MATH-459 Nmerical Mehods for Conservaion Laws by Prof. Jan S. Heshaven Solion se : Scalar Conservaion Laws Eercise. The inegral form of he scalar conservaion law + f ) = is given in Eq. below. ˆ 2, 2 ) d = ˆ 2, ) d + ˆ 2 f, )) d ˆ 2 f 2, )) d ) For f ) = a in he conservaion law we are lef wih he classic advecion eqaion on inegral form. We le ) be any inegrable fncion, i.e., ) may inclde disconiniies. ˆ 2 a 2 ) a ) d = a ˆ 2 a) 2 a) d 2) On he lef side we have an inegral in he variable and on he lef side we have an inegral in he variable. How do we compare? Well, le s ry a change of variables so o make he lef side look like he righ side, le's se a = a and a 2 = 2 a so o ge ˆ 2 2 a) a) d = a ˆ 2 Then we realize ha d d a ) = d d a) d = ad so o ge a ˆ 2 2 a) a) d = a a) 2 a) d 3) ˆ 2 a) 2 a) d 4) Now for his inegral o make sense we also need o change he limis o conform wih he change of variable. For = we ge a = a = while for = 2 we se ha 2 a 2 = 2 a = 2 so o arrive a ˆ 2 ˆ 2 a 2 a) a) d = a a) 2 a) d 5) From which we conclde ha for any inegrable fncion ), a) is a solion o he advecion eqaion on inegral form. The imporan hing o noe here is ha when working wih he original inegral eqaion derived from physical principles, here are no silly diereniabiliy reqiremen and disconinos solions are admissible. These admissible solions conine o eiss as solions o he dierenial eqaion derived from he inegral form nder assmpion of smoohness, if a disconiniy arise in he solion, he assmpion of smoohness is no longer valid and or dierenial model of he inegral eqaion breaks down. Eercise.2 From he heory we know ha he solion of he Cachy problem + f ) = wih, ) = ) can become singlar only when he consrcion of he solion by he mehod of characerisics fail. Ths, we begin by consrcing he solion according o he mehod of characerisics, when we will nd where he procedre may fail. The projecion of he characerisic crves ono he, plane is given by = f ), ξ, ) = ξ. On he characerisics, saises = and ξ, ) = ξ). Therefore, in a neighborhood of he iniial condiion we have ξ, ) = f ξ)) + ξ, ξ, ) = ξ) 6) To ge o he solion, ), we ms erac ξ, ) from Eq. 6. To do so, we employ he inverse fncion heorem which saes ha provided ξ, here eiss a smooh fncion ξ, ) sch ha This procedre fails, if a some ξ and > = f ξ, ))) + ξ, ) 7) = ξ = f ξ)) + 8) Therefore, he solion, ) can become singlar only a ξ and > where = f ξ)) ξ) 9)
Eercise.3 In his problem we considered he inviscid brgers eqaion where f ) = 2 2 so ha + = for wo ses of iniial daa. Firs he solion for se < ) = < < ) > is presened. Figre conains a skech of he iniial daa prole ) wih characerisics evolving. We see ha wo shocks are presen in he iniial daa. The ask is o deermine he eac solion, ) for all >. We can deermine he speed of he shocks by sing he Rankine-Hgonio jmp condiion, his reads Applied o each shock presen in he iniial daa we arrive a he wo speeds s = s = f l) f r ) l r ) 2 2 2 2 = 2, s 2 = 2 2 2 )2 = ) 2 The lef shock is moving righ and he righ shock is moving lef. From his we epec he wo shocks o mee a = a ime = 2. Ok, so for a sarers we know can wrie p he eac solion p nil < 2. < 2 2), ) = 2 2) < < 2 2) 3) > 2 2) When he wo shocks mee we are lef wih a Riemann problem, wha will be he resling speed? Again we rn o he Rankine-Hgonio jmp condiion s 3 = 2) 2 2 2 )2 = 4) ) When he wo shocks mee hey sop moving and for all ime hereafer > 2 we can wrie he solion as <, ) = 5) > Now we move on o he solion for iniial daa se b) given in Eq. 6. < ) = < < > 6) Figre 2 conains a skech of he iniial daa prole ) wih characerisics evolving as a rarefacion wave. Why do we have a rarefacion wave solion and no simply anoher shock moving away from he cener insead of owards i? This wold lead o an enropy-violaing shock, if yo make anoher skech Try i!) yo will noice how he characerisics of sch a solion on his eqaion wold evolve o of he shock and no owards i. On he skech in gre 2 we do no have shocks forming where characerisics cross in ime b rarefacion waves a = ±. A general solion model for rarefacion wave on a Riemann problem was spplied in he eercise descripion for iniial daa cenered a zero, sing he solion model wih f v ξ)) = ξ v ξ) = ξ we arrive a <, ) = < < 7) > for he lef rarefacion wave, and for he righ rarefacion wave we have <, ) = < < + > + 8) 2
