EQUATION SHEET Quiz 2
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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS NUMERICAL FLUID MECHANICS SPRING 05 EQUATION SHEET Quz Nubr Rrstato - Floatg Nubr Rrstato: = b, b - < b 0 Trucato Errors ad Error Aalyss y f (,, 3,..., 3 f ( f ( f '( f ''( f '''(... f ( R! 3!! - Taylor Srs: ( R f (! f (,..., - Th Dffrtal Error (gral rror roagato Forula: y f - Th Stadard Error (statstcal forula: E( sy - Codto Nubr of f(: K f '( f( Roots of olar quatos ( h( f ( - Bscto: succssv dvso of bract half, t bract basd o sg of f ( f ( d-ot f ( U ( L U - Fals-Posto (Rgula Fals: r U f ( L f ( U - Fd Pot Itrato (Gral Mthod or Pcard Itrato: g( or h( f ( - Nwto Rahso: f ( f '( ( - Scat Mthod: f ( f ( f ( - Ordr of covrgc : Dfg, th ordr of covrgc sts f thr st a costat C 0 such that: l C - -
2 Cosrvato Law for a scalar, tgral ad dffrtal fors: d d - d d ( v. da q. da s d dt CM dt C fd CS CS C fd -.( v. q s t Lar Algbrac Systs: - Gauss Elato: rducto, Advctv flus (Adv.& dff. ="covcto" flus followd by a bac-substtuto. b a a - LU dcoosto: A=LU, a - - Othr trasorts (dffuso, tc Su of sourcs ad ss trs (ractos, tc a a a a b b b, (, ( ( (, ( ( ( ( a ( ( (, ( a - Chols Factorzato: A=R R, whr R s ur tragular ad R ts cougat trasos. - Codto ubr of a lar algbrac syst: K( A A A - A badd atr of sur-dagoals ad q sub-dagoals has a badwdth w = + q + ( ( ( - ad 6 - Egdcoosto: A ad dt( A I 0 - Nors: A A A A F a a a a a a Mau Colu Su Mau Row Su A A L or or sctral or Frobus or or Euclda or Itratv Mthods for solvg lar algbrac systs: B c 0,,,... - Ncssary ad suffct codto for covrgc: ( B a, whr gvalu( B Jacob s thod: -D (L U D b Gauss-Sdl thod: ( D L U ( D L b - SOR Mthod: ( D L [ U ( D] ( D L b T - Stst Dsct Gradt Mthod: r r r, r T b A r Ar - Cougat Gradt: v (α such that ach v ar gratd by orthogoalzato of rsduu vctors ad such that sarch drctos ar A-cougat.
3 Ft Dffrcs PDE tys (d ordr, D: A B C F(, y,,, y yy y B AC > 0: hyrbolc; B AC = 0: arabolc; B AC < 0: lltc Ft Dffrcs Error Tys ad Dscrtzato Prorts ( - Cosstcy: ( ˆ ( 0 wh 0 - Trucato rror: ( ˆ ( O( for 0 - Error quato: ( ˆ ( ˆ ˆ ( (for lar systs - Stablty: ˆ - Covrgc: Cost. (for lar systs ˆ O( Ft Dffrcs Gral schs ad Hghr Accuracy u Hghr Ordr Accuracy Ft-dffrc basd o Taylor Srs: Nwto s trolatg olyoal forulas, qudstat salg: ( 0, ˆ ( ˆ 0 s r au Lagrag olyoal: 0 0, f ( L ( f ( wth L ( Hrt Polyoals ad Coact/Pad s Dffrc schs: s q u b a u r Ft Dffrcs No-Ufor Grds, Grd Rft ad Error Estato For a ctrd-dffrc aroato of f ( ovr a D grd, cotractg/adg wth a costat factor r, r, th: r ( r - Ladg tr of th trucato rror s: f ''( ( rh, - Rato of th two trucato rrors at a coo ot s: R r Grd-Rft ad Error stato: - Estat of th ordr of accuracy: u u 4 log u u log h, - 3 -
4 u u - Dscrtzato rror o th grd Δ: Rchardso Etraolato for th Trazodal Rul: Robrg Dffrtato Algorth: D 4 D D,,, 4 Ft Dffrcs Fourr Aalyss ad Error Aalyss f f Fourr trasfor of a grc PDE, : Wth f (,t t f ( t d f ( t dt f ( t f ( t for Ft Dffrc Mthods Effctv wav ubr ad sd d s( Effctv Wav Nubr: ff (for CDS, ordr, ff f f Effctv Wav Sd (for lar covcto q., c 0: t u. df c f t c f t f f dt c u. ff t ( cff t ( ff urcal (, (0 (0, o obtas: ff ff ff Ft Dffrc Mthods Stablty o Nua: (, t ( t, ( t t (γ gral col, fucto of β Strct codto for stablty: t t or for, (for th rror ot to grow t Usful trgootrc rlatos: cos(, s( ad cos( s ( / CFL codto: Nurcal doa of ddc of FD sch ust clud th athatcal doa of ddc of th corrsodg PDE - 4 -
