The point of tangency u* is precisely the post-shock value. The straight line represents the shock jumping from u=0 to u=u* and the segment of f(u) above the tangency point represents the rarefaction wave. Note that the slope f the straight segment is precisely the shock speed The fact that the line is tangent to the curve means that the shock moves at the same speed as the at the edge of the rarefaction fan It can bee seen that if the shock were connected to some point below u*, then he solution would be multi-valued, and if the shock were connected to some point above u* the entropy condition would be violated 6 Total variation 6.1 Definition SLIDE 49 We examine now a property of the scalar conservation laws that will be useful in developing numerical approximations. The total variation of a function u is TV(u) TV(u)=ub-ua+luc-ub +lud=uc +ue -ud|+luy -ue we note that the total variation can be related to the marima and minima of the function as follow TV(u)=ub-ua+luc-ub +lu + -udl+ 2(ub+ud)-2(uc tue) mothma nama) 6.2 Continuous case Let the above function u correspond to the initial condition for our conservationà❀á❝â➀ã☞ä❤å✚æ✖ç❏ä♥è❀ç✽é❤æ❝ê❤â✑æ❇ë✒ìîí✲ï➓å✚ð❏ã❝ñ✍â➌ë✒å❥ð❩â✑ò➣ì➟ç✍á❝â✄ã☞ä❾ð❩ç❩ó↕ð✍á❝ä❳ë✎ô❽õ♠é♥ò✚ö❝â❤÷✙à❀á❝â✄ð❩ç✍ñ✎é♥å✚ê❤á✖ç✔ò✚å➣æ❝â ñ✽â✢ã❝ñ✽â✑ð✍â✢æ✖ç✽ðsç✽á❝â✄ð❩á❇äPë✎ô✄ø❊ö❝ù➓ã❝å➣æ❇ê❚è⑩ñ✍ä❾ùúíüûþý⑨ç✍ä❚í❘ûÿí✲ï❢é❤æ✁￾îç✍á❝â✄ð✍â✢ê❾ù❢â✑æ✖ç✔ä♥è✄✂✆☎❞í✞✝ é✠✟☞ä♠õ❤â✡ç✽á❝â✷ç✎é♥æ❝ê❾â✢æ❇ë✢ì⑨ã☞ä❤å✚æ✖çsñ✽â✢ã❝ñ✽â✑ð✍â✢æ✖ç✽ð✭ç✍á❝â✔ñ✎é♥ñ✽â✒è❞é❾ë↔ç✽å➣ä❾æ☛✡❀é■õ❤â❾÷✌☞❶ä♥ç✽â✷ç✍á✤é♠ç✞ç✽á❝â❏ð✍ò➣ä❾ã✤â ä♥è①ç✍á❝â➀ð❩ç✍ñ✎é♥å✚ê❤á✖ç✷ð❩â✑ê❤ù➓â✢æ✖ç✷å✚ð✡ã❇ñ✍â➌ë✒å❥ð❩â✑ò➣ì✙ç✽á❝â✄ð❩á❇äPë✎ô➟ð✍ã☞â✢â✍￾◆÷❢à❀á❝â➓è❞é❾ë↔ç✡ç✍á✤é♠ç✡ç✍á❝â➀ò✚å✚æ❝â➓å✚ð