au af(u) 0 This equation has applications in oil reservoir simulation where one models the flow of oil and water through porous rock or sand. So varies between 0 and 1 u=0 represents a flow of pure oil, u=l represents pure water f(u)=2+a(1 a: constant w 1 Note 5 The Buckley-Leverett Equations For the most part, we consider equations where f(u) is convex(or a concave function of the unknown variable. In the convex(or concave)case, the solution of an initial discontinuos data distribution(Riemann problem) is always either a shock or a rarefaction(or expansion)wave. When f is not convex(nor concave the solution might involve both. The Buckley-Leverett equati example where this situation may occur 2 Smooth solutions 2.1 Total derivative SLIDE 10 Recall the primitive form of the conservation law +a(u) The total time variation of u(a, t), on an arbitrary curve a=a(t), in the a-t plane, is dt at dt 2.2 Characteristics SLIDE 11 If at=a(u) 0→ The curves r=c(t), such tha dt a(u)are called characteristics a )constant characteristics are straight lines
➆✢➇ ➆✢➈⑨➉ ➆✌➊❈➋s➇✌➌ ➆✢➍ ➎✉➏ ➐✑➑✴➒➔➓❧→❅➣☛↔❪↕❜➙④➒✲➛❜➜❯➑②↕✩➓❧↕➞➝♣➝✑➟➠➒✲➡❅↕✩➙✓➒✲➛✩➜❪➓✠➒✺➜➄➛✩➒✺➟✟➢➞→P➓❏→❏➢P➤➥➛❜➒✺➢➦➓❅➒✺➧⑨↔✆➟⑥↕❜➙④➒✲➛❜➜➩➨⑩➑②→❏➢✐→⑨➛✩➜✑→➫➧✘➛✡➭❝→☛➟➯➓➫➙s➑②→ ➲➛✩➨➳➛♠➵✠➛❜➒✺➟③↕❜➜✢➭✘➨✎↕❜➙➸→❏➢➦➙✺➑✆➢✐➛✩↔♣➺➥➑r➝❪➛✩➢➞➛✩↔✴➓➦➢✐➛✡➡❅➻➼➛✩➢➦➓❏↕✩➜✑➭♣➽❙➾✑➛ ➇ ➤➥↕❜➢P➒✲→P➓❧➚❅→❏➙④➨✎→❅→❏➜➄➪❀↕❜➜✢➭❀➶ ➹ ➇ ➎➳➏ ➢✐→➸➝✑➢✐→☛➓☛→❏➜❪➙④➓✠↕ ➲➛✩➨➘➛❂➵➴➝✢↔✆➢✐→⑨➛✩➒✺➟➠➷ ➇ ➎➮➬ ➢✐→➸➝✑➢✐→☛➓☛→❏➜❪➙④➓➴➝✢↔✆➢✐→✠➨✎↕✩➙➱→☛➢❏➽ ✃r❐ ➊❈➋s➇✌➌ ➎ ➇✑❒ ➇❒ ➉✬❮ ➋ ➬✰❰ ➇✑➌ ❒ ❮◆Ï❡Ð❏Ñ♣Ò❪Ó❂Ô➞Õ♣Ò✸Ô❙Ö×➬ Ø✽Ù✌ÚPÛ❃Ü Ý✰Þ✎Û➼ß✝à✌áPâ✟ã♠Û✡ä③å❂æ✰Û②ç✆Û❜è❝Û②Ú❏Ú➫é❧ê⑩à◆ë✟Ú❂ì➱Ù✆í③î ï Ñ❝ð❙Ô➞ñ②ò▼ó▼Ñ❝Ó❂Ô❿ô❪Õ♣ð❂Ô❞õ✌ö✧ò▼Ð☛Ñ❝Ò❪Ó✐÷✤ø✆ò✡ðùò❞ú✸û❪Õ✩Ô➞÷⑥Ñ❝Ò❪Ó❙ö✰ñ❪ò❏ð➞ò ➊❈➋✲➇✑➌ ÷✤Ó➦Ð❏Ñ♣Ò✴ü♣ò❏ý ➋ Ñ♣ð➫Õ❯Ð☛Ñ♣Ò◆Ð❏Õ➥ü♣ò ➌ þû②Ò❪Ð☛Ô✐÷✤Ñ♣Ò➼Ñþ Ô➞ñ②ò➦û②Ò②ÿ✴Ò②Ñ✩ö✰Ò ü✩Õ❜ð➞÷➔Õ✁✄✂ò✆☎✞✝➱Ò Ô➞ñ②ò➦Ð☛Ñ♣Ò✴ü❝ò☛ý ➋Ñ❝ð➴Ð❏Ñ♣Ò❪Ð✡Õ➥ü♣ò ➌ Ð❏Õ❝Ó❂ò❝õ♣Ô➞ñ②ò➫Ó❂Ñ✂û✆Ô➞÷⑥Ñ❝Ò Ñ þ Õ❜Ò⑨÷⑥Ò②÷⑥Ô✐÷➔Õ✂ ø②÷✤Ó➞Ð☛Ñ❝Ò❝Ô➞÷⑥Ò✴û②Ñ✸Ó❈ø②Õ✩Ô❅Õùø✆÷➔Ó❂Ô✐ð➞÷✁û②Ô✐÷✤Ñ♣Ò ➋✠✟÷✤ò❏ó✘Õ❜Ò❪Òrô②ð➞Ñ✁✡✂ò✡ó➌ ÷➔Ó✁Õ✂ö➴Õ☞☛✆Ó⑩ò❏÷⑥Ô✐ñ②ò✡ð❈Õ Ó✐ñ②Ñ✆Ð❅ÿ❿Ñ❝ð✁Õ❙ð❅Õ❜ð➞òþÕ❝ÐPÔ✐÷✤Ñ♣Ò ➋ Ñ♣ð❈ò☛ý✆ô❪Õ♣Ò❪Ó✐÷⑥Ñ❝Ò➌ öÕ➥ü♣ò✌☎✎✍✉ñ②ò✡Ò ➊ ÷➔Ó❈Ò②Ñ❜Ô✁Ð☛Ñ❝Ò✴ü♣ò☛ý ➋Ò②Ñ♣ð✁Ð☛Ñ❝Ò❪Ð❏Õ➥ü❝ò ➌ õ Ô✐ñ❪ò❃Ó❂Ñ✂û✆Ô➞÷⑥Ñ❝Ò❨ó⑨÷✑✏♣ñ✸Ô ÷✤Ò✴ü♣Ñ✂ü♣ò ✁ Ñ❜Ô✐ñ✎☎✓✒ñ②ò✕✔➴û❪Ð❅ÿ✂ò✖☛✘✗✚✙✄ò❏ü❝ò❏ð➞ò☛Ô❂Ô▼ò✡ú✸û❪Õ❜Ô✐÷✤Ñ♣Ò ÷✤Ó❀Õ➅Ó❂÷✤ó▼ô✂ò ò☛ý②Õ♣ó⑨ô✂ò➦ö✰ñ②ò✡ð✐ò❿Ô✐ñ❪÷✤Ó Ó✐÷⑥Ô✐û❪Õ❜Ô✐÷✤Ñ♣Ò❯ó▼Õ☞☛❀Ñ✴Ð✡Ð☛û②ð✛☎ ✜ ✢✤✣✦✥✧✥✩★✫✪✬✢✭✥✯✮✖✰✤★✫✱✲✥✴✳✶✵ ✷✹✸✻✺ ✼✴✽✫✾❀✿✫❁✯❂❄❃✎❅✡❆❈❇✞✿❉✾❀❆❊❇❋❃ ●■❍✘❏▲❑✎▼❖◆◗P ✟ ò❞Ð❏Õ✂✑✂ Ô✐ñ❪ò➫ô②ð✐÷✤ó▼÷Ô➞÷⑥ü❝ò þÑ♣ð➞ó Ñ þ Ô✐ñ②ò✠Ð❏Ñ♣Ò❪Ó✐ò❏ð➞ü✩Õ✩Ô➞÷⑥Ñ❝Ò ✂Õ➥ö ➆✢➇ ➆✢➈ ➉♦❮ ➋s➇✌➌ ➆✢➇ ➆✢➍ ➎➳➏ ✒ñ②ò➦Ô➞Ñ❜Ô➞Õ✂ Ô✐÷✤ó▼ò✠ü✩Õ❜ð➞÷✤Õ❜Ô✐÷✤Ñ♣Ò❯Ñþ ➇③➋✲➍✎❘✐➈❂➌ õ❪Ñ❝Ò✮Õ❜Ò✝Õ❜ð✁ ÷Ô➞ð➞Õ♣ð❙☛➼Ð☛û②ð➞ü♣ò ➍ ➎ ➍⑩➋✲➈❂➌ õ◆÷⑥Ò✝Ô➞ñ②ò ➍ ❰ ➈ ô ✂Õ❜Ò②ò❝õ✆÷✤Ó ❚ ❚➇ ➈ ➎ ➆✢➇ ➆✢➈ ➉ ❚ ❚➍ ➈ ➆✢➇ ➆✢➍ ✷✹✸❯✷ ❱❳❲❨✿❉❅✡✿✞❩■✾❀❃❬❅❭❆✻❪✌✾✄❆✻❩❫❪ ●■❍✘❏▲❑✎▼❖◆✄◆ ❴✲❵ ❚ ❚➍ ➈ ➎✉❮ ➋s➇✌➌❜❛ ❚ ❚➇ ➈ ➎➘➏ ❛ ➇ ➎ ➇■❝ ➋ Ð☛Ñ❝Ò❪Ó❂Ô➞Õ❜Ò✸Ô ➌ ✒ñ②ò➫Ð☛û❪ð✐ü❝ò✡Ó ➍ ➎ ➍⑩➋s➈❂➌ õ◆Ó✐û❪Ð❅ñ➼Ô✐ñ❪Õ❜Ô ❚ ❚➍ ➈❯➎➘❮ ➋s➇✌➌ Õ❜ð➞ò❿Ð✡Õ✂✑✂ò❞ø❡❞☞❢❉❣❀❤✐❣✄❞◗❥✛❦✘❤☞❧❯♠✖❥✛❧✠❞✌♠ ➇ Ð❏Ñ♣Ò❪Ó❂Ô➞Õ♣Ò❝Ô♦♥ ❮ ➋s➇✌➌ Ð☛Ñ❝Ò❪Ó❂Ô➞Õ❜Ò✸Ô♦♥ ❞☞❢❉❣❀❤✐❣✄❞◗❥✛❦✘❤☞❧❯♠✖❥✛❧✠❞✌♠✧❣❀❤✐❦✕♠♣❥✖❤✐❣✄❧✠q■❢r❥✧s✠❧❯t✉❦✈♠ ✇
