Riemann Solvers and Numerical Methods for Fluid Dynamics

Transcription

1 Riemann Solvers and Nmerical Mehods for Flid Dynamics A Pracical Inrodcion Bearbeie von Eleerio F Toro 3rd ed. 29. Bch. iv, 724 S. Hardcover ISBN Forma (B L): 15,5 23,5 cm Gewich: 269 g Weiere Fachgebiee > Technik > Werksoffknde, Mechanische Technologie > Srömngslehre Z Inhalsverzeichnis schnell nd porofrei erhällich bei Die Online-Fachbchhandlng beck-shop.de is spezialisier af Fachbücher, insbesondere Rech, Seern nd Wirschaf. Im Sorimen finden Sie alle Medien (Bücher, Zeischrifen, CDs, ebooks, ec.) aller Verlage. Ergänz wird das Programm drch Services wie Neerscheinngsdiens oder Zsammensellngen von Büchern z Sonderpreisen. Der Shop führ mehr als 8 Millionen Prodke.

2 2 Noions on Hyperbolic Parial Differenial Eqaions In his chaper we sdy some elemenary properies of a class of hyperbolic Parial Differenial Eqaions (PDEs). The seleced aspecs of he eqaions are hose hogh o be essenial for he analysis of he eqaions of flid flow and he implemenaion of nmerical mehods. For general backgrond on PDEs we recommend he book by John [272] and pariclarly he one by Zachmanoglo and Thoe [596]. The discreisaion echniqes sdied in his book are srongly based on he nderlying Physics and mahemaical properies of PDEs. I is herefore jsified o devoe some effor o some fndamenals on PDEs. Here we deal almos eclsively wih hyperbolic PDEs and hyperbolic conservaion laws in pariclar. There are hree main reasons for his: (i) The eqaions of compressible flid flow redce o hyperbolic sysems, he Eler eqaions, when he effecs of viscosiy and hea condcion are negleced. (ii) Nmerically, i is generally acceped ha he hyperbolic erms of he PDEs of flid flow are he erms ha pose he mos sringen reqiremens on he discreisaion echniqes. (iii) The heory of hyperbolic sysems is mch more advanced han ha for more complee mahemaical models, sch as he Navier Sokes eqaions. In addiion, here has in recen years been a noiceable increase in research and developmen aciviies cenred on he heme of hyperbolic problems, as hese cover a wide range of areas of scienific and echnological ineres. A good sorce of p o dae work in his field is fond in he proceedings of he series of meeings on Hyperbolic Problems, see for insance [87], [184], [213]. See also [326]. Oher relevan pblicaions are hose of Godlewski and Raviar [215], Hörmander [258] and Tveio and Winher [551]. We resric orselves o some of he basics on hyperbolic PDEs and choose an informal way of presenaion. The seleced opics and approach are almos eclsively moivaed by he heme of he book, namely he Riemann problem and high resolion pwind and cenred nmerical mehods. E.F. Toro, Riemann Solvers and Nmerical Mehods for Flid Dynamics, 41 DOI 1.17/b , c Springer-Verlag Berlin Heidelberg 29

3 42 2 Noions on Hyperbolic Parial Differenial Eqaions 2.1 Qasi Linear Eqaions: Basic Conceps In his secion we sdy sysems of firs order parial differenial eqaions of he form m i + a ij (,, 1,..., m ) j + b i(,, 1,..., m )=, (2.1) j=1 for i =1,...,m. This is a sysem of m eqaions in m nknowns i ha depend on space and a ime like variable. Here i are he dependen variables and, are he independen variables; his is epressed via he noaion i = i (, ); i / denoes he parial derivaive of i (, ) wih respec o ; similarly i / denoes he parial derivaive of i (, ) wih respec o. We also make se of sbscrips o denoe parial derivaives. Sysem (2.1) can also be wrien in mari form as wih U + AU + B =, (2.2) 1 b 1 a a 1m 2 U =., B = b 2., A = a a 2m.... (2.3) m b m a m1... a mm If he enries a ij of he mari A are all consan and he componens b j of he vecor B are also consan hen sysem (2.2) is linear wih consan coefficiens. If a ij = a ij (, ) andb i = b i (, ) he sysem is linear wih variable coefficiens. The sysem is sill linear if B depends linearly on U and is called qasi linear if he coefficien mari A is a fncion of he vecor U, ha is A = A(U). Noe ha qasi linear sysems are in general sysems of non linear eqaions. Sysem (2.2) is called homogeneos if B =. For a se of PDEs of he form (2.2) one needs o specify he range of variaion of he independen variables and. Usally lies in a sbinerval of he real line, namely l << r ; his sbinerval is called he spaial domain of he PDEs, or js domain. A he vales l, r one also needs o specify Bondary Condiions (BCs). In his Chaper we assme he domain is he fll real line, <<, and hs no bondary condiions need o be specified. As o variaions of ime we assme <<. AnIniial Condiion (IC) needs o be specified a he iniial ime, which is sally chosen o be =. Two scalar (m = 1) eamples of PDEs of he form (2.1) are he linear advecion eqaion + a = (2.4) and he inviscid Brgers eqaion + =, (2.5)

4 2.1 Qasi Linear Eqaions: Basic Conceps 43 boh inrodced in Sec of Chap. 1. In he linear advecion eqaion (2.4) he coefficien a (a consan) is he wave propagaion speed. In he Brgers eqaion a = a() =. Definiion 2.1 (Conservaion Laws). Conservaion laws are sysems of parial differenial eqaions ha can be wrien in he form U + F(U) =, (2.6) where 1 f 1 2 U =., F(U) = f 2.. (2.7) m U is called he vecor of conserved variables, F = F(U) is he vecor of fles and each of is componens f i is a fncion of he componens j of U. Definiion 2.2 (Jacobian Mari). The Jacobian of he fl fncion F(U) in (2.6) is he mari f 1 / 1... f 1 / m f 2 / 1... f 2 / m A(U) = F/ U =.... (2.8) f m / 1... f m / m The enries a ij of A(U) are parial derivaives of he componens f i of he vecor F wih respec o he componens j of he vecor of conserved variables U, ha is a ij = f i / j. Noe ha conservaion laws of he form (2.6) (2.7) can also be wrien in qasi linear form (2.2), wih B, by applying he chain rle o he second erm in (2.6), namely F(U) = F U U. Hence (2.6) becomes U + A(U)U =, which is a special case of (2.2). The scalar PDEs (2.4) and (2.5) can be epressed as conservaion laws, namely f m + f() =,f() =a, (2.9) + f() =,f() = (2.1)

