第四章 跨音速定常小擾動勢流混合差分方法及隱式近似因式分

第四章跨音速定常小擾動勢流混合差分方法及隱式近似因式分解法

chapter4TheMixedFiniteDifferenceMethod(FDM)forVelocityPotentialFunctionofSteadySmallPerturbationandImplicitApproximateFactorDecompositionMethods主要內(nèi)容：maincontents混合差分解法MixedPDMethod小擾動方程及小擾動激波差分式

Smallperturbationequationandsmallperturbationrelationshipforshockflow小擾動速勢差分方程

Thefinitedifferentialequationofsmallperturbationpotentialfunction邊界條件及邊界條件的嵌入Theinitialconditionandboundarycondition線松弛迭代解法Linearrelaxationiterationmethod升力翼型的跨音速小擾動勢流差分方法FDmethodofvelocitypotentialfunctionforsmallperturbation隱式近似因子分解法ApproximatefactordecompositionmethodAF1方法AF1methodAF2方法AF2method

方法比較Comparisonofthemethod重點：Focus混合差分方法MixedFDMethod

難點：Difficulty隱式近似因子分解法ImplicitApproximatefactorydecomposition第四章跨音速定常小擾動勢流混合差分方法及隱式近似因式分解法

chapter4TheMixedFiniteDifferenceMethodforVelocityPotentialFunctionofSteadySmallPerturbationandImplicitApproximateFactorDecompositionMethods跨音速流：局部超音區(qū)與亞音速同時存在的流場Transonicflow：Localsupersonicflowandsupersonicflowexistsmeantime偏微分方程：混合型方程ThePDE：Mixedtypeequation混合差分方法：用不同的差分方程求解跨聲速流場

MixedFinitedifferencemethodistosolvetransonicflowwithdifferentFDMs混合型方程及流場：采用迭代方法求解，求解之前不知道方程的類型MixedEquationandflowfield,theiterativemethodisusedbecausethetypeoftheequationisunknownbeforeitwassolved小擾動方程：小馬赫（0.6~1.4）流過薄而微變的葉片（機翼或葉柵）時全速勢方程可簡化為小擾動方程

Smallperturbationequation(SPE):whenmachnumberissmall(ie0.6~1.4)thefullvelocitypotentialequationcanbesimplifiedtoSPE混合差分:用混合差分格式求解小擾動方程MixedFDM:TosolveequationusingMFDM混合差分和松弛迭代法求解全速勢方程MixedFDMandRelaxationiteration:Tosolvefullvelocitypotentialequation.優(yōu)缺點：Advantage/disadvantage

跨音速松弛法---速度快，有效Transonicrelaxationmethodfasterefficient

時間推進法：適用范圍廣Timematchingmethods,widelyusage

近似因子分解法：快速Approximatefactordecomposition：faster

多層網(wǎng)格法：收斂性好Multi-gridtechnique：goodconvergence4.1跨聲速小擾動速度勢方程Equationoftransonicsmallperturbationvelocitypotentialfunction跨聲速氣流繞過薄翼的情況Forthecaseoftransonicflowpassathinairfoil二維平面速勢方程2Dvelocitypotentialequation氣流繞過薄翼適用范圍：亞、跨、超音速無旋流動Suitable

case：subsonic，transonic，supersonic

irr-rotational

flow.將流動分解為兩部分：未經(jīng)擾動的流動、擾動流動To

decomposetheflowintounperturbedflowand

perturb

flow未經(jīng)擾動的流動就是無窮遠(yuǎn)前方來流Flow

atunperturbedfieldsisfarfieldflow擾動運動速度勢可以用表示。速度可以用表示Potentialfunctionofperturbationflowis，perturbationvelocitycomponents兩部分的合速度勢Thetotalvelocitypotentialfunction代入速勢方程可得小擾動速度應(yīng)滿足的方程Substitutetheequationandthenthesmallperturbationeq.求得速度場之后，可以得到壓強及壓強系數(shù)為Thepressureandpressurecoefficientcanbeobtainedfromthefollowingequations.再用等熵流動的關(guān)系式可得到其他參數(shù)