5 Fgur 3. Chara ad Caal Forward Dffrcs.9 Nurcal Flud Mchacs PFJL Lctur 0, 5
6 Bacward Dffrcs.9 Nurcal Flud Mchacs PFJL Lctur 0, 6
7 Ctrd Dffrcs.9 Nurcal Flud Mchacs PFJL Lctur 0, 7
8 Ft Dffrc Mthods Schs for scfc PDE tys Hyrbolc, D: u + b uy = 0 Elltc PDEs: D Lalac/Posso Eq. o a Cartsa-orthogoal ufor grd u, u, u, u, h g, SOR, Jacob: u, ( u, 4 u, u, u, u, h g, SOR, Gauss-Sdl: u, ( u, 4-8 -
9 Parabolc PDEs: D Hat Coducto Eq. o a Cartsa-orthogoal grd T, T, T, T, T, T, T, T, Elct: c c t y tc Cra-Ncolso Ilct (for Δ=Δy, wth r : r r ( r T ( r T T T T T T T T T ADI:,,,,,,,,,, (for Δ=Δy: T T T T T T T T / / / /,,,,,,,, c c t / y T T T T T T T T / / / /,,,,,,,, c c t / y rt ( r T rt rt ( r T rt / / /,,,,,, rt ( r T rt rt ( r T rt / / /,,,,,, d Ft olu Mthods: F. da S dt, whr S Cartsa grds d ad S s d Surfac Itgrals: F f da S - D robls (D surfac tgrals Mdot rul ( d ordr: F f da f S f S O y f S ( S Trazod rul ( d ( f fs ordr: F f da S O( y S Sso s rul (4 th ( f 4 f fs 4 ordr: F f da S O( y S 6-3D robls (D surfac tgrals Mdot rul ( d ordr: olu Itgrals: F f da S f O y z S S s d, d (, - D/3D robls, Mdot rul ( d ordr: SP s d sp sp
10 - D, b-quadratc (4 th ordr, Cart.: y SP 6sP 4ss 4s 4sw 4s ss ssw s sw 36 Itrolatos / Dffrtatos (obta flus F= f ( as a fucto of cll-avrag valus P f v. 0 - Uwd Itrolato (UDS: E f v. 0 P - Lar Itrolato (CDS: E P( whr P E P(, wth - Quadratc Uwd trolato (QUICK: U g( D U g ( U UU - Hghr ordr schs: 3 For al, for ( a a a a, E P E P E P E P For ufor grds, U D UU R Covctv flus 48 P E W EE Dffusv Flus, for a ufor Cartsa grd: E P W EE P E 4 For a coact hgh ordr sch: O( 8 P E Soluto of th Navr-Stos Equatos v.( v v v g Nwtoa flud + corssbl + costat: t. v 0 Strog cosrvatv for, gral Nwtoa flud: v u u u.( v v. g t 3 Ktc rgy quato, C for: t v v d ( v. da v. da (. v. da : v. v g. v d C CS CS CS C. D 3 v t...( v v Prssur quato:....( v v. v. g For costat μ ad ρ: - 0 -
11 Prssur-corrcto Mthods Forward-Eulr Elct T: Bacward-Eulr Ilct T: H ( uu + u u t H H ( uu u u t + ( uu Bacward-Eulr Ilct T, larzd otu udat: ( u u ( u u ( u u u u u t Stady stat solvr, atr otato: Outr trato, olar solv: u A u b u δ δ δ u δ u δ u Outr trato, rssur udat:, Ir trato, lar solv: A u A b u δ δ δ u A u b u δ δ Stady stat solvr, atr otato, rssur-corrcto schs: Basd o th abov, but troduc u u u' ' ad furthr slfy to gt vard schs (SIMPLE, SIMPLER, SIMPLEC, PISO, tc. - -
12 Procto Mthods, Prssur-Corrcto For No-Icrtal: ( uu u u t + ; u (bc D Icrtal: ; 0 t u D u u t u u t u ( uu + ; (bc D u ; 0 t u u t Rotatoal Icrtal: ( uu u u t + ; u (bc D u ; 0 t u u t u D D - -
13 MIT OCoursWar htt://ocw.t.du.9 Nurcal Flud Mchacs Srg 05 For forato about ctg ths atrals or our Trs of Us, vst: htt://ocw.t.du/trs.
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