The characteristics are straight lines in the x-t plane along which u is constant lfu(ao, 0)=0, the characteristic passing througha=ao, t=0, is the solution of the following initial value problem dr/dt= a(uo),a(0)=xo; i.e. a Note 6 Characteristics The slope of the characteristic lines is determined by the initial condition uo (a) except for the trivial case in which f is a linear function of u. In this latter case, he slope characteristics is constant i.e. the characteristics are parallel. We note that, for our problems, the characteristics are straight lines, even in the explicitly on a, the characteristics are no longer straight lines on tha.quations linear case, because f is determined by u only. Fo burce term or a flux fu If we solve a problem on a finite domain, the number of boundary conditions to be prescribed, in the non-linear case, depends on the data itself. That is, in tho boundaries with incoming characteristics a boundary condition will be required Similarly, the solution at those boundaries with outgoing characteristics will be determined by the interior. We can see therefore that in the 1D case we can equire, two, one or no boundary condition For nonlinear conservation laws and arbitrary data, the characteristics may cross within finite time. This would suggest a multi-valued solution which does not make any will see that just at the point where th characteristics start crossing, the solution becomes discontinuous. At this point he differential primitive form of the equation, on which we are basing our solution procedure is no longer valid
①r②✄③♦④⑤②✡⑥✐⑦❙⑥✌④✲⑧❈③✲⑦⑩⑨❷❶⑩⑧❸⑨✠④⑩❶♦⑥◗⑦❙③✹❶⑩⑧❸⑦❙⑥✐⑨❺❹☞②✈⑧✉❻❼⑨▲❽■③✲❶✹⑨▲❽✧⑧▲②✡③✫❾➀❿✩➁✎➂r❻➃⑥✐❽r③❨⑥◗❻➃➄✐❽✈❹➆➅✉②❀⑨✠④⑤②✩➇✕⑨❷❶➈④➉➄◗❽✄❶⑩⑧✚⑥◗❽✡⑧❈➊ ➋➍➌➎➇❉➏✠❾✡➐✘➑❙➒✌➓✫➔→➇❭➐✆➣✉⑧➍②✄③↔④➉②✄⑥✐⑦⑤⑥✆④✖⑧✚③✖⑦⑩⑨❷❶⑩⑧❯⑨✠④➀➂✡⑥☞❶➉❶⑩⑨▲❽✘❹✴⑧➍②✈⑦⑤➄✐↕✆❹☞②✩❾✤➔→❾✡➐✌➣❭➁➎➔→➒✄➣✎⑨❷❶❨⑧➍②✄③❨❶✖➄✐❻❼↕❀⑧❸⑨✠➄✐❽ ➄❊➌✶⑧➍②✄③✯➌✲➄✐❻▲❻✑➄✐➅✫⑨▲❽✘❹➙⑨▲❽❭⑨▲⑧❯⑨✠⑥◗❻➜➛✐⑥✐❻❼↕✡③➝➂■⑦❙➄✌➞✲❻✑③✲➟➡➠✌❾r➢✐➠✌➁❡➔➥➤r➏➍➇■➐✛➓✲➑➦❾✉➏❯➒✌➓➧➔➨❾✡➐✌➩➝⑨✠➊▲③♣➊✓❾➫➔ ❾✡➐✹➭➯➤r➏➍➇■➐✛➓❊➁✖➊ ➲■➳✘➵▲➸✎➺❖➻✆➼ ➠✌❾ ➠✆➁ ➔→➤r➏➍➇➐ ➓ ➽ ❾❡➔➾❾➐ ➭➚➤r➏➍➇➐ ➓■➁ ➪♦➶ ➹❳➘❫➴⑩➷➮➬ ➱❐✃✞❒❀❮✐❒■❰✘➴✲➷✐❮☞Ï❯Ð✌➴❙Ï❈❰✛Ð Ñ✹Ò✄Ó➜Ô❙Õ➃Ö✌×❭Ó➈Ö✆Ø❭Ù⑤Ò✄Ó➜Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù❙Ó♣Ü❙Ý❷Ô✻Ù❙Ý❷Ú❋Õ✑Ý➃Þ✡Ó♣Ô✫Ý❷Ô➎ß❀Ó✖Ù❙Ó✖Ü⑤àáÝ➃Þ✡Ó♣ßáâ✈ã✩Ù❙Ò✡Ó➜Ý➃Þ✡Ý❺Ù⑤Ý✑Û✆Õ✡Ú✲Ö✌Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þá➇➐ ➏➍❾r➓⑩ä Ó✲å✄Ú✖Ó✖×❀Ù✫Ø➍Ö✌Ü✞Ù❙Ò✄Ó➈Ù⑤Ü❙Ý✑æ✈Ý✑Û✆Õ✄Ú✖Û✌Ô✻Ó✹Ý✑Þ➝ç➈Ò✄Ý✑Ú➉Ò✧è✧Ý✑Ô➎Û↔Õ✑Ý✑Þ✄Ó♣Û✆Ü✞Ø➍é✄Þ✡Ú⑩Ù⑤Ý➃Ö✌Þ➝Ö◗Ør➇✉ê❉ë❈Þ✴Ù❙Ò✄Ý❷Ô✫Õ❷Û✐Ù❙Ù❙Ó✖Ü➀Ú✖Û✆Ô❙Ó✆ä Ù❙Ò✡Ó➆Ô✻Õ✑Ö✆×■Ó✩Ú➉Ò✡Û◗Ü➉Û✆Ú✲Ù❙Ó✖Ü⑤Ý❷Ô❊Ù⑤Ý✑Ú♣Ô❋Ý❷Ô➜Ú✲Ö✌Þ✡Ô✻Ù⑤Û◗Þ✘Ù➈Ý❸ê Ó✌ê✞Ù❙Ò✄Ó➆Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖ÔìÛ◗Ü⑤Ó↔×✡Û◗Ü➉Û◗Õ✑Õ➃Ó♣Õ❯ê