Consider the total variation of u(a, t)between two points I(t) and a2(t) lying on two char acteristics 2(t If there are no shocks between I(t) and a2(t), then the extrema will not change, and the total variation will stay constant with ti TV(a(a, t))=TV(u(r, 0)) 6.3 Discontinuous Case SLIDE 52 The solution at a shock is determined by yi() and y2(t) provided the shock is entropy satisfying. tA TI(t) y(t) (t) (t) The function in this figure is not ecactly u (a, T), but has the same total variation of u(r, T) since it has the same extrema u(er, T)➟➏➠✛➡▼➢✹➤✖➥❃➦✲➧➩➨❴➫♦➦✿➨❘➠✛➨❴➭✠➯■➲❷➭✛➧❘➤✖➭✓➨❴➤♠➠②➡✢➠✛➳➞➵✆➸➎➺■➻❘➼✹➽✄➾❈➦❵➨❂➚➩➦✲➦✲➡✑➨❂➚➩➠✐➪❈➠✛➤➐➡❊➨❴➢✄➺✙➶✠➸q➼✹➽✄➭✠➡✁➥☛➺▼➹②➸➎➼✹➽➩➯➐➘❚➤♠➡♦➴ ➠✛➡✑➨❂➚➩➠❳➷❦➫▼➭✠➧❦➭✛➷❜➨❴➦✲➧❴➤➐➢✹➨❘➤✖➷✲➢✍➬ ➮❈➱❊✃●❐■❒✗❮✙❰ Ï➳✿➨❘➫♦➦✍➧❘➦✚➭✠➧❴➦❡Ð➞ÑÓÒ✍Ô④Ñ⑧Õ✘Ö⑧Ò❡➾❈➦❵➨❂➚➩➦✲➦✍➡✼➺✞➶✠➸➎➼✹➽P➭✛➡✁➥×➺▼➹②➸➎➼✹➽❵Ø✌➨❘➫♦➦✍➡×➨❘➫♦➦✚➦❵Ù❚➨❘➧❴➦✲Ú♥➭Û➚✌➤➐➯♠➯⑨➡♦➠✠➨ ➷❦➫✁➭✛➡♦➴✛➦②Ø❚➭✛➡✁➥✑➨❘➫♦➦★Ü✘Ñ✁Ü✘Ý♦Þ➏ß♦Ý❃à✓áqÝ❃Ü✘áqÑ❈Ð×â❳á✇ÞqÞ➩Ò✲Ü✘Ý✛ã×Õ✛Ñ❈Ð④Ò✲Ü✘Ý♦Ð✞Ü❳â❳áqÜ✍Ô×Ü✘áqä✭å❈➬ æ⑨ç ➸➎➵✆➸➎➺➞➻✹➼✹➽❘➽❅è æ⑨ç ➸➎➵✆➸➎➺■➻❴é②➽❘➽ ê✄ë✇ì íïî❘ð❊ñ✞ò➏ó❅ô♦î❘ó➩õ✒ò❅õ✒ð✦ö✗÷✆ð❊ø ➮❈➱❊✃●❐■❒✗❮♦ù ú➫♦➦❪➢❘➠✛➯➐û❃➨❴➤♠➠②➡✜➭✠➨⑨➭♥➢❘➫♦➠❃➷❦ü✢➤✖➢✒➥❃➦✲➨❘➦✲➧❴Ú❳➤♠➡✁➦✍➥☛➾❚➘✑ý➶ ➸➎➼✹➽✌➭✛➡✁➥✜ý➹ ➸➎➼✹➽✄➪♦➧❴➠✓➲❚➤➐➥♦➦✍➥✑➨❘➫♦➦✵➢✹➫♦➠❃➷❦ü✑➤➐➢ ➦✲➡❊➨❴➧❘➠②➪❊➘P➢❴➭✓➨❴➤➐➢✹➳➎➘❚➤♠➡✁➴✁➬ ➮❈➱❊✃●❐■❒✗❮♦þ ÿ✁￾✄✂✆☎✞✝✠✟✁✡☞☛✍✌✏✎✑✟✒✌✓✟✔☛✕￾✖✌✘✗✁✙✛✚✜✝✠✢✣✂✤✌✘✗✥✟✁✎✑☛✦✂★✧✪✩✫✡☞☛✍✬✮✭ ➵④➸q➺■➻ æ➽✞✯✦✰ ✝✠☛✱￾✄✩✲✗✤☛✓￾✳✂✤✗✴✩✑✵✒✂✶☛✷✎✑☛✷✩✑✬✪✸✲✩✹✢✺✌✏✩✑☛✍✌✏✎✑✟ ✎✻☎ ➵✆➸➎➺➞➻ æ➽ ✗✺✌✓✟✼✡✞✂✽✌✓☛✾￾✄✩✑✗✿☛✓￾✳✂✽✗✴✩✑✵✒✂❀✂✣✧✫☛✍✢✣✂☞✵✒✩✜❁ ❂✹❃