and stable DNSandLESapplications,includingforcedturbulencesimulations where the time evolution of ﬂows might start with low speed shock-free turbulence and develop into supersonic speeds with shocks.See[21,22,36]fortwoofoursimulations.

Thenext four sectionsgive summaries ofthe fournewmajor developments.

2. Anoverviewofskew-symmetricsplitapproximationsforgas

dynamics

Standardcentereddifference approximationsofnonlinear con-servation laws normally encounter nonlinear instabilities after a

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Fig. 5. Square pulse: Standard nonlinear filter scheme C08+WEBO7fi (left top). Optimized nonlinear filter schemes, DRP4S7+WENO7fi (top right), DRP4S9+WENO7fi (middle left), STO9+WENO7fi (middle right), and standard shock-capturing scheme WENO7 (bottom left). Solutions at t = 3 of the linear advection problem with square pulse initial data. Computed solution plotted in blue color, exact solution shown in black. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

shorttime integration withoutadded numerical dissipation. It is well knownthat the appearance ofthese instabilitiescan be de-layediftheconvectiveﬂuxderivativesarewritteninanequivalent desiredsplit form before the pure central approximation is em-ployed.Hereafterthisisreferredtoasasplitapproximation.

For example, a split approximation starts from rewriting the derivativeoftheproduct(ab)x as

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Fig. 6. 1D Osher-Shu test case. Close up of the density at time 1.8 for C08-DS+WENO7fi, DRP4S7-DS+WENO5fi, DRP4S9-DS+WENO7fi, and STO9-DS+WENO7fi using a grid with 201 points.

Fig. 7. 1D Osher-Shu test case: Close up of the oscillations in density at time 1.8 for C08 Econs _ CK+WENO7ﬁ (left) and WENO7 (right).

beforediscretization.Hereaandbarefunctionsofxand

α

,

γ

and

β

are parameters so chosen tobe still equivalentto the original (ab)x beforediscretization.Acommonsplitderivativeisbysetting

α

=

γ

=

β

=1/2resultingintheform

(

ab

)

x= 1 2

(

ab

)

x+ 1 2abx+ 1 2axb. (2)

Thesemethodshavealonghistoryinﬁnitedifference approxi-mations;see,.e.g.,[1,23].Seealsoageneralizedconservativesplit convective derivative operators study by Pirozzoli [29]. The key mathematicalidea isthatformulasoftype(2)canbe usedto es-timate the L2 _norm _or _the _energy _norm _of _the _computed

solu-tion. From physical considerations some ofthe splittings provide thediscreteconservationofmomentumorpreservationofdiscrete kineticenergy.Awell-knownexampleisthelinearsystemof con-servationlaws

ut+A

(

x

)

ux=0 0<x<L, (3)

whereA(x)isasymmetricmatrix,andwesolvefortheunknown vectoru₌u

(

x_,t

)

fromgiveninitialdatau

(

x_,0

)

₌u0

(

x

)

.Boundary

dataare givenatx=0andx=L. Toshowhow thisisdone,e.g., werewrite(3)inanmathematicallyequivalentform:

ut+ 1 2

(

A

(

x

)

u

)

x+ 1 2A

(

x

)

ux− 1 2A

(

x

)

xu=0 (4)

anddeﬁnethescalarproductandnormby

(

u,v

)

= L

0

uT_v_dx

||

_u

||

2₌

₍

_u_,_u

₎

_. ₍₅₎

Anormestimateisobtainedif(4)ismultipliedbyuandintegrated over[0,L].Weobtain 1 2 d dt

||

u

||

2 ₌₋1 2

(

u,

(

Au

)

x

)

− 1 2

(

u,Aux

)

+ 1 2

(

Axu,u

)

=−1 2u T_A_u

|

L 0+ 1 2

(

Axu,u

)

, (6)

wherethesecond equalityis obtainedfrompartial integrationof (u, (Au)x), and from the symmetry of A which allows it to be

movedbetweentheargumentsofthescalarproduct.Ifthe bound-arydataaresuchthatuT_A_u

_|

L

0≥ 0,thentheestimate

1 2 d dt

||

u

||

2_≤1 2

(

Axu,u

)

(7)

holds,whichundertheassumptionthatmaxx|Ax|isboundedleads

toastabilityestimatebyuseofGronwall’slemma.

Letxj= j

x, j=0,...,N be a grid withspacing

x, and let

uj(t) denote a numerical approximation of u(xj, t). Consider the

semi-discreteapproximationof(4) d dtuj+ 1 2D

(

A

(

xj

)

uj

)

+ 1 2A

(

xj

)

Duj− 1 2A

(

xj

)

xuj=0, (8)

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Fig. 8. 3D compressible Euler equations. Taylor-Green vortex test case. Total kinetic energy (Ekin) vs. time (top) and enstrophy vs. time (bottom) for six different methods.

where D is a centered ﬁnite difference operator approximating

d/dx.Note that A(x) is agiven function,so that theexact deriva-tive Ax canbe used in(8). Thediscrete scalar productand norm

aredeﬁnedby

(

u,v

)

h= N j=0

ω

juTjvj

x

||

u

||

h=

(

u,u

)

h,

where

ω

j>0 are weights that are equal to one at most grid

points,butaregivenspecialvaluesneartheboundaries j=0and

j=N. The boundary modiﬁed norm weights, together with spe-cialboundarymodiﬁcationsofD,leadtothe summation-by-parts property,

(

u,Dv

)

h=−

(

Du,v

)

h+uTNvN− uT0v0,

see [39] for details. Thanks to the summation-by-parts property, the same techniquethat led to the estimate (7) can be used to obtainthesemidiscreteestimate

1 2 d dt

||

u

||

2 h≤ 1 2

(

Axu,u

)

h. (9)

Thepossiblegrowthrateisdetermined byAx inboth (7)and(9),

sothatthediscreteestimatewillhavethesamegrowthrateasthe estimateofthecontinuousproblem.