í❳ÓìÞ✡Ö◗Ù❙Ó✹Ù⑤Ò✡Û✐Ù✛ä✐Ø➍Ö✆Ü✫Ö✌é✄Ü✞×✄Ü⑤Ö✆â✄Õ✑Ó✖à➝Ô♣ä✐Ù❙Ò✄Ó➜Ú➉Ò❭Û◗Ü➉Û✆Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖Ô❉Û◗Ü⑤Ó✹Ô❊Ù⑤Ü⑤Û✆Ý➃î✌Ò✌Ù✞Õ✑Ý➃Þ✄Ó✛Ô✖ä✆Ó✖æ✆Ó♣Þ✯Ý✑Þ✴Ù❙Ò✄Ó Þ✄Ö✌Þ❀ï❸Õ✑Ý✑Þ✄Ó♣Û✆Ü❐Ú♣Û✆Ô❙Ó✆ärâ❭Ó✛Ú✖Û✆é✡Ô✻Ó➝èðÝ❷Ô↔ß❀Ó✲Ù⑤Ó✖Ü⑤àáÝ➃Þ✄Ó✛ß✕â✈ã✕➇ñÖ✆Þ✡Õ➃ã✌êóò✄Ö✌Ü↔Ô❙ã❀Ô❊Ù⑤Ó✖à➝Ô♦Ö◗Ø➀Ó✛ô✘é✡Û✐Ù⑤Ý➃Ö✌Þ✡Ô✖ä Ö✆Ü➎Ø➍Ö✆Ü❋Ô⑤Ú✖Û✆Õ✑Û✆Ü✫Ó♣ô✘é✡Û✐Ù⑤Ý➃Ö✌Þ✡Ô✫ç➈Ý➃Ù❙Ò✧Ó♣Ý❺Ù⑤Ò✄Ó✖ÜìÛ✩Ô✻Ö✌é✄Ü➉Ú✲Ó✹Ù❙Ó♣Ü❙àõÖ✆Ü➀Û❐ö✡é❀å✴Ø➍é✄Þ✡Ú⑩Ù⑤Ý➃Ö✌ÞáÙ❙Ò✡Û◗Ùìß❀Ó♣×❭Ó♣Þ✡ß✄Ô Ó✲å❀×✄Õ✑Ý❷Ú✲Ý➃Ù❙Õ✑ã✧Ö✆Þ✶❾✎ä✄Ù❙Ò✄Ó➆Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖ÔìÛ◗Ü⑤Ó↔Þ✄ÖáÕ➃Ö✌Þ✄î✆Ó♣Ü➜Ô❊Ù⑤Ü⑤Û✆Ý➃î✌Ò✌Ù✹Õ✑Ý➃Þ✄Ó✛Ô✖ê ë✚Ø❬çìÓ♦Ô❙Ö✆Õ✑æ✆Ó❨Û✩×✡Ü❙Ö✌â✄Õ➃Ó♣à÷Ö✆Þ✧Û✩ø❭Þ✄Ý❺Ù⑤Ó♦ß❀Ö✌à➝Û◗Ý✑Þ❬ä✆Ù⑤Ò✄Ó❐Þ✘é✡àóâ■Ó✖ÜìÖ◗Ø❫â❭Ö✌é✄Þ✡ß✄Û✆Ü❙ã➝Ú✖Ö✆Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þ❭Ô✫Ù❙Ö â■Ó➎×✄Ü⑤Ó♣Ô⑤Ú✲Ü⑤Ý➃â■Ó♣ß❫ä☞Ý✑Þ↔Ù❙Ò✄Ó➀Þ✄Ö✆Þ✄ï❸Õ✑Ý➃Þ✡Ó♣Û◗Ü✉Ú✖Û✆Ô❙Ó✆ä☞ß❀Ó♣×❭Ó♣Þ✡ß✄Ô✉Ö✆Þ↔Ù❙Ò✡Ó➀ß✄Û◗Ù⑤Û➈Ý➃Ù⑤Ô❙Ó✖Õ➃Ø❊ê❉Ñ✹Ò❭Û✐Ù✎Ý❷Ô✖ä☞Ý✑Þ↔Ù❙Ò✄Ö✘Ô✻Ó â■Ö✆é✄Þ✡ß✡Û◗Ü⑤Ý➃Ó✛Ô✞ç➈Ý❺Ù⑤Ò✴Ý✑Þ✡Ú✖Ö✆àáÝ➃Þ✡î↔Ú➉Ò❭Û◗Ü➉Û✆Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖Ô❉Û↔â■Ö✆é✄Þ✡ß✡Û◗Ü⑤ãóÚ✲Ö✌Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þ✴ç➈Ý➃Õ✑Õ✄â■Ó➈Ü❙Ó✛ô✌é✡Ý➃Ü⑤Ó♣ß❫ê ùÝ✑àáÝ➃Õ❷Û◗Ü⑤Õ➃ã✌ä✆Ù⑤Ò✄Ó↔Ô✻Ö✌Õ➃é❀Ù⑤Ý➃Ö✌Þ✤Û◗Ù❋Ù❙Ò✡Ö✌Ô❙Ó❨â■Ö✆é✄Þ✡ß✡Û◗Ü⑤Ý➃Ó✛Ô❋ç➈Ý➃Ù❙Ò✤Ö✆é❀Ù⑤î✆Ö✆Ý✑Þ✄îóÚ➉Ò❭Û◗Ü➉Û✆Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖Ô❋ç➈Ý➃Õ✑Õrâ❭Ó ß❀Ó✖Ù❙Ó✖Ü⑤àáÝ➃Þ✡Ó♣ß❳â✈ã➮Ù⑤Ò✄Ó➝Ý➃Þ✘Ù⑤Ó✖Ü⑤Ý➃Ö✌Ü♣ê✤íúÓ✧Ú♣Û◗ÞðÔ✻Ó♣Ó➝Ù❙Ò✡Ó✖Ü⑤Ó✲Ø➍Ö✆Ü⑤Ó✴Ù❙Ò✡Û◗ÙóÝ✑ÞñÙ❙Ò✄Ó➧û✛üýÚ✖Û✌Ô✻ÓáçìÓ✧Ú✖Û✆Þ Ü⑤Ó♣ô✘é✄Ý✑Ü❙Ó✌ä✌Ù❊çìÖ✡ä❀Ö✌Þ✄Ó↔Ö✆Ü➈Þ✄Öáâ■Ö✆é✄Þ❭ß✄Û◗Ü⑤ã✭Ú✲Ö✌Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þ❬ê