3 Discontinuous solution 3.1 Shock formation 19 SHOCK PA When the characteristics cross, the function u(a, t) has an infinite slope. A discontinuity or shock forms, and the differential equation is no longer valid Note 7 Vanishing viscosity approach After the characteristics have crossed, there are some points a where more than one characteristic leads back to t=0. This would imply that the solution is multi-valued at such a point, which in most cases is not physically realisable The correct physical behaviour can be determined by recalling that the inviscid Burgers'equation was a simplified version of a viscous equation with a term ea on the right hand side. If the initial data is smooth and e is very small, then this term is negligible compared to the lefthand-side terms, and the solution of the almost identical to that of the inviscid equation. However, as the discontinuity begins to form, amf becomes very large, and the viscous term becomes important. This term keeps the solution smooth for all time(recall the equation is now parabolic), and determines the correct physical nature of th hown in the figure below This behaviour is evident in the equations governing fluid flow. The Euler equations, which ignore the viscous terms, are hyperbolic and admit discontin lous solutions. Conversely, the Navier-Stokes equations are parabolic, and the viscosity ensures that the solution is always smooth D Exercise 1 (from [LvI) Show that the viscous Burgers'equation has a trav elling wave solution of the form u(a, t)=(as tt) by deriving an ODE for and verifying that this OdE has solutions of the form f (uh s ui )[1 s tanh((uh s ui )y/4e1❵ ❛❝❜✣❞❢❡☞❣✾❤❃✐❥❜✛❤❧❦❂❣❃❦❂❞♥♠✻❣✾♦✛❦✪✐❥❜✣❣❃❤ ♣✝qsr t✝✉✇✈☞①✓②④③✁✈⑥⑤✵⑦⑨⑧❢⑩✥❶s✈❸❷ ❹✲❺✷❻✙❼❾❽➀❿❆➁ SHOCK PATH ➂➄➃✓➅❱➆➀➇✯➃✓➅➉➈✄➃✵➊❆➋✯➊➌➈✆➇✯➅✛➋✯➍➏➎s➇✔➍✖➈✛➎❃➈✣➋✯➐➌➎✯➎✛➑❸➇✔➃✓➅◆➒✎➓✓➆✵➈✆➇✯➍➔➐➌➆➣→❢↔➙↕❾➛s➜s➝❃➃■➊❆➎❧➊❆➆➀➍➏➆✥➞✵➆✓➍➔➇✔➅✬➎s➟➏➐❆➠✲➅❆➡➤➢ ➥➍✖➎✔➈✛➐❆➆✷➇✔➍➏➆▼➓✓➍➔➇➧➦◆➐❆➋❙➨➫➩➯➭❄➲➫➳➀➒✎➐❆➋✯➵✑➎✛➑❾➊❆➆➥ ➇✔➃✓➅ ➥➍➔➸▲➅✛➋✯➅✛➆✷➇✔➍✖➊✳➟❥➅➫➺➌➓■➊✟➇✔➍➏➐❆➆➻➍✖➎❙➆✓➐❂➟➔➐➌➆✓➼❆➅❱➋✱➽✦➊❆➟➔➍➥ ➡ ➾❈➚ ➪✇➶ ➹➻➘➷➴✆➬✤➮ ➱❸✃✥❐❢❒➙❮➫❰❢❒➙❐➯ÏÐ➱✱❒➙❮❆Ñ❆➘✥❮❱❒✔➴➙ÒÔÓ✌Õ❢Õ❥Ö➌➘✓✃✲Ñ✣❰ ➢☞➒✙➇✯➅✛➋❁➇✔➃✓➅×➈✄➃✵➊✳➋✄➊❆➈✣➇✔➅✛➋✯➍✖➎➧➇✯➍➏➈❱➎Ø➃✵➊✦➽❆➅✇➈✛➋✔➐✷➎✔➎✔➅➥ ➑✟➇✔➃✓➅❱➋✔➅×➊✳➋✯➅✇➎s➐➌➵✌➅☞➠■➐➌➍➔➆✷➇✯➎❁↕✻Ù☞➃✵➅✛➋✯➅☞➵❃➐➌➋✔➅✁➇✔➃✵➊❆➆ ➐❆➆✵➅❧➈✄➃✵➊✳➋✄➊❆➈✣➇✔➅✛➋✯➍✖➎➧➇✯➍➏➈✾➟➔➅➫➊➥➎✺Ú✵➊❆➈✄Û❋➇✯➐❋➜❙ÜÞÝ✓➡✻ß✝➃✵➍➏➎✢Ù❁➐➌➓✓➟➥ ➍➏➵✌➠✓➟➔➦➉➇✔➃✵➊✳➇✺➇✔➃✵➅❧➎s➐➌➟➔➓✥➇✯➍➔➐➌➆à➍➏➎ ➵✢➓✵➟✿➇✯➍✿á✴➽✟➊✳➟➏➓✓➅➥ ➊✟➇✺➎✔➓✵➈✄➃✤➊❑➠■➐➌➍➔➆✷➇❱➑➯Ù☞➃✓➍✖➈✄➃➻➍➏➆à➵✌➐➌➎s➇✢➈✛➊❆➎✔➅❱➎❈➍✖➎❙➆✓➐❆➇✺➠✓➃▼➦▼➎✔➍✖➈✛➊✳➟➏➟➏➦◆➋✔➅➫➊✳➟➏➍➏➎✯➊✳Ú✵➟➔➅➌➡ ß✝➃✓➅❈➈✣➐❆➋✯➋✯➅❱➈✆➇⑥➠✵➃✷➦✥➎✔➍➏➈❱➊✳➟■Ú■➅❱➃✵➊✦➽▼➍➔➐➌➓✓➋❁➈❱➊✳➆✑Ú✲➅ ➥➅✛➇✔➅❱➋✔➵✌➍➏➆✓➅➥ Ú▼➦❃➋✔➅➫➈✛➊❆➟➔➟➏➍➔➆✵➼❙➇✔➃✵➊✳➇❸➇✯➃✓➅✇➍➏➆▼➽✷➍✖➎✯➈✣➍➥ âã➓✓➋✯➼❆➅✛➋✄➎❱ä✣➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆❙Ù✝➊❆➎❾➊×➎s➍➏➵✌➠✓➟➏➍✿➞✵➅➥ ➽❆➅❱➋✯➎✔➍➏➐❆➆✱➐✳➒✵➊✇➽✷➍✖➎✯➈✣➐❆➓■➎❄➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆❙Ù☞➍➔➇✔➃✢➊✁➇✔➅❱➋✔➵æå✥ç➫èsé ç➫ê è ➐❆➆❃➇✯➃✓➅✁➋✯➍➔➼➌➃➌➇✯➃✵➊✳➆➥ ➎✔➍➥➅❆➡Øë✴➒▲➇✔➃✓➅✇➍➔➆✓➍➔➇✔➍✖➊✳➟ ➥➊✟➇✄➊❙➍✖➎⑥➎s➵✌➐▼➐✳➇✯➃❧➊✳➆➥ å❁➍➏➎❸➽❆➅❱➋✔➦✢➎✔➵✑➊✳➟➏➟✗➑❆➇✔➃✓➅❱➆✑➇✔➃✵➍➏➎ ➇✔➅❱➋✔➵ì➍✖➎☞➆✓➅✛➼➌➟➔➍➏➼❆➍➏Ú✓➟➏➅✢➈✣➐➌➵✌➠✵➊✳➋✯➅➥ ➇✔➐✑➇✯➃✓➅✺➟➏➅✣➒✙➇✯➃✵➊✳➆➥á✴➎✔➍➥➅❙➇✔➅❱➋✔➵✑➎❱➑■➊✳➆➥ ➇✔➃✓➅✢➎✔➐❆➟➏➓✥➇✯➍➔➐➌➆❋➐✳➒Ø➇✔➃✓➅ ➽▼➍➏➎✯➈✣➐➌➓✵➎➯➅❱➺✷➓✵➊✟➇✯➍➔➐➌➆✢➍✖➎❥➊✳➟➏➵✌➐➌➎s➇❢➍➥➅✛➆✷➇✯➍➏➈❱➊✳➟✷➇✔➐×➇✔➃✵➊✳➇❥➐✳➒✵➇✯➃✓➅ã➍➔➆▼➽▼➍➏➎✯➈✣➍➥ ➅❱➺✷➓✵➊✟➇✯➍➔➐➌➆❄➡➯í✇➐✟Ù❁➅❱➽❆➅❱➋❱➑✟➊❆➎ ➇✔➃✵➅ ➥➍➏➎✯➈✣➐➌➆➌➇✯➍➔➆▼➓✓➍➔➇➧➦✑Ú✲➅✛➼❆➍➏➆✵➎ã➇✯➐✾➒✎➐➌➋✔➵❂➑ ç➫èsé ç➫ê è Ú■➅➫➈✣➐➌➵❃➅➫➎ã➽❆➅✛➋✯➦✑➟✖➊✳➋✯➼❆➅❆➑✥➊❆➆➥ ➇✯➃✓➅✱➽▼➍➏➎✯➈✣➐➌➓✵➎❁➇✔➅✛➋✯➵ Ú✲➅❱➈✣➐➌➵✌➅❱➎Ø➍➏➵❃➠✲➐❆➋✔➇✯➊❆➆✷➇❱➡Øß✝➃✓➍➏➎❥➇✔➅✛➋✯➵îÛ➌➅✛➅❱➠✵➎Ø➇✔➃✵➅✁➎s➐➌➟➔➓✥➇✯➍➔➐➌➆✑➎s➵✌➐▼➐✳➇✯➃✾➒✎➐➌➋⑥➊✳➟➏➟✥➇✔➍➏➵✌➅✺↔✎➋✯➅❱➈❱➊✳➟➏➟✥➇✔➃✓➅ ➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆➻➍➏➎✱➆✵➐✟Ùï➠✵➊❆➋✯➊❆Ú■➐➌➟➔➍✖➈❱➝✣➑❾➊✳➆➥✬➥➅✛➇✔➅❱➋✔➵✌➍➏➆✓➅❱➎✱➇✯➃✓➅❧➈✣➐➌➋✔➋✯➅❱➈✣➇❈➠✵➃✷➦✥➎✔➍➏➈❱➊✳➟❥➆✵➊✳➇✔➓✓➋✯➅✌➐✳➒ã➇✔➃✓➅ ➎✔➦▼➎s➇✔➅❱➵ð➊❆➎☞➎✔➃✓➐✟Ù☞➆❑➍➔➆❑➇✯➃✓➅✱➞✵➼❆➓✵➋✔➅✱Ú✲➅✛➟➏➐✟Ù❙➡ limiting solution ε -> 0 ß✝➃✓➍✖➎✻Ú✲➅✛➃■➊✦➽✷➍➏➐❆➓✵➋✻➍✖➎✪➅❱➽▼➍➥➅❱➆✷➇✪➍➏➆ñ➇✔➃✵➅✬➅➫➺➌➓■➊✟➇✔➍➏➐❆➆■➎❧➼❆➐✟➽➌➅✛➋✯➆✓➍➏➆✓➼àò✵➓✓➍➥ ò✵➐✟Ù❙➡óß✝➃✓➅à➪⑥➓✓➟➏➅✛➋ ➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆✵➎❱➑✥Ù☞➃✓➍✖➈✄➃✪➍➏➼❆➆✵➐❆➋✯➅×➇✯➃✓➅❙➽▼➍➏➎✯➈✣➐➌➓✵➎ã➇✔➅❱➋✔➵✑➎❱➑✓➊✳➋✯➅✱➃✷➦▼➠✲➅✛➋✯Ú■➐➌➟➔➍✖➈✱➊✳➆➥ ➊➥➵✌➍✿➇ ➥➍✖➎✯➈✣➐❆➆✷➇✯➍➔➆✥á ➓✓➐➌➓✵➎✁➎✔➐❆➟➏➓✥➇✔➍➏➐❆➆✵➎❱➡✁ô❁➐❆➆▼➽❆➅❱➋✯➎✔➅✛➟➏➦❆➑▼➇✯➃✓➅✾➾×➊✦➽✷➍➏➅✛➋✔á➧õ✷➇✔➐➌Û❆➅❱➎ã➅➫➺➌➓■➊✟➇✔➍➏➐❆➆■➎✁➊✳➋✯➅❙➠✵➊✳➋✄➊✳Ú✲➐❆➟➏➍✖➈✳➑✵➊❆➆➥ ➇✔➃✓➅ ➽▼➍➏➎✯➈✣➐✷➎s➍➔➇➧➦✑➅✛➆■➎s➓✓➋✯➅❱➎ã➇✯➃✵➊✟➇☞➇✯➃✓➅✺➎✔➐❆➟➏➓✥➇✔➍➏➐❆➆❑➍➏➎☞➊❆➟➔Ù✝➊✦➦✥➎✝➎s➵✌➐▼➐✳➇✯➃❄➡ ö❂÷✇ø➬✳Ö➌Ñ➫❒✗❮✳➬➻ù◆↔✙➒✎➋✯➐❆➵ûúü➌ý✁þ✎➝☞õ▼➃✓➐✟ÙÐ➇✔➃✵➊✳➇☞➇✔➃✓➅✢➽✷➍✖➎✯➈✣➐❆➓■➎☞â❁➓✵➋✔➼➌➅✛➋✄➎✛ä✥➅➫➺➌➓■➊✟➇✔➍➏➐❆➆❑➃✵➊➌➎✁➊✾➇✔➋✄➊✦➽➌á ➅✛➟➏➟➏➍➔➆✓➼✌Ù✝➊✦➽❆➅✱➎s➐➌➟➔➓✓➇✔➍➏➐❆➆❑➐✳➒❾➇✯➃✓➅✱➒✎➐❆➋✯➵ð→➷ÿ✆↔➙↕❾➛s➜s➝❁Ü✁￾✢↔➙↕✄✂✆☎✛➜s➝ãÚ▼➦ ➥➅✛➋✯➍➔➽▼➍➏➆✓➼✑➊✳➆✞✝✠✟×➪➀➒✎➐❆➋✡￾ ➊✳➆➥ ➽➌➅✛➋✯➍✿➒✎➦▼➍➏➆✓➼❃➇✔➃✵➊✳➇☞➇✔➃✓➍✖➎☛✝✠✟×➪➣➃■➊❆➎☞➎✔➐❆➟➏➓✥➇✔➍➏➐❆➆✵➎✝➐❆➒❾➇✯➃✓➅✱➒✎➐❆➋✯➵ ￾✢↔✌☞✓➝❸Ü➄→✎✍✑✏ ➶ ✒ ↔✎→✓✍✔✂➻→✓✕▲➝✛ú✿➶☛✂➉➇✄➊✳➆✓➃❾↔✔↔✎→✓✍✔✂➻→✓✕▲➝✖☞✘✗✚✙➌å✆➝ þ ✛