Thenintroduce

the

isentropy

relationtogetotherparameters比熱比絕熱指數(shù)小擾動條件下，擾動速度遠(yuǎn)小于自由來流速度

on

smallperturbationcondition,theperturbationvelocityless

than

free

stream補充條件：

Supplement

conditions來流不能接近音速incoming

flow

velocitydoes

not

approach

sonic

來流非高超聲速incoming

flow

velocitydoes

not

approach

hypersonic為進一步簡化擾動方程，忽略擾動速度一次項，可得到下列關(guān)系：Simplified

equation最后得到：

Final

equation應(yīng)用范圍:亞、超聲速

Suitable

for

subsonic

and

supersonic不適用于跨聲速區(qū)域:對于跨聲速≈1，必須取消補充假設(shè)條件，即取消來流不能接近音速的假設(shè)，這時速勢方程首項的系數(shù)一次項不能忽略

For

transonic

flow

field

(M≈1)，the

supplement

condition，the

first

item

of

the

potential

function

equation

can

not

be

neglected.

跨聲速小擾動方程應(yīng)為：Thesmall

perturbation

equation

of

velocity

function可以證明：當(dāng)M∞→1時，

It’sproved,whenM∞→1,因此跨聲速條件下，小擾動方程可以寫成Sothatthesmallperturbationequationattransonicflowcanbewrittenas此方程的類型取決于：Typeoftheequationdependson=B2-4AC=4(M2-1)當(dāng)M<1時,<0,不存在實特征根，沒有特征線，為橢圓型WhenM<1,norealeigenvalueexists,thatisnocharacterline,theequ.iselliptic.當(dāng)M>1時,>0,存在兩個特征根，有兩條特征線，為雙曲型WhenM>1,therearetwoeigenvalue,twocharacterlines,theequ.ishyperboliceq.當(dāng)M=1時,=0,存在一個特征根，有一條特征線，為拋物型WhenM=1,thereisoneeigenvalue,onecharacteristicline,theequ.isparabolic

特征線(當(dāng)M>1時)：斜率

Theslopeofcharacteristicline特征線與x軸夾角為局部馬赫角，對稱于x軸。LocalMachangleistheanglebetweenvelocityvectorandthecharacteristiclinexyoqr’pq’r影響區(qū)依賴區(qū)是馬赫角issocallMachangle

影響區(qū)：P點下游由兩條特征線所夾的區(qū)域Influencezone:

upwindzonebetweencharacteristiclines依賴區(qū)：P點上游由兩條特征線所夾的區(qū)域Dependzonedownstreamzonebetweenthecharacteristiclines擾動下的壓強系數(shù)公式

Thepressurecoefficientonsmallperturbationcondition§4-2小擾動激波關(guān)系式Theshockrelationsofsmallperturbation.

等熵激波小擾動激波的熵增是三階小量Forsmallperturbationshock,entropyincreaseisthirdorder，soitisisentropyshock。

激波的精確速度關(guān)系式：Accuratevelocityrelationofshock激波前后的速度關(guān)系式（幾何關(guān)系）Velocityrelationsinfront/rear-shock即

對于直角坐標(biāo)系A(chǔ)tCartesiancoordinates

因此sothat由能量方程可得Fromenergyequation由此得到M∞→1時的方程（跨聲速中）Fromwhere,theequationwhenM∞→1,(transonicflow)超聲速中Atsupersonicflow適用范圍：激波前后小擾動方程，適用于等熵波

Aboveeqs.areavailableforsmallperturbationflowinfront/behindoftheshock,i.e.,iso-entropyflow§4-3跨聲速小擾動速勢差分方程

Smallperturbationequationfortransonicflow

混合性方程，在同一流場中不同點所用的差分方程不同。Mixedequation,differentFDEisusedforthescheme一、中心差分格式

CenteralFDEschemeflowfield

對速度勢Forvelocitypotentialfunction一階導(dǎo)數(shù)的差分格式Firstorderdifferenceequationisobtainedas二階導(dǎo)數(shù)的差分格式Plustwoequations,andget2edorderPD二階精度

