Wave Equation! ( ) with! b = 0; a =1; c = c 2. ( ) = det ( ) = 0. α = ±c. α = 1 2a b ± b2 4ac. c 2. u = f. v = f x ; t c v. t u. x t. t x = 2 f.
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1 Compuaioal Fluid Dyamics p:// Compuaioal Fluid Dyamics Wave equaio Wave Equaio c Firs wrie e equaio as a sysem o irs order equaios Iroduce u ; v ; Gréar Tryggvaso Sprig yieldig u c u sice rom e pde Compuaioal Fluid Dyamics Wave equaio To id e caracerisics u c u u de de A T αi We ca also use α a b ± b 4ac α c α ±c c α α c u wi b ; a ; c c Compuaioal Fluid Dyamics Wave equaio Two caracerisic lies P d dx c ; d dx c d dx c d dx c x To id e soluio we eed o id e eigevecors l u c l u For α c For α c Compuaioal Fluid Dyamics Wave equaio α c α ±c α Take l c l l l c c l l l c l l For α c Compuaioal Fluid Dyamics Wave equaio l l c u c c u u c u c c du d c dv d o dx d c Similarly: For α c l l c du d c dv d o dx d c Add e equaios Relaio bewee e oal derivaive o e caracerisic
2 For cosa c Compuaioal Fluid Dyamics Wave equaio du d c dv d o dx d c du d c dv d o dx d c dr d o dx d c were r u cv dr d o dx d c were r u cv r ad r are called e Riema ivarias Te geeral soluio ca ereore be wrie as: were Compuaioal Fluid Dyamics Wave equaio ( x, r ( x c r ( x c r ( x c r ( x c Ca also be veriied by direc subsiuio Compuaioal Fluid Dyamics Wave equaio-geeral Compuaioal Fluid Dyamics Two caracerisic lies Domai o Iluece P Domai o Depedece d dx c ; d dx c d dx c d dx c x Ill-posed problems Compuaioal Fluid Dyamics Ill-posed Problems Compuaioal Fluid Dyamics Ill-posed Problems Cosider e iiial value problem: Tis is simply Laplace s equaio Here, owever, is equaio appeared as a iiial value problem, were e oly boudary codiios available are a. Sice is is a secod order equaio we will eed wo codiios, wic we may assume are a ad / are speciied a. wic as a soluio i / or are give o e boudaries
3 Compuaioal Fluid Dyamics Ill-posed Problems Te geeral soluio ca be wrie as: x, Look or soluios o e ype: a k (e ikx Subsiue io: a k (e ikx k were e a s deped o e iiial codiios Compuaioal Fluid Dyamics Ill-posed Problems d a k k a d k Geeral soluio a k ( Ae k Be k A, B Tereore: deermied by iiial codiios Geerally, bo A ad B are o-zero a( as a( ; ika( o ge: d a k d k a k Ill-posed Problem Compuaioal Fluid Dyamics Ill-posed Problems Compuaioal Fluid Dyamics Ill-posed Problems Log wave wi sor wave perurbaios Similarly, i ca be sow a e diusio equaio wi a egaive diusio coeicie D ; D < as soluios wi ubouded grow rae or ig wave umber modes ad is ereore a ill-posed problem Compuaioal Fluid Dyamics Ill-posed Problems Compuaioal Fluid Dyamics Ill-posed problems geerally appear we e iiial or boudary daa ad e equaio ype do o mac. Frequely arise because small bu impora iger order eecs ave bee egleced Ill-posedess geerally maiess isel i e expoeial grow o small perurbaios so a e soluio does o deped coiuously o e iiial daa Iviscid vorex see rollup, mulipase low models ad some viscoelasic cosiuive models are examples o problems a exibi ill-posedess. Classical Meods or Hyperbolic Equaios
4 Te wave equaio: c I geeral: Compuaioal Fluid Dyamics Meods or Advecio Wrie as: u c u u u a a a a Mos o e issues ivolved ca be addressed by examiig: Compuaioal Fluid Dyamics Forward i Time, Ceered i Space (FTCS ad pwid Compuaioal Fluid Dyamics Meods or Advecio We will sar by examiig e liear advecio equaio: Te caracerisic or is equaio are: dx d ; d d ; Sowig a e iiial codiios are simply adveced by a cosa velociy x Compuaioal Fluid Dyamics Meods or Advecio A simple orward i ime, ceered i space discreizaio yields ( - Compuaioal Fluid Dyamics Meods or Advecio Tis sceme is O(, accurae, bu a sabiliy aalysis sows a e error grows as ε ε Sice e ampliicaio acor as e orm i( e absolue value o is complex umber is always larger a uiy ad e meod is ucodiioally usable or is case. i si k ε ε i sik Compuaioal Fluid Dyamics Meods or Advecio Aoer sceme or A simple orward i ime bu upwid i space discreizaio yields ( Flow direcio Tis sceme is O(, accurae. -