ò✄Ö✌Ü✤Þ✡Ö✆Þ✄Õ✑Ý➃Þ✡Ó♣Û◗Ü✶Ú✲Ö✌Þ✡Ô❙Ó✖Ü⑤æ☞Û◗Ù❙Ý✑Ö✆Þ→Õ❷Û☞ç➜Ô❡Û◗Þ✡ß✓Û◗Ü⑤â✄Ý❺Ù⑤Ü⑤Û✆Ü❙ã➾ß✄Û◗Ù⑤Û✄ä❨Ù❙Ò✄ÓñÚ➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù❙Ó♣Ü❙Ý❷Ô✻Ù❙Ý❷Ú✖Ô✧à➝Û☞ã Ú✲Ü⑤Ö✌Ô⑤Ô➎ç➈Ý➃Ù❙Ò✡Ý➃Þ✤ø✡Þ✄Ý➃Ù❙Ó❐Ù❙Ý✑àáÓ✆ê➎Ñ✹Ò✄Ý❷Ôìç➀Ö✌é✄Õ❷ß✭Ô❙é✄î✆î✌Ó♣Ô✻ÙìÛ➆à✯é✄Õ➃Ù❙Ý➃ï❸æ✐Û◗Õ✑é✄Ó✛ß✤Ô❙Ö✆Õ✑é❀Ù❙Ý✑Ö✆Þ✭ç➈Ò✄Ý✑Ú➉Ò❡ß❀Ö✈Ó♣Ô Þ✄Ö✆Ù✯à➝Û◗þ✌Ó✤Û◗Þ✈ãñÔ✻Ó♣Þ✡Ô❙Ó✭×✄Ò✈ã❀Ô✻Ý❷Ú✖Û✆Õ➃Õ✑ã✆êúíúÓ❡ç➈Ý✑Õ➃Õ➈Ô❙Ó✖Ó✧Ù⑤Ò✡Û✐Ù♦ÿ❊é✡Ô✻ÙáÛ◗Ù✯Ù⑤Ò✄Ó✭×■Ö✆Ý✑Þ✘Ù✴ç➈Ò✄Ó♣Ü❙Ó✭Ù❙Ò✄Ó Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù❙Ó♣Ü❙Ý❷Ô✻Ù❙Ý❷Ú✖Ô✎Ô✻Ù⑤Û✆Ü✻Ù➎Ú✖Ü❙Ö✘Ô❙Ô❙Ý➃Þ✡î✡ä✛Ù❙Ò✄Ó➈Ô❙Ö✆Õ✑é❀Ù❙Ý✑Ö✆Þ✯â❭Ó✛Ú✲Ö✌à✴Ó✛Ô✫ß❀Ý❷Ô❙Ú✖Ö✆Þ✘Ù❙Ý✑Þ✈é✄Ö✆é✡Ô♣ê✁ìÙ✞Ù⑤Ò✄Ý✑Ô✞×■Ö✆Ý✑Þ✘Ù♣ä Ù❙Ò✡Ó❳ß❀Ý✄✂rÓ✖Ü⑤Ó✖Þ✘Ù❙Ý❷Û◗Õ↔×✄Ü⑤Ý➃àáÝ➃Ù❙Ý✑æ✆Ó➮Ø➍Ö✆Ü⑤à Ö◗Ø✩Ù❙Ò✡Ó❳Ó✛ô✘é✡Û✐Ù⑤Ý➃Ö✌Þ❬ä➜Ö✆Þ✓ç➈Ò✄Ý❷Ú➉Ò çìÓúÛ◗Ü⑤Ó➮â❭Û✆Ô❙Ý➃Þ✄î❖Ö✆é✄Ü Ô❙Ö✆Õ✑é❀Ù❙Ý✑Ö✆Þ❡×✄Ü⑤Ö✈Ú✖Ó♣ß❀é✡Ü❙Ó❐Ý✑Ô➜Þ✄ÖáÕ➃Ö✌Þ✄î✆Ó♣Ü➈æ☞Û✆Õ➃Ý❷ß❫ê ➶
2.3 Examples 2.3.1 Linear Advection Equation SLIDE 13 p(a, t)=po(a-at) +at 2.3.2 Burgers' Equation SLIDE 14 Recall f(u)=2u2, so a(u) at ar 0 Solution: u(, t)=uo(a-ut) The solution is constant along the characteristic lines defined by a -ut=To We note that the above solution is defined implicitly(e.g. the definition of the function requires the function itself and therefore it is often not very useful We can verify however by direct differentiation that it is in fact a solation of the partial diffe Consider the initial data <0 0<x<1 16
