Theory of! Partial Differential Equations-I!

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1 hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and Nonlinear Advecion Equaions! Quasi-linear Second Order Equaions!!- Classiicaion: hyperbolic, parabolic, ellipic! - Domain o Dependence/Inluence! Ill-Posed Problems! Conservaive orm! Navier-Sokes equaions! Eamples o equaions! + U = 0 D = 0 c = 0 Advecion! Diusion! Wave propagaion! Evoluion in ime! Navier-Sokes equaions! Diusion par! + u + v = 1 P ρ + µ u ρ + u Advecion par! + = 0 Laplace equaion! + v = 0 Laplace par! Deiniions! The order o PDE is deermined by he highes derivaives! Linear i no powers or producs o he unknown uncions or is parial derivaives are presen.! + y =, + + = 0 Quasi-linear i i is rue or he parial derivaives o highes order.!! + = =, + y Quasi-linear irs order! parial dierenial equaions!

2 Consider he quasi-linear irs order equaion! a + b = c where he coeiciens are uncions o,y, and, bu no he derivaives o : a = a(, y, ) b = b(, y, ) c = c(, y, ) The soluion o his equaions deines a single valued surace (,y) in hree-dimensional space:! y =(,y) An arbirary change in is given by:! d = d + dy dy y d d The original equaion and he condiions or a small change can be rewrien as:! a + b = c d = d + dy,, 1 (a,b,c) = 0,, 1 (d,dy, d ) = 0 Normal o he surace! =(,y) Boh (a,b,c) and (d,dy,d) are in he surace!! The normal vecor o he curve =(,y) 1! Picking he displacemen in he direcion o (a,b,c)! (d,dy,d ) = ds (a,b, c) -1! Same argumens in he y-direcion. Thus =(,y) n =,, 1 Separaing he componens! d ds = a; dy ds = b; d dy = a b d ds = c;

3 The hree equaions speciy lines in he -y plane! d ds = a; Given he iniial condiions:! dy ds = b; Characerisics! d ds = c; = (s, o ); y = y(s, o ); = (s, o ); he equaions can be inegraed in ime! y d ds = a; dy ds = b; d ds = c; (s); y(s); (s) (0); y(0); (0) slope! d dy = a b Consider he linear advecion equaion! + U = 0 The characerisics are given by:! or! ds = 1; d ds = U; d = U; d = 0; d ds = 0; Which shows ha he soluion moves along sraigh characerisics wihou changing is value! Graphically:! Noice ha hese resuls are speciic or:! + U = 0 (,) = = 0 ( U) The soluion is hereore:! (,) = g U ( ) g() = (, = 0) where! This can be veriied by direc subsiuion:! Se! η(,) = U = g η Then! η = g η ( U ) and! Subsiue ino he original equaion! + U = g η U ( ) + U g = g η η = g η ( 1) η = 0 Disconinuous iniial daa! + U = 0 Since he soluion propagaes along characerisics compleely independenly o he soluion a he ne spaial poin, here is no requiremen ha i is diereniable or even coninuous! (,) = g( U)

4 Add a source! +U = The characerisics are given by:! ds =1; d ds = U; d ds = ; or! d = U; d = = (0)e Moving wave wih! decaying ampliude! Consider a nonlinear (quasi-linear) advecion equaion! + = 0 The characerisics are given by:! or! d =1; ds ds = ; d ds = 0; d = ; d = 0; The slope o he characerisics depends on he value o (,). 3 Muli-valued soluion Why unphysical soluions?! - Because mahemaical equaion neglecs! some physical process (dissipaion)! + ε = 0 Burgers Equaion! 1 Addiional condiion is required o pick ou he physically! Relevan soluion! Correc soluion is epeced rom Burgers equaion wih ε 0 Enropy Condiion hp://users.wpi.edu/~grear/me61.hml! Consrucing physical soluions! + = 0 In mos cases he soluion is no allowed o be muliple valued and he physical soluion mus be reconsruced using conservaion o! The disconinuous soluion propagaes wih a shock speed ha is dieren rom he slope o he characerisics on eiher side! Theory o! Parial Dierenial Equaions-II! Gréar Tryggvason! Spring 010!

5 Ouline! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and Nonlinear Advecion Equaions! Quasi-linear Second Order Equaions!!- Classiicaion: hyperbolic, parabolic, ellipic! - Domain o Dependence/Inluence! Ill-Posed Problems! Conservaive orm! Navier-Sokes equaions! Quasi-linear Second order parial dierenial equaions! Consider! where! a + b + c = d a = a(, y,,, y ) b = b(, y,,, y ) c = c(, y,,, y ) d = d(, y,,, y ) Firs wrie he second order PDE as a sysem o irs order equaions! Deine! hen! v = a + b + c = d = and! w = The second equaion is obained rom! a v + b v + c w = d w v = 0 Thus! Is equivalen o:! where! a + b + c = d a v + b v + c w = d w v = 0 v = w = Any high-order PDE can be rewrien as a sysem o irs order equaions!! a v + b v + c w = d In mari orm! or! v + b a c a v = d a w 1 0 w 0 u + Au y = s w v = 0 u = v w Are here lines in he -y plane, along which he soluion is deermined by an ordinary dierenial equaion?!