5 44 2 Noions on Hyperbolic Parial Differenial Eqaions Definiion 2.3 (Eigenvales). The eigenvales λ i of a mari A are he solions of he characerisic polynomial A λi = de(a λi) =, (2.11) where I is he ideniy mari. The eigenvales of he coefficien mari A of a sysem of he form (2.2) are called he eigenvales of he sysem. Physically, eigenvales represen speeds of propagaion of informaion. Speeds will be measred posiive in he direcion of increasing and negaive oherwise. Definiion 2.4 (Eigenvecors). A righ eigenvecor of a mari A corresponding o an eigenvale λ i of A is a vecor K (i) =[k (i) 1,k(i) 2,...,k(i) m ] T saisfying AK (i) = λ i K (i). Similarly, a lef eigenvecor of a mari A corresponding o an eigenvale λ i of A is a vecor L (i) =[l (i) 1,l(i) 2,...,l(i) m ] sch ha L (i) A = λ i L (i). For he scalar eamples (2.9) (2.1) he eigenvales are rivially fond o be λ = a and λ = respecively. Ne we find eigenvales and eigenvecors for a sysem of PDEs. Eample 2.5 (Linearised Gas Dynamics). The linearised eqaions of Gas Dynamics, derived in Sec of Chap. 1, are he 2 2 linear sysem ρ + ρ =, + a2 ρ ρ =, (2.12) where he nknowns are he densiy 1 = ρ(, ) and he speed 2 = (, ); ρ is a consan reference densiy and a is he sond speed, a posiive consan. When wrien in he mari form (2.2) his sysem reads wih U = [ 1 2 U + AU =, (2.13) ] [ ] [ ρ, A = ρ a 2 /ρ ]. (2.14) The eigenvales of he sysem are he zeros of he characerisic polynomial [ ] λ ρ A λi = de a 2 =. /ρ λ Tha is, λ 2 = a 2, which has wo real and disinc solions, namely λ 1 = a, λ 2 =+a. (2.15) We now find he righ eigenvecors K (1), K (2) corresponding o he eigenvales λ 1 and λ 2.

6 2.1 Qasi Linear Eqaions: Basic Conceps 45 The eigenvecor K (1) for eigenvale λ = λ 1 = a is fond as follows: we look for a vecor K (1) =[k 1,k 2 ] T sch ha K (1) is a righ eigenvecor of A, ha is AK (1) = λ 1 K (1). Wriing his in fll gives [ ρ a 2 /ρ ][ k1 k 2 ] [ ] ak1 =, ak 2 which prodces wo linear algebraic eqaions for he nknowns k 1 and k 2 a 2 ρ k 2 = ak 1, k 1 = ak 2. (2.16) ρ The reader will realise ha in fac hese wo eqaions are eqivalen and so effecively we have a single linear algebraic eqaion in wo nknowns. This gives a one parameer family of solions. Ths we selec an arbirary non zero parameer α 1, a scaling facor, and se k 1 = α 1 in any of he eqaions o obain k 2 = α 1 a/ρ for he second componen and hence he firs righ eigenvecor becomes [ ] K (1) 1 = α 1. (2.17) a/ρ The eigenvecor K (2) for eigenvale λ = λ 2 =+ais fond in a similar manner. The resling algebraic eqaions for K (2) corresponding o he eigenvale λ 2 =+a are a 2 ρ k 2 = ak 1, k 1 = ak 2. (2.18) ρ By denoing he second scaling facor by α 2 and seing k 1 = α 2 we obain [ K (2) 1 = α 2. (2.19) a/ρ Taking he scaling facors o be α 1 = ρ and α 2 = ρ gives he righ eigenvecors K (1) = [ ] ρ a ], K (2) = [ ρ a ]. (2.2) Definiion 2.6 (Hyperbolic Sysem). A sysem (2.2) is said o be hyperbolic a a poin (, ) if A has m real eigenvales λ 1,...,λ m and a corresponding se of m linearly independen righ eigenvecors K (1),...,K (m).the sysem is said o be sricly hyperbolic if he eigenvales λ i are all disinc. Noe ha sric hyperboliciy implies hyperboliciy, becase real and disinc eigenvales ensre he eisence of a se of linearly independen eigenvecors. The sysem (2.2) is said o be ellipic a a poin (, ) ifnoneofhe eigenvales λ i of A are real. Boh scalar eamples (2.9) (2.1) are rivially hyperbolic. The linearised gas dynamic eqaions (2.12) are also hyperbolic, since λ 1 and λ 2 are boh real a any poin (, ). Moreover, as he eigenvales are also disinc his sysem is sricly hyperbolic.

7 46 2 Noions on Hyperbolic Parial Differenial Eqaions Eample 2.7 (The Cachy Riemann Eqaions). An eample of a firs order sysem of he form (2.2) wih replaced by and replaced by y is he Cachy Riemann eqaions v y =, v + =, (2.21) y where 1 = (, y) and 2 = v(, y). These eqaions arise in he sdy of analyic fncions in Comple Analysis [379]. When wrien in mari noaion (2.2) eqaions (2.21) become U + AU y =, (2.22) wih U = [ ], A = v [ ] 1. (2.23) 1 The characerisic polynomial A λi = gives λ =, which has no real solions for λ and hs he sysem is ellipic. Eample 2.8 (The Small Perrbaion Eqaions). In Sec of Chap. 1, he small perrbaion seady eqaions were inrodced a 2 v y =, v y =, (2.24) wih a 2 1 = M 2 1. (2.25) M = consan denoes he free sream Mach nmber and (, y), v(, y) are small perrbaions of he and y velociy componens respecively. In mari noaion hese eqaions read U + AU y =, (2.26) wih [ ] [ ] a 2 U =, A =. (2.27) v 1 The characer of hese eqaions depends enirely on he vale of he Mach nmber M. For sbsonic flow M < 1 he characerisic polynomial has comple solions and hs he eqaions are of ellipic ype. For spersonic flow M > 1 and he sysem is sricly hyperbolic, wih eigenvales λ 1 = a, λ 2 =+a. (2.28) I is lef o he reader o check ha he corresponding righ eigenvecors are K (1) = α 1 [ 1 1/a ], K (2) = α 2 [ 1 1/a ], (2.29) where α 1 and α 2 are wo non zero scaling facors. By aking he vales α 1 = α 2 = a we obain he following epressions for he righ eigenvecors [ ] [ ] K (1) a =, K (2) a =, 1 1

8 2.2 The Linear Advecion Eqaion 2.2 The Linear Advecion Eqaion 47 A general, ime dependen linear advecion eqaion in hree space dimensions reads + a(, y, z, ) + b(, y, z, ) y + c(, y, z, ) z =, (2.3) where he nknown is = (, y, z, ) anda, b, c are variable coefficiens. If he coefficiens are sfficienly smooh one can epress (2.3) as a conservaion law wih sorce erms, namely +(a) +(b) y +(c) z = (a + b y + c z ). (2.31) In his secion we sdy in deail he iniial vale problem (IVP) for he special case of he linear advecion eqaion, namely PDE: + a =, <<,>. (2.32) IC: (, ) = (), where a is a consan wave propagaion speed. The iniial daa a ime = is a fncion of alone and is denoed by (). We warn he reader ha for sysems we shall se a differen noaion for he iniial daa. Generally, we shall no be eplici abo he condiions << ; >onhe independen variables when saing an iniial vale problem. The PDE in (2.32) is he simples hyperbolic PDE and in view of (2.9) is also he simples hyperbolic conservaion law. I is a very sefl model eqaion for he prpose of sdying nmerical mehods for hyperbolic conservaion laws, in he same way as he linear, firs order ordinary differenial eqaion d = β,= (),β= consan, (2.33) d is a poplar model eqaion for analysing nmerical mehods for Ordinary Differenial Eqaion (ODEs). Two sefl references on ordinary differenial eqaions are Brown [81] and Lamber [296]. In Sec of Chap. 15 we sdy nmerical mehods for ODEs in connecion wih sorce erms in inhomogeneos PDEs Characerisics and he General Solion We recall he definiion of characerisics or characerisic crves in he cone of a scalar eqaion sch as ha in (2.32). Characerisics may be defined as crves = () inhe plane along which he PDE becomes an ODE. Consider = () and regard as a fncion of, hais = ((),). The rae of change of along = () is