Fig. 9. 3D DNS of the Taylor-Green vortex test case. Total kinetic energy vs. time (top) and enstrophy vs. time (bottom) for six different methods.

Ducrosetal.typeconservativesplitting:Fornonlinearsystems, suchastheEulerequationsofgasdynamics,splitapproximations havebeenusedforalongtime see,e.g.,Ducrosetal.andBlaisdel etal[2,8]..

The splitapproximationsmakes useof(2)to rewritedifferent termsintheEulerequationsassumsofthreeterms.Thetermsof thesplitform(2)areapproximatedby

1 2D

(

ab

)

+ 1 2D

(

a

)

b+ 1 2aD

(

b

)

, (10)

whereD isaﬁnitedifference operator,anda andb are functions ofx.

As shown in Ducros etal. [8], the approximation (10) can be writteninconservationform.Forexample,withthe secondorder operatorDuj=

(

uj+1− uj−1

)

/

(

2

x

)

,itholdsthat

1 2D

(

ab

)

+ 1 2D

(

a

)

b + 1 2aD

(

b

)

= 1 4

x

+[

(

aj+aj−1

)(

bj+bj−1

)

], (11) where

₊qj=

(

qj+1− qj

)

.

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Eq.(11)canbegeneralizedtostandardcentereddifference op-eratorsof2pthorderofaccuracy,

Dpuj= 1

x p k=1

α

(p) k

(

uj+k− uj−k

)

. (12)

Thecoeﬃcients

α

_k(p)satisfy

k=1 p k

α

(_kp)=1 2 p k=1

α

(p) k k 2n+1₌₀_,_n₌₁_,_._._._,_p_{− 1}_. ₍₁₃₎

Toderive theconservative formofthesplit approximation for anarbitraryoperator,therighthandsideofthealgebraicidentity

aj+kbj+k− aj−kbj−k+

(

aj+k− aj−k

)

bj+aj

(

bj+k− bj−k

)

=

(

aj+k+aj

)(

bj+k+bj

)

−

(

aj+aj−k

)(

bj+bj−k

)

(14)

iswritteninconservativeformby

(

aj+k+aj

)(

bj+k+bj

)

−

(

aj+aj−k

)(

bj+bj−k

)

= k−1 m=0

(

aj−m+aj+k−m

)(

bj−m+bj+k−m

)

− k−1 m=0

(

aj−1−m+aj−1+k−m

)(

bj−1−m+bj−1+k−m

)

. (15)

Theconservativeformofthesplitapproximationbecomes

1 2Dp

(

ab

)

+ 1 2Dp

(

a

)

b+ 1 2aDp

(

b

)

=

1 x p k=1 1 2

α

(p) k

(

aj+kbj+k− aj−kbj−k

)

+aj

(

bj+k− bj−k

)

+

(

aj+k− aj−k

)

bj

=

1 x p k=1

α

(p) k 2

k−1 m=0

(

aj−m+aj+k−m

)(

bj−m+bj+k−m

)

− k−1 m=0

(

aj−1−m+aj−1+k−m

)(

bj−1−m+bj−1+k−m

)

=

1 x

(

hj+1/2− hj−1/2

)

, (16)

wherethenumericalﬂuxisdeﬁnedby

hj+1/2= p k=1 1 2

α

(p) k k−1 m=0

(

aj−m+aj+k−m

)(

bj−m+bj+k−m

)

. (17)

Tosimplifytheformulasoftheconservativeformofsplit approxi-mationsforsystemsofequations,deﬁne

(p) j+1/2

(

a,b

)

= p k=1 1 2

α

(p) k k−1 m=0

(

aj−m+aj+k−m

)(

bj−m+bj+k−m

)

. (18)

ForthethreedimensionalEulerequationsofgasdynamics,the

x-directioninviscidﬂuxis

f=

(

ρ

u,

ρ

u2+p,

ρ

u

v

,

ρ

uw,

(

e+p

)

u

)

T, (19)

where(u,v,w)denotesthevelocitiesinthex-,y-,andz-directions respectively,

ρ

denotesthedensity,pisthepressure,andeisthe

totalenergy. Let

ρ

j,uj,vj,wj,ej, andpj denote thevalues ofthe

discretizedvariablesatgrid pointxj.The ﬂuxcomponentscanbe

writtenasproductsoftwofactorsinmanydifferentways,leading to different split approximations. One Ducros etal.split-type ap-proximationofthegasdynamicsﬂuxderivative thatwillbe used inthisstudyisgivenby

fx

|

x=xj≈

⎛

⎜

⎝

1 2D

ρ

juj+12

ρ

jDuj+12ujD

ρ

j 1 2D

ρ

ju2j+12

ρ

jujDuj+12ujD

ρ

juj+Dpj 1 2D

ρ

juj

v

j+12

ρ

j

v

jDuj+12ujD

ρ

j

v

j 1 2D

ρ

jujwj+12

ρ

jwjDuj+12ujD

ρ

jwj 1 2Duj

(

ej+pj

)