2ndorder在超音速流中，氣流參數(shù)只受上擾動游影響與下游擾動無關(guān)。Atsupersonicflow,theparametersofflowaredependentonupwindperturbationandindependentondownflowperturbation需建立迎風(fēng)一側(cè)差分格式TheupwindonesideFDschemeisneededtobuilt

取上游一側(cè)的點構(gòu)成差分格式TaketheupwindpointtoconstructFDscheme一階精度迎風(fēng)格式1storderupwindscheme二階精度迎風(fēng)格式2ndorderupwindscheme二、一側(cè)差分格式Oneside

FDEofthederivatives三、亞音速點的差分方程Atsubsonicflowequation取網(wǎng)格點如圖：正交等間距網(wǎng)格Thespacenodesareshownas中心差分格式構(gòu)成的差分方程即受周圍四點的影響，這是亞聲速流動的特點iseffectbyaroundfourpoints,thisissubsonicfeature

四、超聲速點的差分方程FDEforsupersonicflow當(dāng)計算點為超音速（M大于1）時，方程為雙曲線型Whenlocalsupersonicflowappear,theequationis

hyperbolic存在依賴區(qū)（上游馬赫錐內(nèi)部）Thedependencezoneexists,(upmachcore)對y的差分可以用中心格式Thecenturialdifferenceisusedforthederivativewithspedtoy對x的差分要用迎風(fēng)格式UpwindschemeisusedforX-direction顯示格式：差分式取，而不用線法Explicitscheme每次都用i網(wǎng)格線上的已知值，可以從左到右逐點計算Theknownvalueisusedtocalculatethevalueateverynodesequently隱式格式：利用當(dāng)前網(wǎng)格線上的值構(gòu)筑差分方程Implicitscheme:usingpresentvaluetoconstructFDE

具有三個未知量（在網(wǎng)格線i上）

Wherethereare3unknownpoints顯式比隱式方便Explicitlyschemeismoreconvenientthanimplicitscheme顯式格式穩(wěn)定區(qū)域小Thestabilityzoneofexplicitissmallerthanthatofimplicitly穩(wěn)定性和收斂性

Stabilityandconvergence收斂性：當(dāng)步長趨于零時，差分方程解趨于微分方程解Convergence:whensteplengthtendstozero,thesolutionofthePDFtendstothesolutionofPDE穩(wěn)定性：差分誤差在傳播過程中有界且逐漸減小Stability:theerrorislimitedordecreased對波動方程（雙曲型）：穩(wěn)定性條件是差分方程依賴區(qū)不小于微分方程的依賴區(qū)Forviberation

Eq,thestabilityconditionisthatthedependentzoneofPDElessthanthatofPDE對超聲速勢函數(shù)

Forpotentialvelocityfuction

差分方程依賴區(qū)半頂角ThehalfconicalangleThedependentzoneoftheFDE微分方程的半頂角theangleofthedependentzone差分方程穩(wěn)定條件為對于跨聲速勢流，不滿足穩(wěn)定條件，因為Fortransonicflow,thestabilityconditionisnotsatisfied跨聲速勢流不能用顯示格式

sotransonicpotentialfunctioncannotsolvewithexplicitmethod隱式格式的依賴范圍大于微分方程的依賴范圍ThedependentzoneofimplicitschemeisgreatthanthatofPEDJ+1JJ-1雙曲方程差分采用一側(cè)隱式格式Forhyperbolicequation,onesideimplicitlyschemeisused五、音速點的差分方程Thefinitediffenceatsonicpoints

當(dāng)M=1時，方程為拋物性，存在一族特征線WhenM=1,theequationisparabolic,thereexistaseriesofcharacteristline速度勢方程化為potentialequationbecome

Subsonic采用差分方程可以寫成UsingFDE六、速度判別式Velocitycriticalcondition

四種情況:

Fourcases

亞聲速sub亞聲速sub

超聲速supe超聲速super

亞聲速sub超聲速super

超聲速super亞聲速subsupersupersonicairfoilⅠⅡⅢ：過渡連續(xù)

continuallychanges

Ⅳ：出現(xiàn)激波參數(shù)不連續(xù)theshockappears,parametersarediscontinous

Ⅲ：有音速線存在Thereexistssonicpoints逐點判別：根據(jù)系數(shù)進行判別Judgeaccordingtothecoefficientof情況的值的值Ⅰ>0>0亞-亞聲速subsonicⅡ<0<0超-超聲速supersonicⅢ>0<0亞-超聲速sonicⅣ<0>0超-亞聲速subsonic中心差分一側(cè)差分A

(i,j)點性質(zhì)對應(yīng)的差分方程any亞音速subsonic超音速supersonic音速點sonic差分方程形式PDEform七.跨聲速小擾動激波的差分方程

PDEfortransonicsmallperturbationshockflow激波處：速度由超聲速過渡到亞聲速Atshock,theflowtransferfromsupersonictosubsonic激波前流場均勻（近似）

Infrontoftheshock,theflowisuniformsupersonicflowi,ji-1ii+1j+1i-1ji,j+1i+1,j+1i+1,ji-1shock激波后流場均勻（近似）Aftertheshock，theflowisalsouniform差分方程（跨聲速小擾動方程的差分形式）FDE(Transonicsmallperturbationflow)對無旋流動（無旋條件）Conditionofirrotationalflow

其差分形式ItsFDform

考慮了無旋條件的擾動速度差分方程Afterconsideringtheirrotatationalconditionthesmallperturbationequationbecomes討論：discussion：

跨聲速區(qū)小擾動激波差分方程與小擾動激波關(guān)系相同八、超音速點差分方程的人工粘性

artificial

viscous

for

supersonicFDE速勢方法假設(shè)了流場均為等熵流The

velocitypotentialmethodassumethattheflowisiso-entropy導(dǎo)致流場間斷解不唯一（可由亞-超，也可由超-亞）Itleadsto

non-unique

solution如果采用迎風(fēng)格式（單側(cè)差分），則只適合壓縮突躍（由超-亞），不可能出現(xiàn)膨脹解。Continuous

solution，if

theupwindschemeisused,thesolutiononlysuitableforcompressiblesharpincrease(shock),notsuitableforsharpdecrease.超聲速點差分方程（迎風(fēng)格式）FDE

of

thepotentialequationatsupersonicflow原因：采用1階迎風(fēng)格式1st

order

upwind

scheme應(yīng)用當(dāng)?shù)豈數(shù)改成相對應(yīng)的微分方程UsinglocalMachnumberMtorewritethePDEthen其中類似于跨音速小擾動粘性流方程中的粘性項。稱為人工粘性

Where

issimilarastheviscousformofsmallpertubationequation,socalleditartificialviscous差分方程的解只含壓縮突躍，即激波（是熵增過程）PDEonlyincludescompressedshapechange(wheretheentropycreases)不可能產(chǎn)生膨脹突躍（即熵減過程）Notsuitableforexpandingshapechange(whereentropydecreases)4.4邊界條件及其嵌入

EmbedingofBoundaryconditions一、邊界條件(BoundaryCondition)1.物面：無粘，無穿透條件onwallnonormalvelocity對于翼型（葉柵），設(shè)物面方程為，則定常流動邊界條件即：若翼型上下表面可表示為則速度分量可寫成

上表面的邊界條件為BConupsurfaceis其中，,為擾動速度Where,istheperturbationvelocitycomponents對于薄翼型Forthinwing小迎角下，時ForsmallAOA,when故上表面(onupsurface)或?qū)懗蒾rbewrittenas