6 The oal derivaive is! dv d = v + v = v +α v where! Rae o change o v wih, along he line y=y()! I here are lines (deermined by α) where he soluion is governed by ODEʼs, hen i mus be possible o rewrie he equaions in such a way ha he resul conains only α and he oal derivaives.! α = Add he original equaions:! v + b v a + c a Is his ever equal o! w + l w v v + α v w + l + α w d = a For some lʼs and α = d a Compare he erms:! v + b v a + c a w + l w v v + α v + l w w +α Thereore, we mus have:! b a l = α c a = l α = d a = d a Characerisic lines eis i:! Or, in mari orm:! b a α c a b a l = α c a = l α 1 l 1 = 0 α 0 l The original equaion is:! v + b a c a v = d a w 1 0 w 0 A! b a α 1 = 0 Rewrie! c a α 0 as! l l b a 1 α 0 = 0 c a 0 0 α 0 A T! l ( A T αi )l = 0 The equaion has a soluion only i he deerminan is zero! b a α 1 c a α l The deerminan is:! A T αi = 0 α b a α + c a = 0 Or, solving or α! α = 1 ( a b ± b 4ac) = 0 0

7 α = 1 ( a b ± b 4ac) b 4ac > 0 Two real characerisics! Eamples! b 4ac = 0 One real characerisics! b 4ac < 0 No real characerisics! Eamples! Eamples! c = 0 Comparing wih he sandard orm! a + b + c = d shows ha! a = 1; b = 0; c = c ; d = 0; b 4ac = c = 4c > 0 Hyperbolic! + = 0 Comparing wih he sandard orm! a + b + c = d shows ha! a = 1; b = 0; c = 1; d = 0; b 4ac = = 4 < 0 Ellipic! Eamples! Eamples! = D Comparing wih he sandard orm! a + b + c = d shows ha! a = 0; b = 0; c = D; d = 0; b 4ac = D = 0 Parabolic! Diusion equaion! Laplace equaion! c = 0 Hyperbolic! = D Parabolic! + = 0 Ellipic!

8 α = α = 1 ( a b ± b 4ac) Wave Equaion! Hyperbolic! Parabolic! Ellipic! c = 0 Firs wrie he equaion as a sysem o irs order equaions! Inroduce! yielding! u = ; v = ; c v = 0 v = 0 since! rom he pde! = To ind he characerisics! c v = 0 v = 0 v ( ) = de de A T αi We can also use! α = 1 a b ± b 4ac α c α = ±c + 0 c α = α c v ( ) = 0 ( ) wih! b = 0; a =1; c = c = 0 Two characerisic lines! P d = + 1 c ; d = 1 c d = + 1 c d = 1 c To ind he soluion we need α = ±c o ind he eigenvecors! v c = 0 α 1 v +l c α = 0 Take =1! For! α = +c For! α = c c l = 0 l = c +c l = 0 l = +c l = 0

9 For! α = +c =1 l = c 1 c v = 0 c v = 0 + c c v + c v = 0 du c dv = 0 on d = +c Similarly:! For! α = c =1 l = +c du + c dv = 0 on d = c Add he equaions! Relaion beween he oal derivaive on he characerisic! du c dv = 0 on d = +c du + c dv = 0 on d = c For consan c! dr 1 = 0 on d = +c where r = u cv 1 dr = 0 on d = c where r = u + cv r 1 and r are called he Rieman invarians! Wave equaion-general! The general soluion can hereore be wrien as:! (,) = r 1 ( c) + r ( + c) Two characerisic lines! d = + 1 c ; d = 1 c where! r 1 ( ) = c r ( ) = + c = 0 = 0 P d = + 1 c d = 1 c Can also be veriied by direc subsiuion! hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-III! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and Nonlinear Advecion Equaions! Quasi-linear Second Order Equaions!!- Classiicaion: hyperbolic, parabolic, ellipic! - Domain o Dependence/Inluence! Ill-Posed Problems! Navier-Sokes equaions!

10 Ill-posed problems! Consider he iniial value problem:! = This is simply Laplaceʼs equaion! + = 0 which has a soluion i / or are given on he boundaries! Here, however, his equaion appeared as an iniial value problem, where he only boundary condiions available are a = 0. Since his is a second order equaion we will need wo condiions, which we may assume are ha and / are speciied a =0.! The general soluion can be wrien as:! ( ) = a k ()e ik, Look or soluions o he ype:! = a k ()e ik k where he aʼs depend on he iniial condiions! Subsiue ino:! + = 0 o ge:! d a k = k a k d a k = k a k General soluion! a k () = Ae k + Be k A, B deermined by iniial condiions! a( 0); ika(0) Long wave wih shor wave perurbaions! Generally, boh A and B are non-zero! Thereore:! a() as! Ill-posed Problem!

11 Ill-posed problems generally appear when he iniial or boundary daa and he equaion ype do no mach.! Frequenly arise because small bu imporan higher order eecs have been negleced! Summary! Ill-posedness generally maniess isel in he eponenial growh o small perurbaions so ha he soluion does no depend coninuously on he iniial daa! Summary! Navier-Sokes equaions! Why is he classiicaion Imporan?! 1. Iniial and boundary condiions!. Dieren physics! 3. Dieren numerical mehod apply! Parabolic par! + u + v = 1 P ρ + µ u ρ + u + v = 0 Hyperbolic par! Ellipic equaion! Summary! The Navier-Sokes equaions conain hree equaion ypes ha have heir own characerisic behavior! Depending on he governing parameers, one behavior can be dominan! The dieren equaion ypes require dieren soluion echniques! For inviscid compressible lows, only he hyperbolic par survives! Summary! In he ne several lecures we will discuss numerical soluions echniques or each class:! Hyperbolic equaions, including! soluions o he Euler equaions! Parabolic equaions! Ellipic equaions! Then we will consider advecion/diusion equaions and he special consideraions needed here.! Finally we will reurn o he ull Navier-Sokes equaions!