9 48 2 Noions on Hyperbolic Parial Differenial Eqaions d d = + d d. (2.34) If he characerisic crve = () saisfies he ODE d = a, (2.35) d hen he PDE in (2.32), ogeher wih (2.34) and (2.35), gives d d = + a =. (2.36) Therefore he rae of change of along he characerisic crve = () saisfying (2.35) is zero, ha is, is consan along he crve = (). The speed a in (2.35) is called he characerisic speed and according o (2.35) i is he slope of he crve = () inhe plane. In pracice i is more common o se he plane o skech he characerisics, in which case he slope of he crves in qesion is 1/a. The family of characerisic crves = () given by Characerisic crve = + a Iniial poin Fig Picre of characerisics for he linear advecion eqaion for posiive characerisic speed a. Iniial condiion a ime = fies he iniial posiion he ODE (2.35) are illsraed in Fig. 2.1 for a> and are a one parameer family of crves. A pariclar member of his family is deermined when an iniial condiion (IC) a ime = for he ODE (2.35) is added. Sppose we se () =, (2.37) hen he single characerisic crve ha passes hrogh he poin (, ), according o (2.35) is = + a. (2.38) This is also illsraed in Fig Now we may regard he iniial posiion as a parameer and in his way we reprodce he fll one parameer family

10 2.2 The Linear Advecion Eqaion 49 of characerisics. The fac ha he crves are parallel is ypical of linear hyperbolic PDEs wih consan coefficiens. Recall he conclsion from (2.36) ha remains consan along characerisics. Ths, if is given he iniial vale (, ) = () a ime =, hen along he whole characerisic crve () = + a ha passes hrogh he iniial poin on he ais, he solion is (, ) = ( )= ( a). (2.39) The second eqaliy follows from (2.38). The inerpreaion of he solion (2.39) of he PDE in (2.32) is his: given an iniial profile (), he PDE will simply ranslae his profile wih velociy a o he righ if a> and o he lef if a<. The shape of he iniial profile remains nchanged. The model eqaion in (2.32) nder sdy conains some of he basic feares of wave propagaion phenomena, where a wave is ndersood as some recognisable feare of a disrbance ha ravels a a finie speed The Riemann Problem By sing geomeric argmens we have consrced he analyical solion of he general IVP (2.32) for he linear advecion eqaion. This is given by (2.39) in erms of he iniial daa (). Now we sdy a special IVP called he Riemann problem PDE: + a =. IC: (, ) = () = { L if <, R if >, (2.4) where L (lef) and R (righ) are wo consan vales, as shown in Fig Noe ha he iniial daa has a disconiniy a =. IVP (2.4) is he simples iniial vale problem one can pose. The rivial case wold resl when L = R. From he previos discssion on he solion of he general IVP (2.32) we epec any poin on he iniial profile o propagae a disance d = a in ime. In pariclar, we epec he iniial disconiniy a = o propagae a disance d = a in ime. This pariclar characerisic crve = a will hen separae hose characerisic crves o he lef, on which he solion akes on he vale L, from hose crves o he righ, on which he solion akes on he vale R ; see Fig So he solion of he Riemann problem (2.4) is simply (, ) = ( a) = { L if a <, R if a >. (2.41) Solion (2.41) also follows direcly from he general solion (2.39), namely (, ) = ( a). From (2.4), ( a) = L if a < and ( a) =

11 5 2 Noions on Hyperbolic Parial Differenial Eqaions () L R = Fig Illsraion of he iniial daa for he Riemann problem. A he iniial ime he daa consiss of wo consan saes separaed by a disconiniy a = R if a >. The solion of he Riemann problem can be represened in he plane, as shown in Fig Throgh any poin on he ais one can draw a characerisic. As a is consan hese are all parallel o each oher. For he solion of he Riemann problem he characerisic ha passes hrogh = is significan. This is he only one across which he solion changes. Characerisic -a = - a < L - a > R Fig Illsraion of he solion of he Riemann problem in he plane for he linear advecion eqaion wih posiive characerisic speed a 2.3 Linear Hyperbolic Sysems In he previos secion we sdied in deail he behavior and he general solion of he simples PDE of hyperbolic ype, namely he linear advecion

12 2.3 Linear Hyperbolic Sysems 51 eqaion wih consan wave propagaion speed. Here we eend he analysis o ses of m hyperbolic PDEs of he form U + AU =, (2.42) where he coefficien mari A is consan. From he assmpion of hyperboliciy A has m real eigenvales λ i and m linearly independen eigenvecors K (i), i =1,...,m Diagonalisaion and Characerisic Variables In order o analyse and solve he general IVP for (2.42) i is fond sefl o ransform he dependen variables U(, ) o a new se of dependen variables W(, ). To his end we recall he following definiion Definiion 2.9 (Diagonalisable Sysem). A mari A is said o be diagonalisable if A can be epressed as A = KΛK 1 or Λ = K 1 AK, (2.43) in erms of a diagonal mari Λ and a mari K. The diagonal elemens of Λ are he eigenvales λ i of A and he colmns K (i) of K are he righ eigenvecors of A corresponding o he eigenvales λ i, ha is λ Λ =..., K =[K(1),...,K (m) ], AK (i) = λ i K (i). (2.44)... λ m A sysem (2.42) is said o be diagonalisable if he coefficien mari A is diagonalisable. Based on he concep of diagonalisaion one ofen defines a hyperbolic sysem (2.42) as a sysem wih real eigenvales and diagonalisable coefficien mari. Characerisic variables The eisence of he inverse mari K 1 makes i possible o define a new se of dependen variables W =(w 1,w 2,...,w m ) T via he ransformaion W = K 1 U or U = KW, (2.45) so ha he linear sysem (2.42), when epressed in erms of W, becomes compleely decopled, in a sense o be defined. The new variables W are called characerisic variables. Ne we derive he governing PDEs in erms of he characerisic variables, for which we need he parial derivaives U and U in eqaions (2.42). Since A is consan, K is also consan and herefore hese derivaives are

13 52 2 Noions on Hyperbolic Parial Differenial Eqaions U = KW, U = KW. Direc sbsiion of hese epressions ino eqaion (2.42) gives KW + AKW =. Mliplicaion of his eqaion from he lef by K 1 and se of (2.43) gives W + ΛW =. (2.46) This is is called he canonical form or characerisic form of sysem (2.42). When wrien in fll his sysem becomes w 1 λ 1... w 1 w 2... w =. (2.47). w m... λ m w m Clearly he i h PDE of his sysem is w i + λ w i i =,i=1,...,m (2.48) and involves he single nknown w i (, ); he sysem is herefore decopled and is idenical o he linear advecion eqaion in (2.32); now he characerisic speed is λ i andherearem characerisic crves saisfying m ODEs d d = λ i, for i =1,...,m. (2.49) The General Iniial Vale Problem We now sdy he IVP for he PDEs (2.42). The iniial condiion is now denoed by sperscrip (), namely U () =( () 1,..., () m ) T, raher han by sbscrip, as done for he scalar case. We find he general solion of he IVP by firs solving he corresponding IVP for he canonical sysem (2.46) or (2.47) in erms of he characerisic variables W and iniial condiion W () =(w () 1,...,w() m ) T sch ha W () = K 1 U () or U () = KW (). The solion of he IVP for (2.46) is direc. By considering each nknown w i (, ) saisfying (2.48) and is corresponding iniial daa w () i we wrie is solion immediaely as w i (, ) =w () i ( λ i ),fori =1,...,m. (2.5) Compare wih solion (2.39) for he scalar case. The solion of he general IVP in erms of he original variables U is now obained by ransforming back according o (2.45), namely U = KW.