☎✝✆✟✞ ✠☛✡✌☞✎✍✑✏✓✒✕✔✗✖ ✘✚✙✜✛✗✙✣✢ ✤✦✥✣✧✩★✫✪✭✬✯✮✱✰✁✲✳★✵✴✷✶✸✥✣✹✺✧✼✻✓✽✿✾✁✪❀✶❁✥✣✹✺✧ ❂✺❃✫❄❆❅✗❇❉❈✷❊ ❋✵●✷❍✜■❀❏▲❑✄●◆▼ ❖✺P✣◗✗❘❚❙❱❯❳❲❨❖✫❩✫P✣◗❭❬❫❪◆❙❱❯ ❴✦❵❜❛✷❝▲❛◆❞❡❏❚❢✸❝❚❑❤❣❱❏❚❑❤❞✐❍✜❑✄▼❜❢✸❣ ◗✯❲❥◗❩❧❦ ❪✫❙ ✘✚✙✜✛✗✙❤✘ ♠♥✾✗✬♣♦✺★✫✬rq◆st✻✉✽✿✾✩✪❀✶❁✥✣✹✺✧ ❂✺❃✫❄❆❅✗❇❉❈♣✈ ✇❢❁❞①❛②❍✜❍✗③ P⑤④✿❯⑥❲⑧⑦⑨ ④⑨②⑩ ❣❚● ❪✺P✣④❶❯⑥❲✼④ ❷ ④ ❷ ❙ ❦ ④ ❷ ④ ❷ ◗ ❲❥❸ ❋❀●✷❍✜■❀❏❚❑✜●✷▼❺❹ ④✩P✣◗✗❘❚❙❱❯❳❲❨④✺❩◆P✣◗❭❬❻④✺❙❱❯ ❼❵✭❢❽❣❚●✷❍✜■❀❏❚❑✜●✷▼☛❑✜❣✐❞❾●✷▼✳❣✕❏❿❛②▼✫❏➀❛②❍✜●✷▼✭➁➂❏❚❵✭❢➃❞❿❵✳❛②❝❿❛✷❞❡❏▲❢①❝▲❑✜❣❱❏❚❑❤❞✉❍✄❑✜▼✭❢✸❣✌➄✭❢❾➅❜▼✭❢❁➄☛➆✫➇ ◗❭❬❺④✳❙t❲✼◗✳❩✷➈ ➉➋➊♥➌✺➍♣➎➏➊♥➎⑤➐✭➑♣➎✌➎⑤➐✭➊➒➑◆➓❿➍♣➔♣➊♥→①➍♣➣↕↔❀➎➛➙✣➍♣➌❫➙❤→➝➜◆➊✣➞❳➌✺➊❿➜✯➙❆➟➀➠❶➣↕➙✣➡①➙❆➎✟➣↕➢❫➤✕➊❁➥➧➦◆➥➝➎⑤➐✭➊➒➜◆➊✣➞❳➌❜➙❆➎➛➙✣➍♣➌➨➍❱➩➫➎⑤➐✭➊ ➩❿↔❀➌✺➡①➎➛➙✣➍♣➌➯➭❚➊❿➲❾↔❀➙❆➭❚➊❾→❭➎⑤➐✭➊✓➩❿↔❀➌✺➡①➎✟➙✣➍②➌❉➙❆➎✟→①➊❾➣➩①➳❻➑②➌✺➜❻➎⑤➐✭➊①➭❚➊✣➩❾➍②➭❚➊☛➙❆➎➵➙❤→✯➍✕➩❿➎➏➊❾➌➯➌✺➍②➎➵➔r➊①➭❡➢➸↔✵→①➊✟➩❿↔❀➣✄➥ ➉➋➊✯➡❿➑♣➌❉➔r➊①➭❡➙➩❿➢☛➐❜➍♣➺❳➊①➔r➊①➭☛➓①➢➸➜♣➙❆➭❚➊❿➡①➎➵➜②➙➻✐➊①➭❚➊❾➌✳➎✟➙✣➑②➎✟➙✣➍②➌❉➎⑤➐✭➑②➎✉➙❆➎✓➙❤→➒➙❆➌✯➩❾➑◆➡❾➎➃➑➼→①➍②➣↕↔❀➎✟➙✣➍②➌➯➍❱➩ ➎⑤➐✭➊✌➠✭➑②➭❡➎✟➙✣➑②➣✩➜♣➙➻✉➊❾➭❚➊①➌❜➎➛➙✣➑♣➣✎➊❿➲❾↔❜➑②➎✟➙✣➍②➌✭➽⑥➙✣➥❆➊❁➥✄➾ ④✺➚➫❲✼④✺➪❩ ❬❺④❶➪❩ ④✺➚✷❙ ➶ ④✺➚➃❲ ④❩➪ ➹ ❦ ④❩➪ ❙ ④✺➘✎❲➴❬✌④❩➪ ④❭❬❺④❩➪ ④✺➘➛❙ ➶ ④✺➘✎❲ ❬✌④❩➪ ④ ➹ ❦ ④❩➪ ❙❀➷ ❂✺❃✫❄❆❅✗❇❉❈✷➬ ❴✦●✷▼✳❣❱❑❤➄❀❢①❝❧❏▲❵✭❢❽❑✄▼❜❑➮❏▲❑✜❛✷❍✗➄✭❛♣❏❿❛ ④✎P⑤◗✗❘▲❸◆❯⑥❲ ➱✃❐ ➹ ◗➋❒❨❸ ➹ ❬❻◗ ❸➝❮➯◗➋❮ ➹ ❸ ◗➋❰ ➹ ❂✺❃✫❄❆❅✗❇❉❈✷Ï Ð
dt SLIDE 17 For t< 1 <t →x=t+x0→a(x,t)=1 For t<z<i: dt=1-to+x=(1-)t+ro For z>1 0→x=x0→ For t<a<l, we first solve for the characteristic lines. In this case they are defined by g=u is constant along each characteristic, we know that on each line dz uo. Once