+21ujD

(

ej+pj

)

+ 1 2

(

ej+pj

)

Duj

⎞

⎟

⎠

, (20)

whichby(17)canbewritteninconservativeformwithnumerical ﬂuxfunction hj+1/2= 1 2 p k=1

α

(p) k k−1 m=1

⎛

⎜

⎝

(

ρ

j−m+

ρ

j+k−m

)(

uj−m+uj+k−m

)

(

ρ

j−muj−m+

ρ

j+k−muj+k−m

)(

uj−m+uj+k−m

)

+pj−m+pj+k−m

(

ρ

j−m

v

j−m+

ρ

j+k−m

v

j+k−m

)(

uj−m+uj+k−m

)

(

ρ

j−mwj−m+

ρ

j+k−mwj+k−m

)(

uj−m+uj+k−m

)

(

ej−m+pj−m+ej+k−m+pj+k−m

)(

uj−m+uj+k−m

)

⎞

⎟

⎠

. (21)

The more compact notationintroduced in (18) allows (21) to be rewrittenas hj+1/2=

⎛

⎜

⎝

(p) j+1/2

(

ρ

,u

)

(p) j+1/2

(

ρ

u,u

)

+

(p) j+1/2

(

p,1

)

(p) j+1/2

(

ρv

,u

)

(p) j+1/2

(

ρ

w,u

)

(p) j+1/2

(

e+p,u

)

⎞

⎟

⎠

. (22)

Anatural nonconservative splitting(not inthe Ducrosetal.

typecategory):

Thehomogeneitypropertyoftheinviscidﬂuxofperfectgas dy-namics implies that f

(

u

)

=A

(

u

)

u, where A(u) is the Jacobian of

f(u).Tomakeuseofthehomogeneityproperty,anon-conservative naturalsplittingis 1 2fx+ 1 2Aux+ 1 2Axu, (23)

wherethediscretizationis

d dtuj+ 1 2Dpfj+ 1 2AjDpuj+ 1 2Dp

(

Aj

)

uj=0. (24)

Here Ax and DpA denote element-wise applicationof

differentia-tionanddifferencing respectively.The approximation (24)canbe rewritteninconservativeformwithnumericalﬂux

hj+1/2= 5 m=1

⎛

⎜

⎝

(p) j+1/2

(

A1,m,um

)

(p) j+1/2

(

A2,m,um

)

(p) j+1/2

(

A3,m,um

)

(p) j+1/2

(

A4,m,um

)

(p) j+1/2

(

A5,m,um

)

⎞

⎟

⎠

,

whereAk,m denoteselement(k,m)ofthematrix-valued function

A(x),andumdenotesthemthcomponentofthevectoru.

A semi-conservative entropy splitting of the Euler ﬂux

derivatives:

AnothersplittingthatgivesentropystabilityoftheEuler equa-tionsofgasdynamicsisbyGerritsenandOlsson[15],Olssonand Oliger[28],Yeeetal.[53].TheymadeuseofHarten’s symmetriz-ableformoftheEulerequationsintermsoftheentropyvariables

[13]toobtainasemi-discretesplittingoftheEulerequationswith a discrete entropy stabilityby the summation-by-parts approach. Duringthecomputations,theentropysplittingiswritteninterms

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Fig. 10. 3D Isotropic turbulence test case. Energy spectra at the ﬁnal time by six schemes using 64 3 grid points. DNS using 256 3 grid points also shown for compar-

ison.

ofthesumofaconservativeportionfortheinteriorscheme (inte-riorgridpoints)andwithasummation-by-partsfortheboundary scheme(boundarypoints).NotethattheHarten[13]andGerritsen &Olssonentropy splittingformselectsthe un-physicalbranch of theinequalityandwaslater correctedbyYeeetal.[53],hereafter referredtoastheentropysplittingoftheEulerequations.Itis con-sideredtobeasemi-conservativesplittingexceptattheboundary gridpoints.TheentropysplittingofOlsson&Oliger,Gerritsenand Olsson,andYee etal.[15,28,53] is a splitting which isof a form thatismoresuitable forthediscretestableenergynormestimate technique,includingboundaryschemeestimateforarbitraryorder ofcentralspatialschemes.See Yeeetal.[54]fortheformulation. Forthe1D Eulerequationstheinviscid ﬂux derivative f(u)x fora

perfectgas issplitintothefollowing viatheentropyvariables W

discussedinHarten[13]. fx=

β

+1fx+ 1

β

+1fWWx,

β

=−1 (25) W =[w1, w2, w3, w3, w5]T = p∗ p

e+

α

_γ

− 1 − 1p, −

ρ

u,−

ρv

,

ρ

w,

ρ

T , (26) where p∗=−

(

p

ρ

−γ

)

α+1γ ₍₂₇₎ and

β

=

α

+

γ

1−

γ

,

α

>0or

α

<−

γ

. (28)

See Yee etal.[32,53,54]for the formulation,the choice for

α

, andnumericalexamples.