5 Compuaioal Fluid Dyamics Meods or Advecio To examie e sabiliy we use e vo Neuma s meod: Te evoluio o e error is govered by: ε ε (ε ε Wrie e error as: ε ε e ikx ε ε ε ( e ik Ampliicaio acor ε ε ε G ( e ik, ε ( eik Or: G e ik Need o id we G < Grapically: G e ik, Sabiliy codiio: < Tis resricio was irs derived by Coura, Fredrik, ad Levy i 9, ad is usually called e Coura codiio, or e CFL codiio. Compuaioal Fluid Dyamics Meods or Advecio Sable Im(G - G k Compuaioal Fluid Dyamics Meods or Advecio Aoer way: Fid e absolue value o e ampliicaio acor G e ik cosk isik G ( cosk si k ( ( cosk cos k si k ( ( cosk ( cosk G ( i cosk G i cosk G ( 4 i cosk G i G cosk cosk Compuaioal Fluid Dyamics Meods or Advecio Te CFL codiio implies a a sigal as o ravel less a oe grid spacig i oe ime sep Flow direcio - No allowable caracerisics Allowable caracerisics Compuaioal Fluid Dyamics Meods or Advecio Compuaioal Fluid Dyamics Meods or Advecio For e liear advecio equaio: Flow direcio ( O(, accurae. Te pwid Sceme - Aloug e upwid meod is excepioally robus, is low accuracy i space ad ime makes i usuiable or mos serious compuaios.5 pwid d.5* 4 8
6 Compuaioal Fluid Dyamics Meods or Advecio Fiie Volume poi o view: (F / F / F / ( x -/ x / F / x Compuaioal Fluid Dyamics Geeralized pwid Sceme (or bo > ad < (, > (, < Deie: (, ( Te wo cases ca be combied io a sigle expressio: [ ( ( ] Or, subsiuig, ( ( ceral dierece umerical viscosiy D um Compuaioal Fluid Dyamics Compuaioal Fluid Dyamics Meods or Advecio Implici (Backward Euler Meod Oer Firs Order Scemes ( - codiioally sable - s order i ime, d order i space - Forms a ri-diagoal marix (Tomas algorim a d b C Lax-Fredrics meod Compuaioal Fluid Dyamics Meods or Advecio ( - sable or < - s order i ime, d order i space - Codiioally cosise Error erm: xx ( xxx Compuaioal Fluid Dyamics Secod Order Scemes
7 Leap Frog Meod Compuaioal Fluid Dyamics Meods or Advecio Te simples sable secod-order accurae (i ime meod: O( Modiied equaio ( xxx - Sable or < - Dispersive (o dissipaio error will o damp ou - Iiial codiios a wo ime levels - Oscillaory soluio i ime (aleraig Compuaioal Fluid Dyamics Meods or Advecio Lax-Wedro s Meod (LW-I Firs expad e soluio i ime ( Te use e origial equaio o rewrie e ime derivaives Subsiuig Compuaioal Fluid Dyamics Meods or Advecio ( O( sig ceral diereces or e spaial derivaives ( ( d order accurae i space ad ime Sable or < Compuaioal Fluid Dyamics Meods or Advecio Two-Sep Lax-Wedro s Meod (LW-II LW-I io wo seps: / / ( / Sep (Lax / / / / / Sep (Leaprog - Sable or / < - Secod order accurae i ime ad space For e liear equaios, LW-II is ideical o LW-I MacCormack Meod Compuaioal Fluid Dyamics Meods or Advecio Similar o LW-II, wiou /, / ( ( - A racioal sep meod - Predicor: orward dierecig - Correcor: backward dierecig - For liear problems, accuracy ad sabiliy properies are ideical o LW-I. Predicor Correcor Compuaioal Fluid Dyamics Meods or Advecio Secod-Order pwid Meod Warmig ad Beam (975 pwid or bo seps ( Combiig e wo: Predicor ( Correcor ( ( ( - Sable i - Secod-order accurae i ime ad space
8 Compuaioal Fluid Dyamics Te oe-sep Lax-Wedro is o easily exeded o o-liear or muli-dimesioal problems. Te spli versio is. I e Lax-Wedro ad e MacCormack meods e spaial ad e emporal discreizaio are o idepede. Oer meods ave bee developed were e ime iegraio is idepede o e spaial discreizaio, suc as e Beam-Warmig ad various Ruge-Kua meods Meods or Advecio Compuaioal Fluid Dyamics FTCS codiioally sable pwid Sable or Implici codiioally Sable Lax-Friedrics Codiioally cosise Sable or x xxx xx xxx xx xxx xx xx xxx / Summary Compuaioal Fluid Dyamics Leap Frog Sable or Lax-Wedro I Sable or Lax-Wedro II Same as LW-I Sable or MacCormack Same as LW-I Sable or x xxx xxxx xxx 8 / / ( / / / / / / / Summary Compuaioal Fluid Dyamics Beam-Warmig Sable or QICK Sable or ENO WENO x 4 xxxx xxx 8 8 A large umber o (codiioally sable ad accurae meods exiss or yperbolic equaios wi smoo soluios Ad May More Summary
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