ç✽é❤æ❝ê❤â✑æ✖ç❢ç✽ä ç✍á❝â➟ë✢ö❝ñ✽õ❤â✙ù➓â✑é♥æ✤ð➓ç✍á❇é♥ç➀ç✍á❝â➟ð✍á❝ä❳ë✎ô❴ù➓ä♠õ❤â➌ð➀é♠ç➀ç✽á❝â➟ð✍é❤ù➓â✙ð✍ã☞â✢â✍￾ é❾ð➓ç✍á❝â ë✎á❇é❤ñ✽é❾ë↔ç✍â✑ñ✍å❥ð❩ç✍å❥ë✞é♠ç✭ç✽á❝â✷â✍￾❝ê❤â✡ä♥è➡ç✍á❝â✷ñ✎é♥ñ✽â✒è❞é❾ë↔ç✍å✚ä❤æ❧è❞é♥æ◆÷ ✎➁çsë✑é♥æ✏✟☞â✢â❼ð✍â✢â✑æ✙ç✍á❇é♥ç✞å➣è✱ç✍á❝â❢ð❩á❝ä❳ë✎ô✑✡①â✑ñ✍â❏ë✢ä❤æ❝æ❇â✑ë↔ç✽â✍￾✩ç✽ä❧ð❩ä❾ù➓â✔ã✤ä❾å➣æ✖ç✒✟☞â✢ò✚ä✓✡ í✲ï✠✔✤ç✍á❝â✑æ ç✍á❇â❧ð❩ä❾ò➣ö❝ç✍å✚ä❤æ✕✡✴ä❤ö❝ò✖￾✗✟✤â⑨ù❏ö❝ò➣ç✍å➣ó⑦õ♠é❤ò➣ö❝â✘￾✙✔✱é♥æ✁￾ å➣è❀ç✍á❝â✩ð❩á❇äPë✎ô✚✡①â✑ñ✍â✄ë✢ä❤æ❝æ❇â✑ë↔ç✽â✍￾✺ç✽ä❽ð❩ä❾ù➓â ã☞ä❤å✚æ❾ç✞é✛✟✤ä♠õ❾âsí◆ï✞ç✍á❇â✷â✢æ✖ç✍ñ✽ä❤ãPì✄ë✒ä❾æ✁￾❳å➣ç✍å✚ä❤æ✜✡✴ä❤ö❝ò✖￾✢✟☞â✷õPå➣ä❾ò✚é♥ç✍â✘￾✲÷ ✣ ✤✦✥★✧✆✩✫✪✭✬✮✩★✯✱✰✲✩✳✧✆✰✲✥✵✴ ✶✄✷✹✸ ✺✼✻✙✽✿✾✒❀❂❁❃❀✹❄❅✾ ❆❈❇❊❉●❋■❍❑❏▼▲ ◆✏❖P❖❘◗❚❙✠❯✵❱●❲✞❖❳❲❈❨✠❩❬❙❪❭✞❫❘❨❴❭✁❖❵❫❜❛❞❝❡❨❂❢✵❛●❣✁❖✐❤✲❥❦❙✓❧♠❙✠❫♥❥❦❨✠❲♦❤✲❖❵❫❜♣✓❙✓❛❞❱q❨✓❲r❧♠❙✓❩✆❤✐❛●❣✁❙✓❛s❩t❱●❧●❧❅✉❦❖❳✈❚❤✲❖✇❢❦✈❃❧ ❱●❲✗①②❖❵♣✓❖✲❧♠❨❴❭✞❱●❲❊③✜❲▼✈❃❯❳❖✲❫❜❱q❥❦❙✓❧④❙❦❭✛❭✞❫❘❨✲◗②❱●❯♥❙✓❛❞❱q❨✓❲♦❤✲⑤✜⑥✞❣♦❖✐❛⑦❨✓❛⑦❙✓❧⑧♣✓❙✓❫❜❱q❙✠❛✇❱q❨✠❲r❨❂❢❪❙⑨❢❦✈❃❲❈❥✲❛✇❱q❨✠❲⑨í⑩❱✖❤ ①✛❖✇❶✱❲✞❖❦①✢❙❷❤✲❸ ❹⑨❺♥☎⑩í✙✝➎û❼❻❾❽❽ ❽ ❽❞❿í ❿❈➀ ❽ ❽ ❽ ❽❘➁➀ ❹⑨❺♥☎❞í✞✝❵û ➂ í✙➃✆➄✺í✞➅❃➂✘➆➇➂ í✙➈✆➄ í✞➃✍➂✍➆➉➂ í✙➊✌➄✺í✞➈✘➂✍➆➉➂ í✞➋❅➄îí✙➊❚➂✲➆➉➂ í✙➌✿➄✺í✞➋✓➂ ◆✏❖♥❲❈❨✠❛➍❖✢❛➎❣♦❙✓❛⑨❛●❣✁❖P❛⑦❨✓❛⑦❙✓❧❅♣❷❙✠❫❜❱q❙✓❛❞❱q❨✓❲✼❥❴❙✠❲⑩✉❦❖♥❫❴❖❵❧♠❙✠❛➍❖❦①❡❛➍❨❡❛●❣✁❖P❯♥❙✲◗②❱●❯♥❙❑❙✓❲✞①✏❯✐❱●❲✁❱●❯♥❙❑❨✹❢ ❛➎❣♦❖➏❢❦✈❃❲❈❥✲❛✇❱q❨✠❲✚❙❷❤✱❢❵❨✓❧●❧➐❨✓❩✆❤✲❸ ❹⑨❺♥☎❞í✞✝❵û ➂ í✙➃✆➄✺í✞➅❃➂✘➆➇➂ í✙➈✆➄ í✞➃✍➂✍➆➉➂ í✙➊✌➄✺í✞➈✘➂✍➆➉➂ í✞➋❅➄îí✙➊❚➂✲➆➉➂ í✙➌✿➄✺í✞➋✓➂ û ➑♦☎❞í✞➃■➆❴í✙➊✍✝✆➄✕➑♦☎❞í✞➈④➆❄í✙➋❜✝✆➄✺í✞➅✄➆❴í✞➌ û ➑❪➒❷➓ ❯❳❙✍◗✛❱●❯♥❙✄➄✭➓ ❯✵❱●❲▼❱●❯❳❙✲➔ ✶✄✷✇→ ➣✚❄❅✾➏❁♦❀✹✾✄↔✒❄❅↔✒↕⑩➣✗➙✆↕❊✻ ❆❈❇❊❉●❋■❍✗➛❃➜ ➝➞❖❵❛✆❛➎❣♦❖✿❙✛✉❦❨✓♣✓❖t❢❦✈❃❲✞❥❵❛❞❱q❨✓❲❢í✭❥❴❨✠❫❜❫❘❖❵❤✇❭✁❨✓❲✞①✵❛➍❨❪❛●❣✁❖✳❱●❲✁❱●❛❞❱q❙✓❧✙❥❦❨✓❲❈①✠❱●❛✇❱q❨✠❲✫❢❵❨✓❫✿❨✓✈❃❫⑨❥❦❨✓❲✁❤✲❖❵❫❜♣✓❙✓❛❞❱q❨✓❲ ❧♠❙✠❩❅⑤ ➑✛➑