Several splitdiscretizations were compared in[14]where dis-cretizationby theentropysplittingformwasshownby numerical experimentsto be one of the best performing for smooth ﬂows. Fortheirskew-symmetricsplittingextensiontotheidealMHD,see Sjögreenetal.[35,36].

3. Generalizationofskew-symmetricsplittingtotheidealMHD

Due to the incomplete hyperbolic nature of the conservative ideal MHD governing equations, not all of the skew-symmetric splittings for gas dynamics can be extended to the ideal MHD. SeeYeeetal.[53]fora discussion.FortheMHDthe Ducrosetal.

Table 1 Coeﬃcients of DRP4S7, optimized over [0, 1.1]. k ak 1 0 .77088238051822552 2 −0.16670590441458047 3 0 .02084314277031176

[8] variants were constructed. In addition, four formulations of theMHDwereconsidered:(a)theconservativeMHD, (b)the Go-dunov/Powell non-conservativeform, (c)the JanhunenMHDwith magneticﬁeldsourceterms[18],and(d)aMHDwithsourceterms of [4]. The different formulation of the MHD equations in con-junction with the variants of Ducros etal. type skew-symmetric splitting have a strong effect on the stability of non-dissipative approximations. For their skew-symmetric splitting extension to theidealMHD,seeYeeetal.,SjögreenandYee andSjögreenetal.

[35,36,53] forthe formulation. Representative test cases for both smoothﬂowsandproblemscontainingdiscontinuitiesfortheideal MHD can be found in [35,36,53]. Their results illustrate the im-proved stabilityby usingthe skew-symmetricsplitting aspartof the central base scheme instead of the pure high order central scheme.

4. DRPschemes

Sinceourobjectiveistoutilizewavenumberoptimizedschemes forgeneralDNSand LESapplications,no attemptis madeto ob-tainoptimizedschemesforspecificIBVPswithspecificinitialdata and boundary data. In this study three different optimized fi-nite difference operators are considered. See Tam [44] and De Roecketal.[31]forthedevelopmentandreferencescitedtherein. Theseare:(a)DRP4S7,theoriginalTam&Webbfourth-order accu-rateDRPoperatorwithseven-pointwide gridstencil,(b)DRP4S9, thefourth-orderaccurate DRPoperatorwithnine-pointwidegrid stencil,and(c)STO9,thefourth-orderaccurateoperatorwith nine-point widestencilby Bogey& Bailly[3].Allthree operatorshave antisymmetriccoefficientsandareoptimizedoverwavenumber in-tervals0_{≤ k}

x_{≤ 1.1}forDRP4S7and

π

/16_{≤ k}

x_≤

π

/2forDRP4S9 and STO9. Here

x is the grid spacing and the integer k is the modenumber.DRP4S7andSTO9werestudiedin[31].

Remark:NumericalexperimentsmadewithDRP4S7optimized

over

π

/16≤ k

x≤

π

/2gaveworseaccuracythanwithDRP4S7 op-timizedoverthemorestandardchoice0≤ k

x≤ 1.1usedhere.Itis reasonabletoexpect thatwithfewerfree parameters,theinterval ofoptimizationshouldbemadeshorter.

DRP4S7andDRP4S9useleastsquareminimizationofthe abso-lute error,i.e., integral ofthe square oftheerrorin wavenumber space.The STO9schemeusesL1 optimizationoftherelativeerror

inwavenumber space,i.e., integral overtheabsolutevalue ofthe errordividedbyk

x,sincek

xistheexactwavenumber.

Theirdifference operators D forthe ﬁrst-order derivative of a gridfunctionujareoftheform

Duj= 1

x q k=1 ak

(

uj+k− uj−k

)

. (29)

Table1 gives the coefficientsof the DRP4S7 scheme, Table2 lists thecoefficientsoftheDRP4S9scheme,andTable3showsthe co-efficientsoftheSTO9scheme.TheSTO9coefficientswereobtained from[31], wherethey are givento 12 decimals.In thiswork we extended the number of decimals by enforcing the fourth order accuracyconstrainttohighprecision.

Note that the centered operators (29) are of the same asym-metricformas(12).Thismeansthat theDucrosetal.splitting

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de-H.C. Yee, B. Sjögreen / Computers and Fluids 0 0 0 (2017) 1–18 11

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Fig. 11. 3D Isotropic turbulence test case. Evolution of kinetic energy (upper left), enstrophy (upper right), temperature variance (lower left), and dilatation (lower right), computed by six schemes, using 64 3 grid points. DNS using 256 3 grid points is also shown for comparison.

Fig. 12. 3D Isotropic turbulence test case: Energy spectra at the final time for en- tropy conserving base scheme (C08EC+WENO7fi, blue), Ducros split base scheme (C08DS+WENO7fi, red) and WENO7fi (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 Coefficients of DRP4S9, optimized over [ π/16, π/2]. k ak 1 0 .846863763009931 2 −0.251240526 84 9904 3 0 .063181723773749 4 −0.008481970157843

scribedinSection2isalsostraightforwardlyapplicabletothe opti-mizedoperatorsdescribedinthissection.TheseDRPformulations areapplicabletotheidealMHDequations.