同理，對于下表面meantimeforlowerside綜合上下表面可以寫成以下小擾動方程翼型上下表面邊界條件Considerupperandlowersideofairfoil,thesmallperturbationssatisfyfollowingcondition2.庫塔條件（后緣邊界條件）Kuttacondition(trailingedgecondition)上下表面流線在后緣尖點平滑匯合thestreamlinesonupsideandLower-sidesmoothlysinksattrailingedge在受氣動載荷時，速度勢在后緣不連續(xù)，形成間斷面。Undertheaerodynamicloads,velocitypotentialfunctionattractingedgeisdiscontinuous在這條間斷面上必須滿足Onthediscontinuitysurface，whatmustsatisfyis。。后上下c

(1)上下壓強相等

the

pressureonup

andlowersideofairfoilisequal

(2)速度方向相同，大小不同

the

direction

ofvelocityareconsistent,butthevalueofthevelocityisnotequal

小擾動條件下，因此上述方程可寫成：

forsmallperturbation,aboveequationscanbewrittenas:

經(jīng)間斷面速度勢變化稱為環(huán)量

through

the

section

surface

the

velocity

potential

function

changes

is

circulation.3.遠(yuǎn)場條件Farfieldcondition用有限遠(yuǎn)代替無限遠(yuǎn)場，擾動速度勢的近似條件為：usinglimitedfarfieldreplacetherealfarfieldperturbationvelocitypotentialfunctionBCcanbewrittenas:二、邊界條件的嵌入Embedingoftheboundarycondition邊界點上速度勢應(yīng)同時滿足邊界條件和速勢方程OnboundarythevelocitypotentialfunctionsatisfyboththeBCandthepotentialEq.1.物面邊界嵌入Embedingofwallboundarycondition翼型上表面Ontheairfoilsurface將速勢拓延到邊界的另一側(cè)（i，j-1）Extendthevelocitypotentialfunctiontoothersideofboundary即Or邊界點的中心差分Thecentraldifferenceonboundary利用邊界條件得到：UsingBCthenget2.庫塔條件的嵌入EmbeddingofKuttacondition增加新方程使上下表面上相同，即Additionalnewequationtomakeconsistentonupandlowersurface3.遠(yuǎn)場條件的嵌入Embeddingoffarfieldcondition根據(jù)具體問題特點建立運動場的計算方法Tofoundthecomputationmethodaccordingtothecharacterofcertainproblem對于自由繞流，運動速度為，自由來流的速度勢為forafreeflowaroundtheairfoil，thefarfieldvelocityis，andthevelocitypotentialfunctionoffreeflowis擾動速度勢應(yīng)滿足Thereforetheperturbationvelocitypotentialsatisfy§4.5線松弛迭代解法

Thelinerelaxationiterationmethod一、非線性代數(shù)方程的迭代解法Iterativemethodfornon-linearequations跨聲速小擾動速勢方程是非線性的TransonicsmallperturbationequationisnonlinearPDE其差分方程為非線性代數(shù)方程，即系數(shù)是與函數(shù)值或其導(dǎo)數(shù)有關(guān)ItsFDEisalsononlinearequationthatisitscoefficientsarerelatedtothevariables迭代求解：

Iterationmethod把系數(shù)假設(shè)成已知量，每次求解之后再重新計算系數(shù),再次求解直到得出收斂解為止.Assumethecoefficientareknownatfirstiteration,thenrecalculatethecoefficientsagainafteronceiteration,repeatiterationuntiltheiterationconvergences二、高階代數(shù)方程的線松弛解法

ThelinerelaxationiterationmethodforHighorderarithmeticlinearequations

高階線性方程組，線性化后的差分方程

Highorderlinearequations,linearizedFDE階數(shù)為,M為網(wǎng)格點數(shù),n為問題的維數(shù).或階數(shù)M*N*L（M,N,L為空間三坐標(biāo)方向的網(wǎng)格點數(shù)）Theorderoflinear-algebraequationis,whereMisthenumberofthegrids,nisthenumberofdimension.Theorderoflinear-algebraequationisM*N*L,whereM,N,Larenumberofgridsincoordinatesx,yandz松弛迭代：Relaxationiteration