14 2 2.3 Linear Hyperbolic Sysems 53 Eample 2.1 (Linearised Gas Dynamics Revisied). As a simple eample we now sdy he general IVP for he linearised eqaions of Gas Dynamics (2.12), namely [ ] [ ][ ] 1 ρ + 1 a 2 =, /ρ 1 ρ, 2, wih iniial condiion We define characerisic variables 2 [ ] [ ] 1 (, ) () = 1 () 2 (, ) () 2 () W =(w 1,w 2 ) T = K 1 U, where K is he mari of righ eigenvecors and K 1 is is inverse, boh given by [ ] ρ ρ K =, K 1 = 1 [ ] a ρ. a a 2aρ a ρ Since λ 1 = a and λ 2 = a, in erms of he characerisic variables we may wrie [ ] [ ][ ] w1 a w1 + =, w 2 a w 2 or in fll w 1 a w 1 =, w 2 + a w 2 =. The iniial condiion saisfies [ ] [ ] = K 1 or in fll w () 1 w () 2 () 1 () 2 [ ] w () 1 () = 1 2aρ a () 1 () ρ () 2 (), [ ] w () 2 () = 1 2aρ a () 1 ()+ρ () 2 (). Each eqaion involves a single independen variable and is a linear advecion eqaion of he form (2.48). The solion for w 1 and w 2 in erms of heir iniial daa w (), according o (2.5) is 1, w() 2 w 1 (, ) =w () 1 ( + a), w 2(, ) =w () 2 ( a),,. or in fll w 1 (, ) = 1 2aρ [ a () 1 ( + a) ρ () 2 ( + a) ],

15 54 2 Noions on Hyperbolic Parial Differenial Eqaions w 2 (, ) = 1 [ ] a () 1 2aρ ( a)+ρ () 2 ( a). This is he solion in erms of he characerisic variables. In order o obain he solion o he original problem we ransform back sing U = KW. This gives he final solion as 1 (, ) = 1 2a [ a () 1 ( + a) ρ () 2 ( + a) ] [ ] + 1 2a a () 1 ( a)+ρ () 2 ( a), 2 (, ) = 1 2ρ [ a () 1 ( + a) ρ () 2 ( + a) ] [ ] + 1 2ρ a () 1 ( a)+ρ () 2 ( a). Eercise Find he solion of he general IVP for he Small Perrbaion Eqaions (2.24) sing he above mehodology. Solion (Lef o he reader). We rern o he epression U = KW in (2.45) sed o recover he solion o he original problem. When wrien in fll his epression becomes or 1 2. = w 1 m or more sccincly 1 = w 1 k (1) 1 + w 2 k (2) w m k (m) 1, i = w 1 k (1) i + w 2 k (2) i w m k (m) i, m = w 1 k m (1) + w 2 k m (2) w m k m (m), k (1) 1 k (1) 2.. k (1) m + w 2 U(, ) = k (2) 1 k (2) 2.. k (2) m w m k (m) 1 k (m) 2.. k (m) m, (2.51) m w i (, )K (i). (2.52) i=1 This means ha he fncion w i (, ) is he coefficien of K (i) in an eigenvecor epansion of he vecor U. B according o (2.5), w i (, ) =w () i ( λ i ) and hence m U(, ) = w () i ( λ i )K (i). (2.53) i=1 Ths, given a poin (, ) inhe plane, he solion U(, ) a his poin depends only on he iniial daa a he m poins (i) = λ i. These are he

16 2.3 Linear Hyperbolic Sysems 55 inersecions of he characerisics of speed λ i wih he ais. The solion (2.53) for U can be seen as he sperposiion of m waves, each of which is adveced independenly wiho change in shape. The i h wave has shape w () i ()K (i) and propagaes wih speed λ i The Riemann Problem We sdy he Riemann problem for he hyperbolic, consan coefficien sysem (2.42). This is he special IVP PDEs: U + AU =, <<,>, { (2.54) IC: U(, ) = U () UL <, () = U R > and is a generalisaion of he IVP (2.32). We assme ha he sysem is sricly hyperbolic and we order he real and disinc eigenvales as λ 1 <λ 2 <...<λ m. (2.55) The General Solion The srcre of he solion of he Riemann problem (2.54) in he plane is depiced in Fig I consiss of m waves emanaing from he origin, one for each eigenvale λ i. Each wave i carries a jmp disconiniy in U propagaing wih speed λ i. Narally, he solion o he lef of he λ 1 wave is simply he iniial daa U L and o he righ of he λ m wave is U R. The ask a hand is o find he solion in he wedge beween he λ 1 and λ m waves. As he eigenvecors K (1),...,K (m) are linearly independen, we λ λ λ i 2 λm-1 1 λ m Lef daa U L Righ daa U R Fig Srcre of he solion of he Riemann problem for a general m m linear hyperbolic sysem wih consan coefficiens

17 56 2 Noions on Hyperbolic Parial Differenial Eqaions can epand he daa U L, consan lef sae, and U R, consan righ sae, as linear combinaions of he se K (1),...,K (m), ha is m m U L = α i K (i), U R = β i K (i), (2.56) i=1 wih consan coefficiens α i, β i,fori =1,...,m. Formally, he solion of he IVP (2.54) is given by (2.53) in erms of he iniial daa w () i () for he characerisic variables and he righ eigenvecors K (i). Noe ha each of he epansions in (2.56) is a special case of (2.53). In erms of he characerisic variables we have m scalar Riemann problems for he PDEs w i + λ w i i =, (2.57) wih iniial daa obained by comparing (2.56) wih (2.53), ha is i=1 w () i () = { αi if <, β i if >, (2.58) for i =1,...,m. From he previos resls, see eqaion (2.5), we know ha he solions of hese scalar Riemann problems are given by w i (, ) =w () i ( λ i )= { αi if λ i <, β i if λ i >. (2.59) For a given poin (, ) here is an eigenvale λ I sch ha λ I < <λ I+1, ha is λ i > i sch ha i I. We can hs wrie he final solion o he Riemann problem (2.54) in erms of he original variables as U(, ) = m i=i+1 α i K (i) + I β i K (i), (2.6) where he ineger I = I(, ) is he maimm vale of he sb inde i for which λ i >. i=1 The Solion for a 2 2 Sysem As an eample consider he Riemann problem for a general 2 2 linear sysem. From he origin (, ) in he (, ) plane here will be wo waves ravelling wih speeds ha are eqal o he characerisic speeds λ 1 and λ 2 (λ 1 <λ 2 ); see Fig The solion o he lef of d/d = λ 1 is simply he daa sae U L = α 1 K (1) + α 2 K (2) and o he righ of d/d = λ 2 he solion is he consan daa sae U R = β 1 K (1) + β 2 K (2). The wedge beween he λ 1 and λ 2 waves is sally called he Sar Region and he solion here is denoed by U ; is vale is de o he passage of wo waves emerging from

