e distance between cars to increase. Therefore we see a gradual acceleration and spreading out of cars(the rarefaction wave Note also there is another weak solution to this problem. This solution contains a shock and corresponds to drivers accelerating instant aneously from uL=0 to uR>0 as the cars in front move out of the way. This solution is not physically viable in terms of the original problem. In terms of mathematical iability, although it satisfies the governing equations, it is inadmissible since the characteristics run out of the shock and therefore the entropy condition is violated 5.13“ Sound Speed SLIDE 4 For smooth solutions information travels with speed f'(e Pt+f(pPx=0 Consider a nearly constant solution p(r, t)=p0 +EPi(a, t) If a is small we can model pi (a, t) with the linear equation The initial data simply propagates unchanged with velocity f(po) 10=Umax pmax Pma ∫(p)≤ wo cars travel ahead of disturbances P0<5Pmax f(po)>0 disturbances move for ward po>2Pmax f'(po)<0 distur bances move backward =Pmax:“ sonic point” 5.2 Buckley-Leverett Equation SLIDE 47 +a(1 Here we consider the Riemann problem consisting of a discontinuos initial data for a<0 and u=0 fora>0➟✎➠✈➡✠➢②➢⑦➤✕➥❚➦②➧✯➨✦➩✴➫✯➭s➯✯➦②➲✪➩✴➡✵➧❱➳✄➭s➟✜➭✗➩⑤➥➌➭☛➭✗➧❏➳☛➡✠➵✓➲❼➩✴➤✻➦②➧❱➳✄➵✴➭☛➡✁➲✪➭✁➸❩➺❸➫❱➭✗➵✴➭✄➻❺➤✵➵✴➭✖➥➐➭➎➲★➭✗➭s➡✻➨✁➵✴➡✁➯✚➼❱➡✵➢ ➡✵➳☛➳✄➭☛➢⑦➭☛➵✴➡✠➩★➦②➤✵➧✑➡✵➧❱➯✥➲✪➽✯➵✴➭☛➡✁➯✚➦②➧✯➨➎➤✁➼✚➩❼➤✠➻➾➳✗➡✵➵✴➲❄➚❺➩★➫✯➭❄➵✓➡✠➵✴➭✄➻➆➡✁➳✣➩★➦②➤✵➧✑➥➐➡➏➪✁➭❭➶✄➸ ➹➤✵➩★➭❼➡✵➢②➲★➤❷➩★➫✯➭☛➵★➭❚➦❻➲✭➡✵➧✯➤✠➩✴➫✯➭✗➵➘➥➐➭☛➡✵➴➎➲★➤✵➢②➼✚➩★➦②➤✵➧s➩✴➤❷➩★➫✯➦❻➲❂➽✯➵✴➤✵➟❱➢⑦➭☛➷✥➸➾➺❸➫✯➦❻➲✭➲★➤✵➢②➼✚➩★➦②➤✵➧✦➳✗➤✵➧✤➩✴➡✵➦⑦➧❱➲ ➡✈➲★➫✯➤✚➳✓➴❊➡✵➧❱➯❶➳✄➤✁➵★➵✴➭☛➲★➽✜➤✁➧❱➯✯➲❩➩✴➤❏➯✚➵✴➦⑦➪✁➭✗➵✓➲✞➡✵➳☛➳✄➭✗➢②➭✗➵✓➡✕➩✴➦⑦➧❱➨✥➦②➧❱➲✪➩✴➡✵➧✁➩✓➡✠➧✯➭☛➤✵➼❱➲★➢②➠❡➻❺➵✴➤✵➷➮➬✘➱➃✃❒❐ ➩★➤❴➬✘❮➮❰➜❐Ï➡✁➲❀➩✴➫✯➭♠➳✗➡✠➵✓➲✑➦⑦➧➃➻❺➵✴➤✵➧✤➩♣➷✲➤✕➪✵➭❡➤✁➼✚➩✥➤✵➻❄➩★➫✯➭❊➥❸➡➏➠✵➸❥➺❸➫✯➦②➲♣➲★➤✵➢②➼✚➩★➦②➤✵➧Ð➦②➲✑➧✯➤✠➩ ➽✯➫✎➠✚➲✪➦❻➳✗➡✵➢⑦➢②➠❏➪✎➦❻➡✠➟✯➢②➭✻➦②➧❶➩✴➭✗➵✴➷✲➲✖➤✠➻❚➩✴➫✯➭✻➤✁➵★➦②➨✵➦②➧❱➡✠➢✭➽✯➵★➤✁➟✯➢②➭✗➷♣➸❡Ñ✰➧♠➩★➭☛➵★➷✦➲➎➤✠➻❚➷✦➡✕➩✴➫✯➭✗➷✦➡✕➩✴➦②➳☛➡✠➢ ➪✎➦②➡✵➟✯➦②➢⑦➦⑦➩⑤➠✵ÒÓ➡✠➢⑦➩★➫✯➤✁➼✯➨✵➫♠➦⑦➩➎➲★➡✠➩★➦❻➲⑤Ô✜➭☛➲❩➩★➫❱➭❀➨✵➤✕➪✁➭✗➵✴➧✯➦⑦➧❱➨✑➭❭Õ✁➼✜➡✕➩★➦②➤✵➧✜➲✗Ò▼➦④➩✖➦❻➲❄➦②➧❱➡✵➯✯➷s➦❻➲✴➲✪➦②➟✯➢②➭❀➲✪➦②➧❱➳✄➭ ➩★➫❱➭s➳✓➫❱➡✵➵✴➡✁➳✣➩✴➭✗➵✴➦②➲✪➩★➦❻➳✗➲❚➵✴➼✯➧❡➤✁➼✚➩❷➤✠➻❂➩✴➫✯➭s➲★➫✯➤✚➳✓➴♣➡✠➧❱➯❡➩★➫✯➭☛➵★➭✗➻❺➤✵➵✴➭❄➩★➫✯➭s➭✗➧✤➩✴➵★➤✁➽✤➠✈➳✗➤✵➧❱➯✚➦⑦➩★➦②➤✵➧❏➦②➲ ➪✎➦⑦➤✁➢②➡✠➩★➭❭➯●➸ Ö❪×➆Ø✘×②Ù Ú✤Û❛Ü✜ÝÓÞÓßÐÛ●à✭á✎á✎ßÓâ ã✘ä✤å✌æ❛ç❏è✜é ê➤✁➵❼➲✪➷✲➤✎➤✠➩✴➫✥➲★➤✵➢②➼✚➩✴➦⑦➤✁➧❱➲❸➦⑦➧✯➻❺➤✵➵✴➷✲➡✠➩★➦②➤✵➧✑➩★➵✓➡➏➪✵➭☛➢②➲➐➥❚➦⑦➩★➫✈➲✪➽✘➭✗➭❭➯✈ë✶ì➆➚❺í✚➶ í✤î●ï❴ë ì ➚❺í✚➶✰í✎ð✖✃⑨❐ ñ➤✵➧✜➲✪➦❻➯✚➭✗➵❚➡✲➧✯➭❭➡✠➵✴➢⑦➠❀➳✗➤✵➧❱➲✪➩✴➡✵➧✤➩❚➲★➤✵➢②➼✚➩✴➦⑦➤✁➧ í✶➚❺ò▼ó✪ô✪➶✭✃➃í✁õ➐ï❶ö✕í✚÷✠➚❺ò▼ó✪ô✪➶ Ñ✢➻Óö✖➦②➲↕➲✪➷✦➡✠➢②➢●➥➐➭❷➳☛➡✠➧✥➷✲➤✎➯✯➭✗➢❪í÷ ➚➆ò❛ó✪ô✪➶❚➥❚➦④➩✴➫✑➩✴➫✯➭❄➢②➦⑦➧✯➭❭➡✠➵❚➭❭Õ✁➼✜➡✕➩★➦②➤✵➧ í✯÷✰î●ïÏë ì ➚❺í✤õ➏➶⑤í✚÷✪ð✖✃Ð❐ ø✶ù✯ú✖û✌ü❱û✌ý✧û➆þ✕ÿ✁￾✵þ✕ý✰þ✄✂✓û✆☎✞✝✘ÿ✠✟✡✝☞☛✍✌✎✝✯þ✑✏✁þ✠ý✢ú✒✂✔✓✚ü☞✕✓ù✯þ✠ü✖✏✤ú✎￾✄✗➘û✌ý❺ù✙✘✕ú✄ÿ✚✌✑✕✗û✌ý✛✟ ë✶ì➑➚➆íõ ➶✢✜ ã✘ä✤å✌æ❛ç❏è✤✣ ë ì ➚❺í✤õ❭➶❂✃⑨➬✤✥✔✦✢✧✩★✫✪✭✬✯✮í✤õ í✥✔✦✒✧✱✰ ó ➬✘õ③✃➃➬☞✥✔✦✒✧✙★✲✪✭✬ í✁õ í✥✔✦✢✧✱✰ ë✶ì✧➚❺í✤õ➏➶✴✳✫➬✜õ❄➳☛➡✠➵✓➲➐➩★➵✓➡➏➪✵➭✗➢●➡✵➫✯➭☛➡✁➯✻➤✵➻Ó➯✚➦❻➲⑤➩✴➼✯➵✴➟❱➡✠➧❱➳✗➭☛➲ ✵➼✚➩✞➸✗➸✗➸ íõ✷✶ ÷ ✸ í✥✔✦✢✧ ë ì ➚➆íõ ➶➐❰✫❐ ➯✚➦②➲✪➩★➼❱➵★➟❱➡✵➧❱➳✄➭❭➲❸➷✲➤✕➪✵➭❩➻❺➤✵➵✴➥➐➡✵➵✴➯ íõ ❰ ÷ ✸ í✥✔✦✢✧ ë✶ì➑➚➆íõ ➶ ✶ ❐ ➯✚➦②➲✪➩★➼❱➵★➟❱➡✵➧❱➳✄➭❭➲❸➷✲➤✕➪✵➭❷➟❱➡✵➳✓➴✎➥❸➡✠➵✓➯ íõ ✃ ÷ ✸ í✥✔✦✒✧✺✹✷✻ ➲★➤✵➧✯➦❻➳↕➽✘➤✵➦②➧✤➩✽✼ ✾❀✿✛❁ ❂❄❃✞❅❇❆✭❈❊❉✲❋✴●■❍❏❉✲❑▲❉◆▼❇❉✲❖P❖❘◗❚❙✴❃✞❯❱❖❳❲❩❨▲❬ ã✘ä✤å✌æ❛ç❏è❪❭ ë➾➚❺➬●➶➾✃ ➬✸ ➬✸ ï❴❫✶➚❊✪✭✬❊➬●➶ ✸ ó❵❫s✃❛✪ ❜✖ú✢☛★ú✡✗✭ú❏✕✽✌✠ü❝✂✣û❞￾✵ú✢☛❷ý✌ù❱ú✞❡③û➆ú✢☎✲þ✠ü❱ü❢✝☞☛✽✌❤❣✗ÿ⑦ú✒☎✐✕✎✌✕ü❝✂✣û❥✂✣ý✧û✌ü✖✏✄✌❩❦❷þ❧￾✕û❥✂✢✕✎✌✕ü❱ý✧û✌ü❇✓❇✌♠✂❩û✌ü✜û✌ý➑û➆þ✠ÿ✫￾✵þ✠ý✢þ ➬✑✃❛✪ ❦✒✌♠☛ ò ✶ ❐ þ✠ü☞￾ ➬♣✃⑨❐ ❦✒✌♥☛ ò❡❰◗❐✤✜ ✮ ❐