Table 3

Coeﬃcients of STO9, optimized over [ π/16, π/2], from [31] . k ak 1 0 .841570216389881 2 −0.244678789340406 3 0 .059463699920073 4 −0.007650934367322

5. Highorderentropyconservativenumericalﬂuxes

Weconsiderthesystemofconservationlaws,

ut+f

(

u

)

x=0, −∞<x<∞t>0 (30)

wheretheunknownu=u

(

x,t

)

isgivenatt=0.Entropy conserv-ingschemesfor(30)were introducedinthe1980s.See,e.g.,[43]. Theseschemesareinconservationform,andadmitadiscrete con-servation law for the entropy. An entropy, E(u), and an entropy ﬂux,F(u),aretwofunctionssatisfying

ETuA

(

u

)

=FuT.

Here,Eu denotesthe gradientofE withrespectto u,andA(u) is

theJacobianoftheﬂuxfunctionf(u).Furthermore,E(u)isassumed tobeaconvexfunction.Theentropyvariablesare deﬁnedby v₌ Eu

(

u

)

.Multiplying(30)byvTandusing

vT_u

t+vTAux=E

(

u

)

t+FuTux=E

(

u

)

t+F

(

u

)

x

givestheadditionalconservationlawfortheentropy,

E

(

u

)

t+F

(

u

)

x=0.

Theentropyﬂuxpotential,deﬁnedby

ψ

=vT_f_{− F}_,

hasthepropertythatf=

ψ

v.

Ifthenumericalﬂuxfunctionh_j₊₁_/₂₌h

(

u_j₊₁_,u_j

)

satisﬁes

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Fig. 13. 3D Isotropic turbulence test case: Evolution of kinetic energy (upper left), enstrophy (upper right), temperature variance (lower left), and dilatation (lower right), computed by entropy conserving base scheme (C08EC+WENO7fi, blue), Ducros split base scheme (C08DS+WENO7fi, red) and WENO7fi (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

thenthesemi-discreteapproximation

duj

dt +

1

x

(

hj+1/2− hj−1/2

)

=0

isentropyconserving,see[42].Thisresultcanbegeneralized,by deﬁning

h(_jk₊)_k_/₂=h

(

uj+k,uj

)

, k=1,2,...

where h

(

uj+1,uj

)

is an entropy-conserving numerical ﬂux

func-tion.Thedifferencescheme

duj dt + 1

x p k=1

α

k

(

h(jk+)k/2− h (k) j−k/2

)

=0, (32)

is then entropy conserving for arbitrary coeﬃcients

α

_k. It is straightforwardto verifythat (32)can bewritten inconservative form,withnumericalﬂuxfunction

h(_jec₊₁)_/₂= p k=1 2

α

k k m=1 h

(

uj+m,uj+m−k

)

.

Early entropy conserving schemes were second-order accurate. High order entropy-conserving schemes can be constructed by using the scheme (32) with suitable coeﬃcients,

α

_k. The 2p th-orderaccurate standard centered finite difference operatoris de-fined by (12) with coefficients

α

_k(p) satisfying (13). Let hj+1/2=

h

(

u_j₊₁_,u_j

)

beasecond-orderaccurateentropy-conserving numer-icalﬂuxfunction.Thedifferencescheme

duj dt + 1

x p k=1 2

α

_k(p)

(

h

(

uj+k,uj

)

− h

(

uj,uj−k

))

, (33)

is then 2pth-order accurate and entropy-conserving, see [34,37]. Furthermore,(33) can be written in conservative form with nu-mericalﬂuxfunction

hecp_j₊₁_/₂= p k=1 2

α

_k(p) k m=1 h

(

uj+m,uj+m−k

)

. (34)

Thescheme(33)isboth2pth-orderaccurateandentropy conserv-ing.

Similarly, anyﬁnitedifference operator ofthe form(29),have an entropy-conservingcounterpart fornonlinear systems of con-servationlaws,approximatingthe ﬂuxderivative f(u)x.For

exam-ple, it is possible to deﬁne entropy conserving DRP schemes, by substituting thecoeﬃcients

α

_k(p) in(34)forthe coeﬃcientsa_k of

(29).

Examplesofentropyconservingnumericalﬂuxes:

Numericalfluxesforthe3DEulerx-directionflux(19).The Eu-ler equations of compressible gas dynamics have several differ-ent entropies. The different entropies lead to different entropy-conserving schemes. Furthermore, even for a fixed entropy, the entropy-conserving numerical flux function is not unique, since theentropy conservationcondition(31) isonlyone constrainton the five(in the caseof 3D Eulerx-direction fluxes(19)) numeri-cal flux components. Eq.(31) can be satisfied by first expressing theentropyfluxpotential,

ψ

intermsofentropyvariables,v,and secondly rewrite thedifference

ψ

j+1−

ψ

j in termsofdifferences

vj+1− vj.This canbecome algebraically involved.To simplifythe

algebra,aparametervector, zcanbeintroduced.Thederivationis thencarriedoutbyexpressingboth

ψ

j+1−

ψ

jandvj+1− vjas

dif-ferencesz_j₊₁_{− z}_j.Foranexample,see[48],wherethederivation isexpressedindetailfortheequationsofMHD.

