1 Scalar Conservation laws 1.1 Definit 1.1.1 Conservative form ID af (u) u(a, t): is the unknown conserved quantity f(u): is the flux 1.1.2 Primitive form t at du Note 1 More g n some applications, the fux function f may depend explicitly (not through u) on i.e. f(u, r). In such cases, the primitive form of the equation becomes where a(u)=2f; and g(u)=-2f plays the role of a source term The procedures presented here will be generally applicable, sometimes with small modifications, to this more general form. However, for clarity of preser tation we will restrict ourselves to the case where f can be determined once u is k❉ ❊☎❋✫●✌❍■●✣❏▲❑✂▼✗◆✲❖✜P✛❏✴◗✣●❙❘❅❚✑▼❯◆❲❱❳●✫❨✰❖ ❩❅❬❭❩ ❪❴❫✡❵✮❛❝❜❡❞❢❜❭❣❂❛❝❤ ✐✽❥❦✐✽❥❦✐ ❧❯♠✽♥✜♦✭♣rq✭s✉t✉✈①✇②s✽♣④③✺♠⑤q⑦⑥ ⑧✽⑨r⑩❷❶✺❸❀❹ ❺❙❻■❼❽❻■❾➀❿➂➁➄➃②➅➆❾➈➇➊➉❭➋①➌✮➍✎➎ ➏✽➐ ➏✽➑✗➒ ➏✡➓ ➉➐ ➍ ➏✽➔ →❴➣ ➐ ➉➔✜↔❭➑ ➍↕➎❢➙➜➛↕➝➟➞❽❻✣➠✉❼✉➡➢❼✉➅⑦➤✫❼✲➥✑➅➆❼⑤➛❭❻①❾➟➦➧❻①➨☎➩r➠❽❿➆❼r➝➟➙➫➝❡➭ ➉❦➇➯❿➆➛➈➛■➲❢➇➳➅➆➇➳❻①❼➧➝➈➠✉➇✲➲✉➞✉❻✭❿⑦➝①➲✺➵■➵✑➵➀➍ ➓ ➉➐➍✟➎✉➙➜➛✟➝➟➞❽❻✮➸❽➠❢➺ ✐✽❥❦✐✽❥➜➻ ➼✮q⑦✇❦⑥➽✇②✈①✇❦s⑤♣➾③✜♠❽q⑦⑥ ⑧✽⑨r⑩❷❶✺❸➪➚ ➶✟❿➂❼✲❿➂➁➜➛➟➅✗➹✽❻✣➤✫❾➈➙➘➝➟➝➟❻■❼➪➵■➵①➵ ➏✽➐ ➏✽➑ ➒ ➏➄➓ ➉➐➍ ➏✽➔ → ➏✽➐ ➏⑤➑ ➒✶➴➓ ➴➐ ➏✽➐ ➏✽➔ →❴➣ ➏✽➐ ➏✽➑ ➒❀➷ ➉➐ ➍ ➏✽➐ ➏✽➔ →❴➣ ➤✫➞✉❻①❾➟❻ ➷ ➉➐ ➍ →➬➴➓ ➴➐ ➵ ➮ ➋ ➱➪✃✡❐✎❒✙❮ ❰Ï✃➢Ð➧❒✝ÑÒ❒➂ÓÔ❒➂Ð⑦Õ❢Ö✣×Ø✃➢Ó❆Ù➆❒⑦Ð➢Ú➂Õ✡❐➟ÛÜ✃❢ÓÞÝ✹Õ⑤ß❙Ù à❼✪➛➟➅➆➇➳❻✛❿➂á✉á❽➁➫➙➜➥■❿➂➝➟➙â➅➆❼❽➛①➲➂➝➈➞✉❻✫➸❽➠❢➺➳➃②➠✉❼⑤➥✎➝➟➙â➅➆❼ ➓ ➇➯❿ã➭❯➨✉❻■á✽❻■❼❽➨➯❻■➺❢á✉➁➫➙➜➥✑➙➫➝➟➁â➭✲➉②❼✉➅➆➝Ô➝➟➞✉❾➈➅➆➠❽ä➆➞ ➐➍ ➅➆❼ ➔✺å ➙æ➵ ❻➆➵ ➓ ➉➐✜↔➟➔➍✎➵ à❼➾➛➟➠❽➥➀➞✲➥■❿➆➛➟❻①➛①➲➢➝➟➞❽❻✣á✉❾➟➙â➇➳➙➘➝➈➙➫➦➧❻❙➃②➅➆❾➈➇ç➅➂➃❆➝➟➞✉❻✣❻✭➩r➠❽❿⑦➝➈➙➫➅➧❼Ò➹✽❻①➥■➅➆➇➳❻①➛ ➏✽➐ ➏⑤➑ ➒❳➷ ➉➐➍ ➏⑤➐ ➏⑤➔ →❴è ➉➐ ➍ ➤✫➞✉❻①❾➟❻ ➷ ➉➐ ➍ →✶éãê é✭ë å ❿➂❼⑤➨ è ➉➐ ➍ →íì☎éãê é①î á❽➁â❿ã➭❢➛✟➝➟➞❽❻✣❾➟➅➧➁➫❻✮➅➂➃✴❿➳➛❭➅➧➠✉❾➈➥■❻❙➝➟❻■❾➈➇✲➵ ï➞✉❻➪á✉❾➈➅➢➥■❻①➨❢➠❽❾➟❻✭➛✪á✉❾➈❻①➛➟❻■❼r➝➟❻✭➨❴➞✉❻①❾➟❻✝➤✫➙➫➁â➁✮➹⑤❻➪ä➆❻①❼✉❻■❾➀❿➂➁â➁➫➭❴❿➆á✉á✉➁â➙â➥①❿➂➹✉➁â❻➆➲❝➛➟➅➆➇➳❻■➝➟➙â➇✗❻✭➛✥➤✫➙➘➝➈➞ ➛➟➇➳❿➆➁➫➁✜➇➳➅➢➨✉➙➘ð⑤➥①❿⑦➝➈➙➫➅➧❼❽➛■➲⑤➝➟➅☎➝➟➞✉➙➜➛✬➇✗➅➧❾➟❻Øä➆❻■❼❽❻■❾➀❿➂➁❇➃②➅➧❾➟➇✲➵❝ñ❝➅⑦➤✹❻①➦➆❻①❾①➲❽➃②➅➆❾✬➥■➁â❿➆❾➟➙➫➝❡➭✥➅➂➃Ôá✉❾➈❻①➛➟❻■❼❢ò ➝➈❿➂➝➟➙â➅➆❼➾➤✟❻Ø➤✫➙â➁➫➁✜❾➟❻✭➛❡➝➈❾➟➙➜➥✎➝✬➅➆➠✉❾➀➛❭❻①➁➫➦➧❻①➛↕➝➈➅✪➝➈➞✉❻❯➥①❿➆➛➟❻✌➤✫➞✉❻■❾➈❻ ➓ ➥■❿➆❼➾➹⑤❻✗➨❢❻✑➝➈❻■❾➈➇➳➙➫❼✉❻✭➨➾➅➆❼⑤➥✑❻ ➐ ➙➜➛↕➡r❼❽➅⑦➤✫❼❇➵ ➋