輪流放松流場中的的部分速勢，將其假設(shè)為未知，其余部分看成已知的,利用線性方程組聯(lián)立求解

Relaxatethepotentialfunctionsequently,assumethatthepresentpointisunknown,andothersareknown.松弛迭代點松弛：每次把一個點作為未知點Pointrelaxation:onlyonepointisassumedtobeunknown線松弛：每次把一條網(wǎng)格線上的所有點作為未知Linerelaxation:allpointononegridlineareassumedtobeunknown線松弛linerelaxationj線松弛linerelaxationi線松弛法：要求內(nèi)存較多，方程組的個數(shù)減少到一維點數(shù)Linerelaxation:requiremorememorysource,thenumberofequationequalstothenumberof1Dpointsij點松弛pointrelaxation逐點松弛：要求內(nèi)存較少（為線性松弛的倍）,掃描流場中的各個網(wǎng)格點，把周圍點均看成是已知點。Sequentpoint:requirelessmemoryresource,onlytimesoflineelaxation.Scanallthegridpointssequently.線松弛方程組可采用三對角矩陣快速解法Forlinerelaxationmethod,thetri-diagonalarraycanbesolvewithquickmethod.三、簡單迭代和改進迭代Improvemethodofsimpleiterationmethod簡單迭代：迭代公式右端的速度勢全部采用前次迭代結(jié)果Simpleiteration,allparametersonrighthandareoldvalueoflastiteration.改進迭代：每次迭代都用最新速度勢值代替右端項。速度判別式要用簡單迭代方式計算，則會導(dǎo)致超臨界氣流計算振蕩發(fā)散。Theimprovediterationmethodalwaysusesthenewestvalue,andthevelocitycriteriamustbecalculatedaccordingtothesimpleiterationway,otherwise

thedivergencewilloccuratcriticalstate.四、追趕法Thechasemethod求解三對角矩陣線性方程快速方法Itisafastmethodtosolvetri-diagonalmatrix線松弛方法求解方程組Equationforlinerelaxationmethod對于邊界點：forboundarypoints上邊界upboundary下邊界lower

boundaryj=1,2……Ni-1,ji,ji,j-1i,j+1i+1,j對應(yīng)的系數(shù)矩陣為三對角矩陣

追趕法：順著消去，逆著帶入。從上至下消去首項，從下而上代入末項。Thechasemethod：eliminatingsequently，substitutinginversely.Eliminatefromtoptodown，substitutefromdowntoup.五、初場Initialfield

初始值分布：影響收斂速度Initialfielddistributionofparameterswillinfluenceconvergence對亞音速流場：可以選全場速度勢為0，即未經(jīng)擾動Subsonicflowfield：globefieldcanbeputas0，thatistheflowisnotdisturbed對跨音速流場：初值選取需謹(jǐn)慎，合理初場能加速收斂Fortransonicflowinitialvaluemustgivencarefully，thereasonableinitialvaluemightaccelerateconvergence

一般應(yīng)選用與流場相近的速度勢分布Usuallyselectanearsolutionofpotentialfunction可以用相近的亞聲速計算結(jié)果Theapproximatelysubsonicresultcanbeused六、收斂標(biāo)準(zhǔn)Criterionofconvergence所有點相鄰兩次計算所得的速勢差別的最大值Themaximumdifferencebetweentwoimmediatevicinityiterative可以用與初始比值判別收斂

theratioofcurrentandinitialcanbethecriterion七、超松弛法Superrelaxationmethod加速速度勢函數(shù)的修正步伐Toacceleratetheconvergence

超松弛

superrelaxation

弱松弛weakrelaxation

八、加密網(wǎng)格法Meshrefinemethod計算精度增加,計算網(wǎng)格數(shù)增加Toincreasetheprecision,toincreasethemeshNo.問題復(fù)雜度增加TheincreaseofcomplexitytoincreasethemeshNo.計算機時與網(wǎng)格總點數(shù)以正比增加computationtimeincreaseasthemeshNo.采用疏密結(jié)合的方法可以減少計算時間