Thissubsectionwilldenotetheaverageofafunctionqby

{

q

}

=

(

qj+1+qj

)

/2

andthelogarithmicaverageby

qln₌lnqj+1− lnqj

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Thesecond-orderaccuratenumericalﬂuxfunction

h

(

uj+1,uj

)

=

⎛

⎜

⎝

ρ

ln

{

_u

}

ρ

ln

{

_u

}

2₊ {ρ} {ρ/p}

ρ

ln

{

_u

}{

_v

}

ρ

ln

{

_u

}{

_w

}

h5

⎞

⎟

⎠

, (35)

whereh5 isthelongerexpression

h5 =

{

u

}

{

ρ}

{

ρ

/p

}

+ 1

γ

− 1

ρ

ln

(

ρ

/p

)

ln − 1 2

ρ

ln

₍

{

_u2

}

₊

{

_v

2

}

₊

{

_w2

}

₎

+

ρ

ln

₍

_{

_u

_}

2₊

_{

_v

_}

2₊

_{

_w

_}

2

₎

(36)

isentropy-conservingfortheentropy

E=−

_γ

ρ

− 1lnp

ρ

−γ.

Foraderivationof(35),see[38].

Anotherexample,derivedin[34],isthenumericalﬂuxfunction

h

(

uj+1,uj

)

=

⎛

⎜

⎝

{

u

}{

ρ

(

p

ρ

)

−γγ+1

}

_Q

{

u

}{

ρ

u

(

p

ρ

)

−γγ+1

}

_Q+

{

1 ρ

(

p

ρ

)

γ γ+1

}{

(

_p

ρ

)

γ1+1

}

{

u

}{

ρv

(

p

ρ

)

−γγ+1

}

_Q

{

u

}{

ρ

w

(

p

ρ

)

−γγ+1

}

_Q

{

u

}{

e

(

p

ρ

)

−γγ+1

}

_Q+

{

u ρ

(

p

ρ

)

γ γ+1

}{

(

_p

ρ

)

γ1+1

}

⎞

⎟

⎠

, (37) where Q=

(

γ

− 1

)

(

pj+1

ρ

j+1

)

1 γ+1−

(

_p_j

ρ

_j

)

γ1+1

(

pj+1

ρ

j+1

)

1−γ γ+1−

(

_p j

ρ

j

)

1−γ γ+1 .

When

(

p

ρ

)

j+1−

(

p

ρ

)

jis small Qapproaches

(

p

ρ

)

γ

γ+1_._The

numer-icalﬂuxfunction(37)conservestheentropy

E= 1₁+

γ

−

γ

(

ρ

p

)

1 γ+1.

The entropy conservative highorder baseschemes in the nu-mericalexperimentsinthispaperusethenumericalﬂuxfunction

(34)togetherwith(35).

Entropy conservativeschemesare centerednon-dissipative ap-proximations. For flows where shockwaves are present, entropy conservationis unphysicalandentropyconservative schemeswill generatestrongoscillationsarounddiscontinuities.Tobeusefulfor compressible flows, it is necessary to add some shock-capturing dissipation to the entropy conservative approximation. This is sometimesdonebyusinglineardissipation,appliedtotheentropy variables.InthenonlinearfiltermethodbyYeeandSjögreen (de-scribedinthenextsubsection),itisstraightforwardtouseentropy conservingschemes.Allthatisneededistosubstitutethecentered schemeofthebaseschemestepbyanentropy-conservingscheme. Thedescriptionaboveismadeforthecaseofstandardgas dy-namics. Entropyconserving schemesforthe equationsofMHDis asubjectwheretherehasbeenrecentprogress,see,e.g.,[38].The ideas presented here, for example, the generalization of second-order accuracy to higherorder by the numericalflux (34), apply equallywelltotheequationsofMHD.

6. Classicalcentral,entropystableandDRPasbaseSchemes withskew-symmetricsplittingasthepreprocessingstepinthe

frameworkofthenonlinearﬁltermethodofYeeandSjögreen

[50]

Thissection givesa briefoverviewofthehigh-ordernonlinear ﬁlter schemeof Yeeetal.and Yee andSjögreen[49,50,52,53] for

accuratecomputationsofDNSandLESofcompressibleturbulence for a wide range of ﬂow types by introducing as little shock-capturingnumericaldissipationaspossible.Forsimplicity,the dis-cussionusesthe3DinviscidEulerequations.

Preprocessingstep by skew-symmetric splittingfor gas dy-namics:Beforetheapplicationofahigh-ordernon-dissipative spa-tial base scheme, a preprocessing step is employed to improve numericalstability. The inviscid ﬂux derivatives of the governing equationsare split in the following two ways,depending on the ﬂow types and the desire for rigorous mathematical analysis or physicalargument.

• Entropy splitting of [53] or the natural splitting described previously. These are non-conservative splittings andthey are among some of the best in improving numerical stability for non-dissipative central schemes,especially forlong time inte-grationofshock-freeturbulence.Ithasbeenutilizedextensively inDNSofshock-freeturbulence.See[32] andtheir laterwork fortheirwideapplications.

• The Ducros etal. splitting [8] for systems (or variants of the conservativeskew-symmetricsplittingdescribedearlier):These are conservative splittings and are suitable forproblems with discontinuities.

Remark. For problems containing discontinuities, conservative

skew-symmetricsplittingsshouldbeused.