18 2.3 Linear Hyperbolic Sysems 57 λ1 λ 2 U * : solion in sar region P * (,) UL U R (2) (1) Fig Srcre of he solion of he Riemann problem for a 2 2 linear sysem wih consan coefficiens he origin of he iniial disconiniy. From he poin P (, ) we race back he characerisics wih speeds λ 1 and λ 2. These are parallel o hose passing hrogh he origin. The characerisics hrogh P pass hrogh he iniial poins (2) = λ 2 and (1) = λ 1. The coefficiens in he epansion (2.6) for U(, ) are hs deermined. The solion a a poin P has he form (2.6). I is a qesion of choosing he correc coefficiens α i or β i. Selec a ime and a poin L o he lef of he slowes wave so U( L, )=U L, see Fig The solion a he saring poin ( L, ) is obviosly λ 1 Sar region λ 2 * L Lef daa Righ daa Fig The Riemann problem solion fond by ravelling along dashed horizonal line = 2 U L = α i K i = α 1 K (1) + α 2 K (2), i=1

19 58 2 Noions on Hyperbolic Parial Differenial Eqaions i.e. all coefficiens are α s, ha is, he poin ( L, ) lies o he lef of every wave. As we move o he righ of ( L, ) on he horizonal line = we cross he wave d/d = λ 1, hence λ 1 changes from negaive o posiive, see (2.59), and herefore he coefficien α 1 above changes o β 1. Ths he solion in he enire Sar Region, beween he λ 1 and λ 2 waves, is U (, ) =β 1 K (1) + α 2 K (2). (2.61) As we conine moving righ and cross he λ 2 wave he vale λ 2 changes from negaive o posiive and hence he coefficien α 2 in (2.6) and (2.61) changes o β 2, i.e he solion o he righ of he fases wave of speed λ 2 is, rivially, U R = β 1 K (1) + β 2 K (2). Remark From eqaion (2.56) i is easy o see ha he jmp in U across he whole wave srcre in he solion of he Riemann problem is ΔU = U R U L =(β 1 α 1 )K (1) +...+(β m α m )K (m). (2.62) I is an eigenvecor epansion wih coefficiens ha are he srenghs of he waves presen in he Riemann problem. The wave srengh of wave i is β i α i and he jmp in U across wave i, denoed by (ΔU) i,is (ΔU) i =(β i α i )K (i). (2.63) When solving he Riemann problem, someimes i is more sefl o epand he oal jmp ΔU = U R U L in erms of he eigenvecors and nknown wave srenghs δ i = β i α i The Riemann Problem for Linearised Gas Dynamics As an illsraive eample we apply he mehodology described in he previos secion o solve he Riemann problem for he linearised eqaions of Gas Dynamics (2.12) U + AU =, wih U = [ 1 2 ] The eigenvales of he sysem are [ ] [ ρ, A = λ 1 = a, λ 2 =+a, and he corresponding righ eigenvecors are [ ] K (1) ρ =, K (2) = a ρ a 2 /ρ [ ] ρ. a ].

20 2.3 Linear Hyperbolic Sysems 59 Firs we decompose he lef daa sae U L =[ρ L, L ] T in erms of he righ eigenvecors according o eqaion (2.56), namely [ ] [ ] [ ] ρl ρ ρ U L = = α 1 + α L a 2. a Solving for he nknown coefficiens α 1 and α 2 we obain α 1 = aρ L ρ L, α 2 = aρ L + ρ L. 2aρ 2aρ Similarly, by epanding he righ hand daa U R =[ρ R, R ] T in erms of he eigenvecors and solving for he coefficiens β 1 and β 2 we obain β 1 = aρ R ρ R, β 2 = aρ R + ρ R. 2aρ 2aρ Now by sing eqaion (2.61) we find he solion in he sar region as [ ] [ ] [ ] U ρ ρ ρ = = β 1 + α a 2. a Afer some algebraic maniplaions we obain he solion eplicily as ρ = 1 2 (ρ L + ρ R ) 1 2 ( R L )ρ /a, (2.64) = 1 2 ( L + R ) 1 2 (ρ R ρ L )a/ρ. Fig. 2.7 illsraes he solion for ρ(, ) and(, ) a ime =1forhe parameer vales ρ =1,a = 1 and iniial daa ρ L =1, L =,ρ R = 1 2 and R =. The wo symmeric waves ha emerge from he iniial posiion of he disconiniy carry a disconinos jmp in boh densiy ρ and velociy. 1 Densiy profile a = 1 Posiion of iniial disconiniy Velociy profile a = 1 Posiion of lef wave Posiion of righ wave -1 1 Fig Densiy and velociy solion profiles a ime =1 Remark The eac solion (2.64) can be very sefl in esing nmerical mehods for sysems wih disconinos solions.

21 6 2 Noions on Hyperbolic Parial Differenial Eqaions Some Usefl Definiions Ne we recall some sandard definiions associaed wih hyperbolic sysems. Definiion 2.15 (Domain of Dependence). Recall ha for he linear advecion eqaion he solion a a given poin P =(, ) depends solely on he iniial daa a a single poin on he ais. This poin is obained by racing back he characerisic passing hrogh he poin P =(, ).As a maer of fac, he solion a P =(, ) is idenical o he vale of he iniial daa () a he poin. One says ha he domain of dependence of he poin P = (, ) is he poin.fora2 2 sysem he domain of dependence is an inerval [ L, R ] on he ais ha is sbended by he characerisics passing hrogh he poin P =(, ). λ 1 λ 2 * P Domain of L deerminacy * Domain of dependence R Fig Domain of dependence of poin P and corresponding domain of deerminacy, for a 2 by 2 sysem Fig. 2.8 illsraes he domain of dependence for a 2 2 sysem wih characerisic speeds λ 1 and λ 2, wih λ 1 <λ 2. In general, he characerisics of a hyperbolic sysem are crved. For a larger sysem he domain of dependence is deermined by he slowes and fases characerisics and is always a bonded inerval, as he characerisic speeds for hyperbolic sysems are always finie. Definiion 2.16 (Domain of Deerminacy). For a given domain of dependence [ L, R ], he domain of deerminacy is he se of poins (, ), wihin he domain of eisence of he solion U(, ), inwhichu(, ) is solely deermined by iniial daa on [ L, R ]. In Fig. 2.8 we illsrae he domain of deerminacy of an inerval [ L, R ] for he case of a 2 2 sysem wih characerisic speeds λ 1 and λ 2, wih λ 1 <λ 2.

