y+>9SIntch thi-ola tio A a Ap iapicatmhor it varina 3.2 The riemann Problem SLIDE 20 In order to understand the behaviour of the solution at discontinuities it is useful to start with a simplified problem. The Riemann problem is a conservation law together writh piecewise constant data having a single discontinuity at u(x,0) x<0 3. 3 Shock path To determine the shock path we need to go back to the integral form of the (f(a)-f(us)) u(a, t) u(a, t+ot The boundaries a and as are taken sufficiently close to the shock, so that spatial variations of the solution away from the shock become unimportant. They are also taken sufficiently far apart from the shock so that the boundary will not inter fere wnith the shock motion over a time interval ot✜✡✢✤✣✦✥✆✧✩★✫✪✌✬✓✭✔✮✯✬✓✰✲✱✦✳✵✴✷✶✹✸✻✺✼✣✦✽✾✣✦✥✘✿✵✣❁❀✹✪✌❂✷✱❄❃❅✬✓✰❆✿✼❇❄❂✔❃❉❈❄❊❋✿✵●✘❍✔❀✹✪✌❂✷✱■❃❏✬✓✭❑✿✼❇ ❂✩❃▲✮❁❊✁✶◆▼P❖◗✽❘✣❚❙❯✥❱✣✦✥✘✢❲❇✻❇✦✺✼❳❲❨❩✣❚✢✤✺◗●❬✿✼●✘❍❬✢✤●✘❍✷✢❲❙❭✿❪✣✦✽■✥✷✺❪✜❫✢❴✣✻❵❪✿✼❛✦✢❲✽❭❇✡✿◗❇✡❜❝❃❡❞✘✶ ❢❤❣❥✐ ❦♠❧✻♥♣♦✁q✖♥sr✉t✇✈☛✈✫①✔②✷③⑤④❄⑥✖♥⑦r ⑧✎⑨❶⑩❸❷s❹♠❺❩❻ ❼❾❽➀❿✵➁✦➂✼➃➄➁❤➅➆❿❁➇❩❽✎➂✼➃➄➁❯➈❾➅➆➉✵❽✎➂✠➅✌➊✷➃❄➋❚➃❯➊✷➉✵➌❭➍➎❿✵➇❩➁☛❿✖➏❝➅✌➊✷➃❤➈➄❿✵➐➑➇❩➅❥➍➎❿✵❽❱➉❪➅➒➂❪➍➓➈➄➔❯❿❪❽✘➅→➍❸❽✘➇❩➍❸➅→➍➎➃❘➈❤➍❸➅s➍➓➈❝➇P➈➄➃❥➏❯➇❩➐ ➅➣❿↔➈❾➅➆➉✵➁❾➅⑤↕✇➍❸➅✌➊✔➉➙➈❾➍❸➛✻➜✓➐➑➍➝➞➃❯➂❁➜✎➁✦❿◗➋❘➐✤➃➄➛➙➟➀➠✓➊✷➃■➡❄➍➎➃❘➛➙➉❪❽➢❽➙➜✎➁❚❿✼➋➄➐✤➃❘➛➤➍➓➈✹➉➀➔❯❿✵❽✷➈➄➃❘➁❾➌❪➉❪➅→➍➎❿❪❽♠➐✤➉✵↕ ➅➣❿✦➥❶➃❘➅✌➊✷➃➄➁❁↕✇➍❸➅✌➊✾➜✎➍➎➃❯➔❯➃➄↕✇➍➓➈❘➃➙➔❯❿✵❽✷➈❾➅➆➉✵❽✘➅❤➂✼➉❪➅➣➉✾➊✘➉❪➌➦➍❸❽❶➥➧➉↔➈❾➍❸❽P➥✵➐✤➃↔➂✵➍➓➈❘➔❯❿✵❽✘➅→➍❸❽✘➇❩➍❸➅→➨✼➟ ➩ ✬ ➩✎➫ ✮ ➩✲➭ ✪✌✬✲✱ ➩✎➯ ★✁❞ ✬➲✪➯s➳ ❞◗✱◆★➸➵ ✬✰ ➯✑➺ ❞ ✬✓✭ ➯✑➻ ❞ ❢❤❣❥❢ ➼❤❧☛③✡➽✷➾➚①➧t➒➪✷❧ ⑧✎⑨❶⑩❸❷s❹♠❺✲➶ ➠➢❿➹➂◗➃❘➅➣➃❘➁❾➛✩➍❸❽✎➃❱➅✌➊✷➃➀➈✦➊✷❿➦➔❚➘➙➜✘➉❪➅✌➊➴↕⑤➃➀❽✓➃❯➃❚➂✞➅➆❿❬➥❶❿➹➋❯➉✼➔❯➘✔➅➆❿✔➅✌➊✷➃❱➷➣➬✇➮✦➱❾✃✘❐➦❒✎❮◆❰◗Ï➢❐◗ÐÑ❿Ò➏✄➅✌➊✷➃ ➔❯❿❪❽✘➈❘➃➄➁❾➌❪➉❪➅→➍➎❿❪❽✔➐❲➉❪↕❑↕⑦➊❩➍➎➔❚➊✞➍➓➈■➈❯➅→➍❸➐❸➐s➌❪➉❪➐❴➍➎➂❱➉◗➔❘➁✦❿❪➈❯➈Ó➅❸➊✘➃❁➈✦➊✷❿➦➔❯➘✼➟ ➩ ➩➢➫⑤Ô➴Õ➦Ö Õ✚× ✬✹Ø➯ ★Ù❈✆✪➭ ✪✌✬✓✭✇✱➲❈ ➭ ✪➎✬✎✰✲✱✦✱ ➠✓➊✷➃Ú➋❯❿❪➇❩❽✓➂✼➉❪➁❾➍➎➃❘➈ ➯✭ ➉❪❽✎➂ ➯ ✰ ➉❪➁✦➃➹➅➣➉❭➘◗➃❘❽❫➈❾➇❘Û✯➔➄➍➎➃❘❽➢➅❥➐❴➨Ü➔❘➐❲❿✚➈➄➃➹➅➆❿➴➅❸➊✘➃✑➈✦➊✘❿❭➔❯➘✚Ý➙➈➄❿➴➅❸➊✘➉❪➅ ➈→➜✷➉✵➅❥➍➎➉✵➐❩➌✚➉✵➁❾➍➎➉❪➅→➍➎❿❪❽✘➈✻❿Ò➏❝➅❸➊✘➃❝➈➄❿❪➐➑➇❩➅→➍➎❿❪❽➀➉✵↕⑤➉❪➨◆➏❯➁✦❿✵➛Ù➅❸➊✘➃❝➈✦➊✷❿➦➔❚➘✹➋❯➃❚➔❯❿✵➛↔➃✻➇❩❽✘➍❸➛☛➜✷❿✵➁❾➅➆➉✵❽✘➅➣➟✹➠✓➊✷➃➄➨ ➉❪➁❚➃❁➉❪➐Þ➈➄❿✩➅➣➉❭➘◗➃❘❽➧➈❾➇❘Ûß➔❘➍➎➃➄❽✘➅→➐➑➨➞➏❘➉✵➁■➉❯➜✷➉✵➁❾➅✘➏❯➁✦❿✵➛à➅❸➊✘➃❄➈✦➊✷❿➦➔❚➘✾➈➄❿✾➅✌➊✷➉❪➅⑤➅❸➊✘➃❁➋❚❿✵➇❩❽✎➂◗➉❪➁❾➨✩↕✇➍❸➐❸➐✎❽✓❿❪➅ ➍❸❽✘➅➣➃➄➁✌➏❘➃❘➁✦➃✹↕✇➍❸➅✌➊➧➅✌➊✷➃Ó➈Ò➊✘❿❭➔❯➘➀➛↔❿✵➅❥➍➎❿✵❽♠❿✵➌✚➃➄➁✾➉➙➅→➍❸➛➙➃Ó➍❸❽➢➅➆➃➄➁❾➌✚➉✵➐✘á➫ ➟ ⑧✎⑨❶⑩❸❷s❹♠❺✷❺ â ❞