Base scheme stepusing the preprocessing step: A full time step is advanced using a high-order non-dissipative or very low dissipersivespatially centralschemeonthesplitformofthe gov-erningpartial differential equations(PDEs) (i.e., afterthe prepro-cessing step). Forthe current study,fourth-order to eighth-order classicalcentralschemes,andthethreeDRP4S7,DRP4S9andSTO9 DRPschemesareconsideredasbaseschemes.

Thefulltimestepofhigh-ordertemporaldiscretizationsuchas thefourth-orderRunge–Kutta(RK4)methodisused.Itisremarked that other DRPtemporaldiscretizations can be usedforthe base schemestep.SeeTam[44,45],Brambley[6],andHarasandTaasan

[11].

Baseschemestepusingthehighorderentropyconservative numerical ﬂuxesThe preprocessingstep isleft out if thespatial discretizationofthebaseschemeismadebyanentropy conserv-ingmethod.Inthiscase,againthebaseschemestepadvancesthe non-dissipativediscretizationonefulltimestepbyanexplicittime integrator.

Post-processing(Nonlinearfilterstep):Tofurtherimprovethe accuracyofthecomputedsolutionfromthebaseschemestep, af-ter a full time step of a non-dissipative high-order spatial base schemeon the splitform ofthe governingequation(s), the post-processingstepisusedtononlinearlyfilterthesolutionbya dissi-pativeportionofahigh-ordershock-capturingschemewithalocal flow sensor.Comparable order ofaccuracy of thenonlinear filter dissipation withthe basescheme usually isconsidered. For non-entropysatisfyingshock-capturingschemesitisassumedthat en-tropysatisfyingfixes forboth1D andmulti-Dare employed[55]. For extreme flows positivity-preserving shock-capturing schemes should be used.See Kotov etal.[19,20] for some performance of positivity-preservingnonlinearfilterschemes.

The flow sensor provides locations and amounts of built-in shock-capturingdissipation that can befurther reducedor elimi-nated. Ateach gridpoint a local flow sensor isemployed to an-alyze the regularity of the computed flow data.Only the strong discontinuity locations would receive the full amount of shock-capturingdissipation.Insmoothregions noshock-capturing dissi-pationwouldbeaddedunlesshighfrequencyoscillationsdevelop, owningtothepossibilityofnumericalinstabilityinlongtime inte-grationsofnonlineargoverningPDEs.Inregionswithstrong

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PRESS

lence,ifneeded,asmallfractionoftheshock-capturingdissipation wouldbeaddedtoimprovestability.

Note that the filter numerical fluxes only involve the invis-cid flux derivatives, regardless if the flow is viscous or inviscid. Ifviscous termsarepresent,a matchinghighordercentral differ-enceoperator(as theinviscid difference operator)is includedon thebasescheme step.Forease ofsummation-by-parts numerical boundaryclosureimplementationfortheviscous flux derivatives, thesameinviscidcentraldifferenceoperatorforthefirstderivative isemployedtwicefortheviscousfluxderivatives.

Remark. Forthegasdynamicsthepost-processing(nonlinearfilter step)isemployed foralloftheequation setforbothnon-reacting andreactingflows. For theMHD on a uniformCartesian grid, in ordertoobtainzerodiscretedivBerrorwithoutanydivBcleaning, the nonlinear filterstep is not employed forthe three magnetic fieldequations.SeeYeeandSjögreen[52]fordetails.

Forsimplicityofpresentation,considerthe3D Eulerequations

∂

U

∂

t +

∂

E

∂

x+

∂

F

∂

y+

∂

G

∂

z =0, (38)

whereE, Fand Gare inviscid ﬂuxesin the x,y andz directions, respectively.

LetU∗bethesolutionafterthecompletionofthefulltimestep ofthebaseschemestep.Theﬁnalupdateofthesolutionafterthe ﬁlterstepis Un_j,k,l+1=U∗_j_,k,l−

t x[H ∗(x) j+1/2,k,l− H∗j−1(x)/2,k,l] −

t

y[H ∗(y) j,k+1/2,l−H∗( y) j,k−1/2,l]−

t

z[H ∗(z) j,k,l+1/2−H∗( z) j,k,l−1/2],(39)

H_j∗₊₁(x)_/₂_,k,l and H_j∗₋₁(x)_/₂_,k,l are “ﬁlter” numerical ﬂuxes in the x -directionin terms of Roe’s average states based on U∗. Similarly

H∗_j_,k(y₊₁)_/₂_,l and H∗_j_,k,l+1(z) _/₂ are numerical ﬁlter ﬂuxes in the y- and

z-directions respectively. From here on, the simpliﬁed notation

H∗_j₊₁_/₂ willbe usedfor thex-direction ﬁlterﬂux H∗_j(x)

+1/2,k,l,and the

gridpoint indicesk, lwillbesuppressedonallquantitiesdeﬁned below.Thediscussionwillfocusonthex-directionﬂux,they-and

z-directionfluxesaredefinedsimilarly.Thefilterfluxisdefinedin characteristicscomponentsby