we have determined the characteristic lines. we use the fact that u is constant along each line to determine the overall solutio a(a, t) At t= l the solution develops a discontinuity. This corresponds to the time at which the characteristics first cross The procedure breaks down for t>1
Ñ✺Ò✫Ó❆Ô✗Õ❉Ö✫× Ø✭Ù◆ÚÜÛ➫ÝßÞ Ø✭Ù◆Ú✌à➋á❉â✉ã äà ä â➋åçæ✑è à å â✿éêà❜ë è ì✎íà✁î❱â❱ï åðæ Ø✭Ù◆Ú✌â✦ñ❉àòñ æ ã äà ä â➋åçæ✝ó à❜ë è à åôí✕æ✌ó à❜ë♣ï➏â✗éêà❜ë ì✩íà✗î❚â❱ï åçæ✝ó à❜ë å æ✌ó à æ✌ó â Ø✭Ù◆Ú✌à➋õ æ ã äà ä â å❥ö÷è à å àë è ì✎íà✁î❱â❱ï åøö ù✁ú♣û✓â✉ñüàêñ æ✷ý❧þ❳ÿ✁û✄✂✆☎✝✂①ú✟✞✡✠ ÿ☞☛ú♣û✌☎✎✍ ÿ✑✏✍✓✒②û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✗✕ ✏ ✞✡✕✙✘ ÿ ✂✛✚✢✜✣✘✤☎✙✍✥✕✖✂ ✏✒✦✂ ÿ ☎✎✍ ÿ✛✧ ✒♣ûÿ ★ ÿ✗✘ ÿ★✪✩ ✧✬✫✄✭✫✯✮ å❥ì ✚✱✰✲✕✙✘✏❿ÿ✉ì ✕✖✂ ✏ú✳✘✓✂✆☎✴✒✳✘✵☎☞✒✟✞✄ú✳✘✷✶ ÿ✒ ✏✍ ✏✍✵✒♣û✔✒ ✏ ☎ÿ û✆✕✖✂✄☎✸✕ ✏❾ý❧þ❳ÿ✺✹✘✺úþ ☎✙✍✵✒✟☎ ú✟✘ ÿ✒ ✏✍✪✞✡✕✙✘ ÿ✻✫✄✭✫✯✮ å❥ìë✷✚✺✼✝✘✏❿ÿ➵þ❳ÿ ✍✓✒✟✠ ÿ ★ ÿ ☎ ÿ û✆✽✾✕✙✘ ÿ★ ☎✎✍ ÿ✢✏✍✵✒♣û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✸✕ ✏ ✞✿✕✙✘ÿ ✂ ýtþ❳ÿ✱❀✂ ÿ ☎✎✍ ÿ❁☛✒ ✏ ☎❁☎✎✍✓✒✳☎ ì ✕✖✂ ✏ú✳✘✓✂✆☎✴✒✳✘✵☎✝✒✟✞✄ú✳✘✷✶ ÿ✒ ✏✍❂✞✡✕✙✘ ÿ ☎➏ú ★ ÿ ☎ÿ û✆✽❃✕✙✘ÿ ☎✎✍ ÿ ú✟✠ ÿ û✯✒✟✞✙✞❄✂①ú✳✞❀ ☎✗✕✣ú✳✘✲✚ Ñ✺Ò✫Ó❆Ô✗Õ❉Ö❆❅ Ø✭Ù◆ÚÜÛ❈❇ Þ ì✩íà✗î❱â❱ï å ❉ æ à➋ñ æ ö à❋❊ æ ● ☎✩â å æ ☎✎✍ ÿ ✂①ú✟✞❀ ☎✸✕✣ú✟✘ ★ ÿ ✠ ÿ ✞✜ú✯❍■✂✑✒ ★✕✖✂ ✏ú✳✘✵☎✸✕✙✘❀ ✕✙☎✧ ✚❑❏▲✍✥✕✖✂ ✏ú♣û❡ûÿ ✂✸❍✭ú✳✘★✂✢☎ú✻☎✎✍ ÿ ☎✗✕✙✽ÿ ✒✟☎ þ ✍▼✕ ✏✍◆☎✙✍ ÿ✾✏✍✓✒②û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✗✕ ✏ ✂ û✄✂✄☎ ✏ û❚ú✟✂✄✂✣✚ ❖✝P✓◗❙❘✭Ú▲Ù✥❚✣◗❱❯✓❲✭Ú✔◗❈❳✭Ú✯◗❱❨❆❩▼❬✝❯❀Ù✟❭☞❪❑❫⑤Ù◆Ú✌â❁❊ æ ❴