22 2.4 Conservaion Laws 61 Definiion 2.17 (Range of Inflence). Anoher sefl concep is ha of he range of inflence of a poin Q =(, ) on he ais. I is defined as he se of poins (, ) in he plane in which he solion U(, ) is inflenced by iniial daa a he poin Q =(, ). Fig. 2.9 illsraes he range of inflence of a poin Q =(, ) for he case of a 2 2 sysem wih characerisic speeds λ 1 and λ 2, wih λ 1 <λ 2. λ 1 2 Range of inflence λ of poin Q Q Fig Range of inflence of poin Q for a 2 by 2 sysem 2.4 Conservaion Laws The prpose of his secion is o provide he reader wih a sccinc presenaion of some mahemaical properies of hyperbolic conservaion laws. We resric or aenion o hose properies hogh o be essenial o he developmen and applicaion of nmerical mehods for conservaion laws. In Chap. 1 we applied he physical principles of conservaion of mass, momenm and energy o derive ime dependen, mlidimensional non linear sysems of conservaions laws. In his secion we resric orselves o simple model problems. In Sec. 2.1 we advanced he formal definiion of a sysem of m conservaion laws U + F(U) =, (2.65) where U is he vecor of conserved variables and F(U) is he vecor of fles. This sysem is hyperbolic if he Jacobian mari A(U) = F U has real eigenvales λ i (U) and a complee se of linearly independen eigenvecors K (i) (U), i =1,...,m, which we assme o be ordered as λ 1 (U) <λ 2 (U) <,..., <λ m (U), K (1) (U), K (2) (U),..., K (m) (U).

23 62 2 Noions on Hyperbolic Parial Differenial Eqaions I is imporan o noe ha now eigenvales and eigenvecors depend on U, alhogh someimes we shall omi he argmen U Inegral Forms of Conservaion Laws As discssed in Sec. 1.5 of Chap. 1, conservaion laws may be epressed in differenial and inegral form. There are wo good reasons for considering he inegral form (s) of he conservaion laws: (i) he derivaion of he governing eqaions is based on physical conservaion principles epressed as inegral relaions on conrol volmes, (ii) he inegral formlaion reqires less smoohness of he solion, which paves he way o eending he class of admissible solions o inclde disconinos solions. The inegral form has varians ha are worh sdying in deail. Consider a one dimensional ime dependen sysem, sch as he Eler eqaions inrodced in Sec. 1.1 of Chap. 1. Choose a conrol volme V =[ L, R ] [ 1, 2 ] on he plane as shown in Fig The inegral form, see Sec. 1.5, of he 2 Conrol volme 1 L R Fig A conrol volme V =[ L, R] [ 1, 2]on plane eqaion for conservaion of mass in one space dimension is R d ρ(, )d = f( L,) f( R,), d L where f = ρ is he fl. For he complee sysem we have d d R L U(, )d = F(U( L,)) F(U( R,)), (2.66) where F(U) is he fl vecor. This is one version of he inegral form of he conservaion laws: Inegral Form I. The corresponding differenial form reads as (2.65). Anoher version of he inegral form of he conservaion laws is obained by inegraing (2.66) in ime beween 1 and 2, wih 1 2. See Fig Clearly,

24 Conservaion Laws 63 [ d R ] R R U(, )d d = U(, 2 )d U(, 1 )d d L L L and hs (2.66) becomes R L U(, 2 )d = R L U(, 1 )d F(U( R,)) d, F(U( L,)) d (2.67) which we call: Inegral Form II of he conservaion laws. Anoher version of he inegral form of he conservaion laws is obained by inegraing (2.65) in any domain V in space and sing Green s heorem. The resl is [U d F(U)d] =, (2.68) where he line inegraion is performed along he bondary of he domain, in an aniclock wise manner. We call his version Inegral Form III of he conservaion laws. Noe ha Inegral Form II of he conservaion laws is a special case of Inegral Form III, in which he conrol volme V is he recangle [ L, R ] [ 1, 2 ]. A forh inegral form resls from adoping a more mahemaical approach for eending he concep of solion of (2.65) o inclde disconiniies. See Chorin and Marsden [112]. A weak or generalized solion U is reqired o saisfy he inegral relaion + + [φ U + φ F(U)] d d = + φ(, )U(, ) d, (2.69) for all es fncions φ(, ) ha are coninosly differeniable and have compac sppor. A fncion φ(, ) has compac sppor if i vanishes oside some bonded se. Noe ha in (2.69) he derivaives of U(, ) and F(U) have been passed on o he es fncion φ(, ), which is sfficienly smooh o admi hese derivaives. Remark The inegral forms (2.66) (2.69) corresponding o (2.65) are valid for any sysem (2.65), no js for he Eler eqaions. Eamples of Conservaion Laws Scalar conservaion laws (m = 1) in differenial form read + f() =,f() : fl fncion. (2.7) To be able o solve for he conserved variable (, ) he fl fncion f() ms be a compleely deermined algebraic fncion of (, ), and possibly some era parameers of he problem. As seen in Sec. 2.2 he linear advecion

25 64 2 Noions on Hyperbolic Parial Differenial Eqaions eqaion is he simples eample, in which he fl fncion is f() =a, a linear fncion of. The inviscid Brgers s eqaion has fl f() = 1 2 2, a qadraic fncion of. Anoher eample of a conservaion law is he raffic flow eqaion ρ + f(ρ) =, f(ρ) = m (1 ρ )ρ. (2.71) ρ m Here he conserved variable ρ(, ) is a densiy fncion (densiy of moor vehicles), m and ρ m are parameers of he problem, namely he maimm speed of vehicles and he maimm densiy, boh posiive consans. For deails on he raffic flow eqaion see Whiham [582], Zachmanoglo and Thoe [596], Toro [528] and Haberman [232]. An eample of pracical ineres in oil reservoir simlaion is he Bckley-Levere eqaion 2 + f() =,f() = 2 + b(1 ) 2, (2.72) where b is a parameer of he problem. More deails of his eqaion are fond in LeVeqe [38]. Sysems of conservaion laws are consrced, as obvios eamples, from linear sysems U + AU =, wih consan coefficien mari A. The reqired conservaion law form is obained by defining he fl fncion as he prodc of he coefficien mari A and he vecor U, namely Trivially, he Jacobian mari is A. U + F(U) =, F(U) =AU. (2.73) Eample 2.19 (Isohermal Gas Dynamics). The isohermal eqaions of Gas Dynamics, see Sec of Chap. 1, are one eample of a non linear sysem of conservaion laws. These are U = [ 1 2 ] U + F(U) =, [ ] ρ, F = ρ [ f1 f 2 ] [ ] ρ ρ 2 + a 2, ρ (2.74) where a is posiive, consan speed of sond. The Jacobian mari is fond by firs epressing F in erms of he componens 1 ρ and 2 ρ of he vecor U of conserved variables, namely [ ] [ ] f1 F(U) = 2 f 2 2 2/ 1 + a 2. 1 According o (2.8) he Jacobian mari is