b) Figre : Eercise.3a. Iniial daa ) Eq. applied o he inviscid brgers eqaion. The wo shocks move owards each oher and merge a =. A = hey a new saionary shock is formed. Iniial daa. b) Characerisics. b) Figre 2: Eercise.3b. Iniial daa ) Eq. 6 applied o he inviscid brgers eqaion. The wo shocks move away from each oher wih eqal speed and never merge. Iniial daa. b) Characerisics. = T c? b) Figre 3: Eercise.4. Iniial daa ) Eq. 9 applied o he inviscid brgers eqaion. The rarefacion wave and he shock boh move in posiive direcion, he rarefacion wave moves faser han he shock and a some poin in ime = T c > hey mee and merge. Iniial daa. b) Characerisics. 3
Eercise.4 In his problem we again considered he inviscid brgers eqaion, now wih he se of iniial daa 2 < < ) = 9) oherwise For eercise.4 he gre 3 conains a skech of he iniial daa prole ) wih characerisics evolving. A he shock a = we epec a rarefacion wave o arise. Wha direcion will hings be moving? To deermine he speed of he shock we rern o he Rankine-Hgonio jmp condiion o ge s shock = f l) f r ) l r = 2 22 2 )2 = 2) 2 ) How abo he rarefacion? From he general solion o he rarefacion wave problem we have ha his wave will move wih a speed of s rf = f r ) = 2. The rarefacion wave hs moves o he lef wice as fas as he shock wave and a some poin in ime he waves ms mee. b) Firs we seek o deermine he eac solion for < < T c, where T c is he ime when he rarefacion wave caches p wih he shock as given in he eercise. The ime T c is rivial o nd by wriing wo eqaions for he posiion of he rarefacion fron wave and he posiion of he shock, shock = +, rf = 2 from which we have T c + = 2T c T c =. Up nil ime T c we can nd he eac solion sing he same approach as in eercise.3 o ge <, ) = < < 2 2 2 < < + > + c) B wha abo when > T c? We can again se he Rankine-Hgonio jmp condiion, now o consrc an ODE for he posiion of he shock s afer he rarefacion and shock wave have merged. 2) d s ) d = f l) f r ) l r = 2 ) 2 s) 2 )2 s) ) = s ) 2 22) This ODE has he general solion s ) = C. A = we know ha s = 2 so C = 2. B wha abo he prole of, )? The inegral over, ) for a any ed ms be conserved, remember we are working wih a conservaion law. When he wo waves mee hey merge and conine wih he speed s /2, on he righ side here is a shock and on he lef side he rarefacion. If s ), ), > T c remains consan he inegral over, ) will no be conserved a 2. Ths 2 = 2 s ) s ), ) s ), ) = 2. Wih his informaion, and a he paramerizaion 2 2 =,we can wrie he solion for ime > T c as <, ) = < < 2 > 2 The above is he same solion as one wold have obained by applying he general epression for he solion of he conve scalar eqaion rarefacion wave problem, insead we obained he solion by remembering ha is a conserved variable. 23) 4
Eercise.5 In his problem we considered he rac ow eqaion where f q) = ma 2 ) q + ma 2 )) = 24) where q, ) is he densiy of cars and ma he maimm speed of hese cars a any given poin. We wish o deermine wha he condiions are for a shock o be admissible in his eqaion. Firs we apply he enropy condiion, he speed of sch a shock across a disconiniy q l, q r ) is again given by he Rankine-Hgonio jmp condiion s = f q l) f q r ) q l q r we apply he enropy condiion o ge f q l ) > s > f q r ) which redces o = ) ma ql ql 2 ma qr qr) 2 = ma q r + q l )) 25) q l q r ma q l ) > ma q r + q l )) > ma q r ) 26) q l q r < 27) from which i is clear ha we ms have q l < q r for a shock o be admissible. b) The fncion for or rac eqaion is f q) = ma 2 ), choosing an enropy fncion η q) = q 2, by ψ q) = η q) f q) we dedce he enropy o be ψ ) = ma q 2 4 3 q3). Insering in s η q r ) η q l )) > ψ q r ) ψ q l ) 28) we nd ha ma q r + q l )) qr 2 ql 2 ) > ma qr 2 4 ) 3 q3 r ma ql 2 4 ) 3 q3 l 29) which afer qie a few operaions redces o > 3 q l q r ) 3 3) and we reach he same conclsion ha for a shock o be admissible we ms have q l < q r. As an eample of sch a non admissible shock, we presen Figrer 4. Figre 4 conains a skech of a weak solion of he rac ow eqaion on he Riemann problem Eq. 3 wih q l > q r..5 < ) = 3).5 > Noice how, on his pariclar weak solion, he characerisics ow o of he shock. Sch a shock formaion is called an enropy-violaing shock. If ime permis, yo shold creae his skech yorself by remembering how he speed of he characerisic for he rac ow eqaion is given by dx) d = f q) = ma 2q). ) b) Figre 4: Eercise.5. Iniial daa Eq. 3 applied o he rac ow eqaion. As seen from he behavior of he characerisics, his pariclar weak solion conains an enropy-violaing shock. This is no srprice as he enropy-violaion for sch a se of iniial daa was already prediced in he eercise. Iniial daa. b) Characerisics. 5