H∗_j₊₁_/₂=Rj+1/2Hj+1/2, (40)

where Rj+1/2,k,l is the matrix of right eigenvectors of the

Jaco-bian of the inviscid flux vector in terms of Roe’s average states basedonU∗.Denotetheelementsofthefilternumericalflux vec-torHj+1/2,k,l byh

l

j+1/2,l=1,2,...,5,whereh

l

j+1/2hastheform

hl_j₊₁_/₂=

κ

l j+1/2 2 w l j+1/2

φ

l j+1/2. (41) Here wl

j+1/2 isa ﬂow sensorto activatethe nonlinear

numer-ical dissipation portion of a high order shock-capturing scheme

1

2

φ

lj+1/2,and

κ

l

j+1/2 isapositiveﬂowdependentparameterthatis

lessthanorequaltoonetocontroltheamountofshock-capturing dissipationtobeused.Thenonlineardissipativeportionofa high-resolution shock-capturing scheme “1

2

φ

lj+1/2” can be any

shock-capturingscheme.The choiceofthe parameter

κ

l

j+1/2 canbe

dif-ferentfordifferentﬂowtypesandisautomaticallychosenbyusing thelocal

κ

l

j+1/2 describedin[50].Theﬂowsensorwlj+1/2 canbea

varietyofformulaeintroducedintheliteratureorcanbeswitched fromoneflowsensortoanother,dependingonthecomputedflow data at that particular location. For a variety of local flow sen-sors with automatic selection of the proper parameter, depend-ingondifferentflowtype,see[50].TheformofTauber–Sandham

[47] forthe filternumerical flux uses the Ducros etal.flow sen-sor[9]as

κ

l

j+1/2andtheHartenartiﬁcialcompressionmethod

for-mula(ACM)[12]astheflowsensorindicatedin[54]andsimilarly in[26]ispartoftheYeeandSjögreenadaptivenumerical dissipa-tioncontrolgeneralizationfilterformulae.Forthenumerical exper-imentspresented,wemainlyconcentrateonthewaveletflow sen-sorofYee andSjögreen,the Ducrosetal.flowsensor[9]andthe artificialcompressionmethodflowsensorof[54].Forthewavelets and ACM flow sensors, see the aforementioned references cited. TheDucrosetal.flowsensorwasdesignedmainlytocaptureflows containing shocks andvorticity withthe divcurl tolerance of the form:

sw=

₍

(

∇

· u

)

2

∇

· u

)

2₊

ω

2₊

ε

. (42)

Hereuisthevelocityvector,

ω

isthevorticitymagnitudeand

ε

is a smallnumbertoavoiddivision by zero(e.g.,10−6).The Ducros etal. ﬂow sensor consists of a cut off parameter

δ

as an input parameter based on the value of swthat can be used to switch onoroff thedissipativeportionofthehighordershock-capturing scheme.If

δ

issettobeone,thedissipationonlyswitchesonwhen itencountersashockwave.Foralowervalue ofthecut off

δ

pa-rameter,vorticitycanbedetected.The

δ

parameterisusedasthe

κ

l

j+1/2fortheDurcrosetal.ﬂowsensor.

Thelow Machnumber

κ

curvewasdevelopedinYeeand Sjö-green [50] and detail is omitted here. Local ﬂow sensors for a wide spectrum of ﬂow speed and shock strength developed in

[21,22,50]arealsoomittedhere.

Theaforementionedhighordernonlinearfiltermethodisvalid for the four forms of the MHD formulation and the four skew-symmetricsplittings oftheMHDtobe usedasthepreprocessing step. In addition, the aforementioned high order nonlinear filter methodisvalidforthefourformsoftheMHDformulationandthe differenthighorderentropyconservativenumericalfluxessuchas thespatialbaseschemesdiscussedinSections4and5ofSjögreen andYee[37,38].

From here on, without loss of generality, the term “a split scheme” refers to the use of a high order central scheme to discretize a skew-symmetric splitting form of the inviscid flux derivatives. If the three considered DRP4S7, DRP4S9 and STO9 schemes are used as the base schemes, and the dissipative por-tionoftheseventh-orderWENO(WENO7)isusedasthenonlinear filter, they are denoted by DRP4S7+WENO7fi, DRP4S9+WENO7fi, and STO9+WENO7fi respectively. Similarly if WENO5fi is used, they are denoted by DRP4S7+WENO5fi, DRP4S9+WENO5fi, and STO9+WENO5fi.Ifaneighth-orderclassicalcentraldifference oper-atorisusedasthebaseschemefortheaforementionedthreeDRP schemes,itisdenotedbyC08+WENO7fi.IfDucrosetal.splittingis used,e.g.,itisdenotedbyC08-DS+WENO7fi.

Notethatanygoodhigh-resolution highordershock-capturing methods are suitable as the dissipative portion of the nonlin-ear ﬁlter approach. Here standard Jiang and Shu [17] WENO5 andWENO7 arechosenforthenumericalexperiments.Optimized WENOschemesarenotasrobustforournonlinearﬁlterapproach.

7. Numericalresults

Thissectionshowssomenumericalresultsforcompressiblegas dynamics.Extensive gridrefinement andscheme comparison, in-cluding 3D forced turbulence, LES and MHD simulations can be found by the authors and collaboratorsin [21,22,35–38] and ref-erencescitedtherein.Thetestcasesshownhereincludeproblems with smooth flows, problems containing shockwaves, shock-free turbulenceandturbulencewithweakshocks.Thesetestcasesare well knowntest cases intheliterature andwill be usedto illus-tratetheperformance oftheproposedmethods.Thefirsttwotest