Usingcoarse/finemeshmaydecreasecomputationtime加密網(wǎng)格法：先用疏網(wǎng)格數(shù)算初始場，加密之后獲得精確解meshesrefiningmethod多重網(wǎng)格：先疏后密、再疏；交替使用疏密相間的網(wǎng)格multiplegrid*§4-6繞升力翼型的跨聲速小擾動勢流差分計算方法FDMforpotentialfunctionoftransonicsmallperturbationflowaroundairfoil一、繞升力翼型的跨聲速小擾動方程勢流的差分方程Theequationforpotentialfunctionoftransonicsmallperturbationflowaroundairfoil4—7隱式近似因式分解法的基本思想

ThebasicconceptofApproximateDecompositionMethod求解速度勢方程的快速收斂解法ItisafastworkingmethodforPotationequationSLOR是顯式迭代方法，因此收斂慢SLORisfullexplicitmethod,thusitworksslowly全隱式松弛算法：每次迭代中流場中的任意一點能受到它的依賴區(qū)中全部其點的影響

Fullimplicitrelaxationmethod,anypointinflowfieldcanbeinfluencedbyallotherpointsADI(AlternatingDirectionImplicit)交替方向隱式迭代，分為AF1和AF2UsingAF1andAF2基本差分算子：Somebasicfinitedifferencecalculator迎風(fēng)差分（前差）upwindFD順風(fēng)差分（后差）backward/rearwardFD二階中心差分：2ndordercentralFD二階一側(cè)迎風(fēng)差分：upwind2ndonesideFD位移算子：displacement(FD)calculator用位移算子表示差分算子

TheFDexpressedusingdisplacementcalculators差分算子位移：ThedisplacementofFDcalculator差分算子的分解與組合：ThedecompositionandcombinationofFDs差分方程可以用算子表示TheexpressionofFDEusingFDcalculatorL代表未經(jīng)松弛的差分算子

FDcalculatorrelaxation松弛差分算子N，第n次迭代的修正值為TheFDcalculatorofrelaxationiteration,thecorrectionofnthiteration.算子表達(dá)式:

Calculatorexpression當(dāng)松弛迭代收斂時，，即兩者相同Whentheiterationconverged,bothcalculatorarethesame

.

當(dāng)時，表示其不是差分方程的解，因此表示差分方程滿足微分方程的程度。When，denotesthesolutionofFDEisnotthesolutionoftheoriginalPDE，thereforedenotesthedegreeofhowdosetheFDEsatisfythePDE.差分松弛迭代算子的選取原則TheprincipleforseleetingFDcalculator

便于求解，線性，有快速解法convenienceforsolvingequation，linearmethod，fastsolver

穩(wěn)定，能達(dá)到收斂標(biāo)準(zhǔn)stable

高效，N盡可能接近L。

higherefficiency，NapproachingL差分算子的用途：可以清晰的顯示差分方程的結(jié)構(gòu)UsageofthePDcalculator，itmakestheFDEsimpleandevident近似因式分解的基本思路ThebasicconsiderationofapproximatedecomposeLaplace方程的差分格式（簡單迭代法）TheFDschemeofLaplaceequation（simpleiterationmethod）

改進的迭代法

improvediterationmethod松弛迭代格式

Relaxationiteration

scheme中間值由此then還原為（n）和（n-1）表達(dá)式后差分方程還原為ExpresstheFDEusing（n）and（n-1）改進的差分格式為

ImprovementofFDE或（隱式）or(ImplicitForm)引入差分算子，采用差分算子表示，并令I(lǐng)ntroducetheFDcalculator,usingFDcalculatorexpression.則松弛迭代法的差分格式為：TheFDschemeofrelaxationiterationis線松弛迭代對應(yīng)的差分格式FDSrelatedtolinerelaxationiterationis因此超松弛差分算子Thus,thesuper-relaxationFDcalculatorisN分解成兩個因式的乘積，則IffactorizeNintotwofactors,then