26 A(U) = F [ U = 1 ( 2 / 1 ) 2 + a / 1 I is lef o he reader o verify ha he eigenvales of A are 2.4 Conservaion Laws 65 ] [ ] 1 = a λ 1 = a, λ 2 = + a (2.75) and ha he righ eigenvecors are [ ] K (1) 1 = a [, K (2) = 1 + a ], (2.76) where he scaling facors for K (1) and K (2) have been aken o be niy. The isohermal eqaions of Gas Dynamics are hs hyperbolic. Eample 2.2 (Isenropic Gas Dynamics). Anoher non linear eample of a sysem of conservaion laws are he isenropic eqaions of Gas Dynamics U + F(U) =, [ ] [ ] [ ] [ ] (2.77) 1 ρ f1 ρ U =, F = ρ ρ 2, + p 2 ogeher wih he closre condiion, or eqaion of sae (EOS), See Sec of Chap. 1. f 2 p = Cρ γ, C = consan. (2.78) Eercise (i) Find he Jacobian mari, he eigenvales and he righ eigenvecors for he isenropic eqaions (2.77) (2.78). (ii) Show ha for a generalized isenropic EOS, p = p(ρ), he sysem is hyperbolic if and only if p (ρ) >, ha is, he pressre ms be a monoone increasing fncion of ρ. (iii) Show ha he sond speed has he general form a = p (ρ). Solion The eigenvales are λ 1 = a, λ 2 = + a, (2.79) and he righ eigenvecors are [ ] K (1) 1 = a wih he sond speed a as claimed. [, K (2) = 1 + a ], (2.8)

27 66 2 Noions on Hyperbolic Parial Differenial Eqaions Non Lineariies and Shock Formaion Here we sdy some disingishing feares of non linear hyperbolic conservaion laws, sch as wave seepening and shock formaion. We resric or aenion o he iniial vale problem for scalar non linear conservaion laws, namely + f() =,(, ) = (). (2.81) A corresponding inegral form of he conservaion law is d d R L (, )d = f(( L,)) f(( R,)). (2.82) The fl fncion f is assmed o be a fncion of only, which nder cerain circmsances is an inadeqae represenaion of he physical problem being modelled. Relevan physical phenomena of or ineres are shock waves in compressible media. These have viscos dissipaion and hea condcion, in addiion o pre advecion. A more appropriae fl fncion for a model conservaion law wold also inclde a dependence on, so ha he modified conservaion law wold read + f() = α, (2.83) wih α a posiive coefficien of viscosiy. The conservaion law in (2.81) may be rewrien as + λ() =, (2.84) where λ() = df d = f () (2.85) is he characerisic speed. In he sysem case his corresponds o he eigenvales of he Jacobian mari. For he linear advecion eqaion λ() = a, consan. For he inviscid Brgers eqaion λ() =, ha is, he characerisic speed depends on he solion and is in fac idenical o he conserved variable. For he raffic flow eqaion λ() = m (1 2 ρ m ). The behavior of he fl fncion f() has profond conseqences on he behavior of he solion (, ) of he conservaion law iself. A crcial propery is monooniciy of he characerisic speed λ(). There are essenially hree possibiliies: λ() is a monoone increasing fncion of, i.e. dλ() d = λ () =f () > (conve fl) λ() is a monoone decreasing fncion of, i.e. dλ() d = λ () =f () < (concave fl)

28 2.4 Conservaion Laws 67 λ() has erema, for some, i.e. dλ() d = λ () =f () = (non conve, non concave fl). In he case of non linear sysems of conservaion laws he characer of he fl fncion is deermined by he Eqaion of Sae. One speaks of conve, or oherwise, eqaions of sae. See he review paper by Menikoff and Plohr [349]. For he inviscid Brgers eqaion λ () =f () =1>, he fl is conve. For he raffic flow eqaion λ () =f () = 2 m /ρ m <, he fl is concave. Eercise Analyse he characer of he fl fncion for he Bckley Levere eqaion and show ha i is non conve, non concave. Solion (Lef o he reader). We sdy he inviscid IVP (2.81) and for he momen we assme ha he iniial daa (, ) = () is smooh. For some finie ime he solion (, ) will remain smooh. We rewrie he IVP as + λ() =,λ() =f (), (2.86) (, ) = (). Noe ha he PDE in (2.86) is a non linear eension of he linear advecion eqaion in (2.32) in which he characerisic speed is λ() = a = consan. We consrc solions o IVP (2.86) following characerisic crves, in mch he same way as performed for he linear advecion eqaion. Consrcion of Solions on Characerisics Consider characerisic crves = () saisfying he IVP d d = λ(),() =. (2.87) Then, by regarding boh and o be fncions of we find he oal derivaive of along he crve (), namely d d = + λ() =. (2.88) Tha is, is consan along he characerisic crve saisfying he IVP (2.87) and herefore he slope λ() is consan along he characerisic. Hence he characerisic crves are sraigh lines. The vale of along each crve is he vale of a he iniial poin () = and we wrie (, ) = ( ). (2.89)

29 68 2 Noions on Hyperbolic Parial Differenial Eqaions Fig shows a ypical characerisic crve emanaing from he iniial poin on he ais. The slope λ() of he characerisic may hen be evalaed a so ha he solion characerisics crves of IVP (2.87) are = + λ( ( )). (2.9) Relaions (2.89) and (2.9) may be regarded as he analyical solion of IVP (2.86). Noe ha he poin depends on he given poin (, ), see Fig , and hs = (, ). The solion given by (2.89) and (2.9) is implici, which is more apparen if we sbsie from (2.9) ino (2.89) o obain (, ) = ( λ( ( ))). (2.91) Noe ha his solion is idenical in form o he solion (2.39) of he linear advecion eqaion in (2.32). Fig Typical characerisic crves for a non linear hyperbolic conservaion law Ne we verify ha relaions (2.89) and (2.9) acally define he solion. From (2.89) we obain he and derivaives = ( ) From (2.9) he and derivaives are fond o be From (2.93) we obain, = ( ). (2.92) λ( ( )) + [1 + λ ( ( )) ( )] =, [1 + λ ( ( )) ( )] =1. (2.93) λ( ( )) = 1+λ ( ( )) ( ) (2.94) and

30 2.4 Conservaion Laws 69 = 1 1+λ ( ( )) ( ). (2.95) Sbsiion of (2.94) (2.95) ino (2.92) verifies ha and saisfy he PDE in (2.86). Wave Seepening Recall ha in he case of he linear advecion eqaion, in which he characerisic speed is λ() = a = consan, he solion consiss of he iniial daa () ranslaed wih speed a wiho disorion. In he non linear case he characerisic speed λ() is a fncion of he solion iself. Disorions are herefore prodced; his is a disingishing feare of non linear problems. (a) () (1) (2) (3) (4) (5) (1) (2) (3) (4) (5) (b) Fig Wave seepening in a conve, non linear hyperbolic conservaion law: (a) iniial condiion and (b) corresponding picre of characerisics To eplain he wave disorion phenomenon we consider iniial daa () as shown in Fig A smooh iniial profile is shown in Fig. 2.12a along wih five iniial poins (i) and heir corresponding iniial daa vales (i) = ( (i) ). For he momen le s assme ha he fl fncion f() is conve, ha is λ () =f () >. In his case he characerisic speed is an increasing fncion of. Fig. 2.12b shows he characerisics (i) () emanaing from he iniial poins (i) and carrying he consan iniial vales (i) along hem. Given he assmed conve characer of he fl, higher vales of () will ravel faser han lower vales of (). There are wo inervals on he ais where disorions are mos eviden. These are he inervals I E =[ (1),(3) ]andi C =[ (3),(5) ]. In I E he vale (3) will propagae faser han (2) and his in rn will propagae faser ha (1). The orienaion of he respecive characerisics in Fig. 2.12b makes his siaion clear. A a laer ime he iniial daa in I E will have been ransformed ino a broader