4-8AF1方法AF1method小擾動速勢方程Equationofpotentialfunctionforsmallperturbationflow其中，對亞音速小擾動?？捎弥行牟罘指袷交螂[式方程Where,forsubsonicpoint,thecentralschemecanbeused其中where令Let

則小擾動速勢方程的隱式差分格式為Then，theimplicitFDEschemeofthesmallperturbationpotentialis分解第一項系數(shù)

Tofactorizethefirstcoefficientterm原系數(shù)origin誤差error松弛差分算子N可分解成為N1和N2，為加速收斂參數(shù)，求解可分兩步，

RelaxationPDEcalculatorNcanbefactorizedIntoN1andN2,thesolvingcanbedecomposeintotwosteps

代表中間結(jié)果Whereisamiddleresult交替方向隱式差分格式（ADI,or,ApproximateFactorization）

Step1:全場沿X方向線松弛，解三對角矩陣

Xdirectionlinerelaxation,tosolvetrianglematrixStep2:全場方向沿y方向線松弛，也解三對角矩陣

Ydirectionlinerelaxation,tosolveatrianglematrix全隱式格式，對亞聲速區(qū)適用，稱為AF1，

fullimplicitscheme,Forsubsonicit’scallAFI

對超音速點，采用迎風(fēng)格式Forsupersonicpointstheupwindschemeisused對應(yīng)兩步

Correspondingtwostepsare1.

2.二、AFI的收斂性TheconvergenceofAFI亞音速點（中心差分格式）等價于時間相依方程（將迭代過程看成時間推進）Forsubsonic(centralPDEscheme),thesolvingprecedingisequivalenttoatime-dependentproblem設(shè)和，當(dāng)與系數(shù)異號時，差分方程的解收斂于微分方程的解。andhavedifferentsign,thenthePDEconvergestwothePDE三、AF1的穩(wěn)定性StabilityofAFI采用Vonneumann方法分析誤差UsingVoneumannanalyticmethod代入AF1差分方程SubstituteintothePDEofAFI其中where假設(shè)，為實數(shù)Assume，isrealnumber收斂條件：Theconditionofconvergence即or穩(wěn)定性條件：

stabilitycondition兩個可選參數(shù)和，適當(dāng)選取可以加快收斂Twoparametersandcanbechosencarefullytogetquickconvergence取得到最短迭代次數(shù)（）Take,thengetminimiterationtimes對應(yīng)的最佳選擇是：CorrespondingopticalchoiceisAFI中所有的誤差Fourier分量均可以同速度下降

AFIerrorcomponentsofFourierseriesmaydecrease超聲速點的AF1差分方程等價于（4-8-12），但沒有阻尼項

TheAFIforsupersonicpointitequaltoequation（4-8-12）

亞，超聲速流混合問題，AF1第一個算子需四對角矩陣求逆Forsub-super-sonicmixedproblem,AFIhastosolvefour-diagonalmatrix,thustheefficiencyislower對此類流動，AF1不是最有效的方法Forsuchflow,AFIisnotthemostefficiencyone§4-9AF2方法

methodofAF2對超聲速流增加時間阻尼項，取Forsupersonicflow,thetunedampingistraduced差分算子FDcalculatorwithupwindscheme對超聲速點，?。篎orsupersonicflowpoint等價的一階微分方程（時間依存）Equivalent1storderPDE(timedependent)其中，為超聲速阻尼項，當(dāng)時，與系數(shù)同號，大小取決于

Where,issupersonicdamping,whenthecofficentofandhavethesamesignandthequantitydependson更加高效（有阻尼）ItishigherefficiencyAF2格式對亞聲速流收斂性比AF1差A(yù)F2convergenceforsubsonicisworstthanAF1亞，超聲速點的兩步AF2格式如下：Forsub-supersonicflow,twosteps,AF2havefollowingscheme亞：（y方向線松弛x方向迎風(fēng)格式）

sub

LinerelaxationinYupwindschemeinX

（x向線松弛x方向順風(fēng)格式）