31 7 2 Noions on Hyperbolic Parial Differenial Eqaions and flaer profile. We say ha I E is an epansive region. In he epansive region he characerisic speed increases as increases, ha is λ >. By conras he inerval I C is compressive and λ < ; he vale (3) will propagae faser han (4) and his in rn will propagae faser ha (5),asshown by he orienaion of he respecive characerisics in Fig. 2.12b. The compressive region will end o ge seeper and narrower as ime evolves. The wave seepening mechanism will evenally prodce folding over of he solion profile, wih corresponding crossing of characerisics, and riple valed solions. Noe ha he compressive and epansive characer of he daa js described reverses for he case of a concave fl, λ () =f () <. Before crossing of characerisics he single valed solion may be fond following characerisics, as described previosly. When characerisics firs inersec we say ha he wave breaks; he derivaive becomes infinie and his happens a a precise breaking ime b given by b = 1 λ ( ). (2.96) This is confirmed by eqaions (2.94) (2.95). Breaking firs occrs on he characerisic emanaing from = for which λ ( ) is negaive and λ ( ) is a maimm. For deails see Whiham [582]. This is an anomalos siaion ha may be resced by going back o he physical origins of he eqaions and qesioning he adeqacy of he model frnished by (2.81). The improved model eqaion (2.83) says ha he ime rae of change of is no js de o he advecion erm f() b is a compeing balance beween advecion and he diffsion erm α.asshown in Fig. 2.12a in he inerval [ (3),(4) ]hewave seepening effec of f() is opposed by he wave-easing effec of α, which is negaive here. In he inerval [ (4),(5) ] he role of hese conradicory effecs is reversed. The more complee descripion of he physics does no allow folding over of he solion. B raher han working wih he more complee, and herefore more comple, viscos descripion of he problem, i is acally possible o insis on sing he inviscid model (2.81) by allowing disconiniies o be formed as a process of increasing compression, namely shock waves. Frher deails are fond in La [31], Whiham [582] and Smoller [451]. Shock Waves Shock waves in air are small ransiion layers of very rapid changes of physical qaniies sch as pressre, densiy and emperare. The ransiion layer for a srong shock is of he same order of magnide as he mean free pah of he molecles, ha is abo 1 7 m. Therefore replacing hese waves as mahemaical disconiniies is a reasonable approimaion. Very weak shock waves sch as sonic booms, are an ecepion; he disconinos approimaion

32 2.4 Conservaion Laws 71 here can be very inaccrae indeed, see Whiham [582]. For a discssion on shock hickness see Landa and Lifshiz [297], pp We herefore insis on sing he simplified model (2.81) b in is inegral form, e.g. (2.82). Consider a solion (, ) sch ha (, ), f() and heir derivaives are coninos everywhere ecep on a line s = s() on he plane across which (, ) has a jmp disconiniy. Selec wo fied poins L and R on he ais sch ha L <s() < R. Enforcing he conservaion law in inegral form (2.82) on he conrol volme [ L, R ] leads o f(( L,)) f(( R,)) = d d s() L (, )d + d d R s() (, )d. Direc se of formla (1.68) of Chap. 1 yields f(( L,)) f(( R,)) = [(s L,) (s R,)] S + s() L (, )d + R s() (, )d, where (s L,) is he limi of (s(),)asends o s() fromhelef,(s R,) is he limi of (s(),)asends o s() from he righ and S = ds/d is he speed of he disconiniy. As (, ) is bonded he inegrals vanish idenically as s() is approached from lef and righ and we obain f(( L,)) f(( R,)) = [(s L,) (s R,)] S. (2.97) This algebraic epression relaing he jmps Δf = f(( R,)) f(( L,)), Δ = ( R,) ( L,) and he speed S of he disconiniy is called he Rankine Hgonio condiion and is sally epressed as Δf = SΔ. (2.98) For he scalar case considered here one can solve for he speed S as S = Δf Δ. (2.99) Therefore, in order o admi disconinos solions we may formlae he problem in erms of PDEs, which are valid in smooh pars of he solion, and he Rankine Hgonio condiions across disconiniies. Two Eamples of Disconinos Solions Consider he following iniial vale problem for he inviscid Brgers eqaion + f() =,f() = 1 2 2, { (2.1) L if <, (, ) = () = R if >.

33 72 2 Noions on Hyperbolic Parial Differenial Eqaions Firs assme ha L > R. As he fl is conve λ () =f () > he characerisic speeds on he lef are greaer han hose on he righ, ha is λ L λ( L ) > λ R λ( R ). Based on he discssion abo Fig he iniial daa in IVP (2.1) is he ereme case of compressive daa. Crossing of characerisics akes place immediaely, as illsraed in Fig. 2.13b. The L (a) R (b) (c) L R Shock of speed S Fig (a) Compressive disconinos iniial daa (b) picre of characerisics and (c) solion on plane disconinos solion of he IVP is (, ) = { L if S <, R if S >, (2.11) where he speed of he disconiniy is fond from (2.99) as S = 1 2 ( L + R ). (2.12) This disconinos solion is a shock wave and is compressive in nare as discssed previosly and as observed in Fig. 2.13a; i saisfies he following condiion λ( L ) >S>λ( R ), (2.13) which is called he enropy condiion. More deails are fond in Chorin and Marsden [112], LeVeqe [38], Smoller [451], Whiham [582]. Now we assme ha L < R in he IVP (2.1). This daa is he ereme case of epansive daa, for conve f(). A possible mahemaical solion has idenical form as solion (2.11) (2.12) for he compressive daa case. See

34 2.4 Conservaion Laws 73 Fig However, his solion is physically incorrec. The disconiniy has no arisen as he resl of compression, λ L < λ R ; he characerisics diverge from he disconiniy. This solion is called a rarefacion shock, or enropy violaing shock, and does no saisfy he enropy condiion (2.13); i is herefore rejeced as a physical solion. Compare Figs and 2.14; in he compressive case characerisics rn ino he disconiniy. Given he (a) L R (b) Rarefacion shock (c) L Fig (a) Epansive disconinos iniial daa (b) picre of characerisics and (c) rarefacion shock solion on - plane R epansive characer of he daa and based on he discssion on Fig. 2.12, i wold be more reasonable o epec he iniial daa o break p immediaely and o broaden wih ime. This acally gives anoher solion o be discssed ne. Rarefacion Waves Reconsider he IVP (2.1) wih general conve fl fncion f() + f() =, (, ) = () = { L if <, R if >, (2.14) and epansive iniial daa, L < R. As discssed previosly, he enropy violaing solion o his problem is