Numerical schemes for Scalar one- dimensional Conservation laws Lecture 12
✂✁☎✄✝✆✟✞✡✠☞☛✍✌✏✎✒✑✓☛✕✔☎✆✖✄✗✆✟✘✂✙✛✚✜✞✢✑✓☛✣✌✏✎✤✌✥✞✧✦✩★✪✆✬✫✡✭✢✠☞✄✗✆✟★☎✘✮✠☞✚✯★✰✌✏✎ ✱✚✯★☎✘✲✆✖✞✴✳✵✌✍✶✲✠☞✚✯★✸✷✹✌✻✺✼✘ ✷✽✆✖☛✾✶✕✁☎✞✿✆✗❀✣❁
1 Finite volume discretization 1.1 Computational Cells tn=n△t j-13j1lJ 1.2 Cell averages Recall that in finite differences un a u(a, tn) We think of u; as representing cell averages u (r, t") This "new"interpretation can be easily extended to irregular grids 2 Conservative methods 2.1 Definition SLIDE 3 Applying integral form of conservation law to a cell d(+ud=-f(a(x+,t)-f(u(x1-+,) We consider here only explicit schemes, but implicit schemes are also possible
❂ ❃❅❄❇❆❈❄❊❉✕❋❍●❏■▲❑❇▼❈◆✗❋P❖◗❄☞❘✕❙✣❚✾❋❯❉✾❄☞❱✣❲❳❉✬❄❨■▲❆ ❩✾❬❭❩ ❪❴❫✻❵❜❛❳❝✣❞❢❡✕❞❢❣❭❫✻❤✍❡✾✐✓❪❴❥✮✐❦✐❭❧ ♠✴♥♣♦rq✮s✹t ✉❢✈❳✇②①♣③▲✉ ④⑥⑤ ✇❅⑦✮③④ ❩✾❬⑨⑧ ❪❴❥✮✐❦✐⑩❡✿❶✻❥✿❷❸❡✬❹❺❥✮❧ ♠✴♥♣♦rq✮s❼❻ ❽❿❾❊➀❊➁➃➂r➂✿➄➆➅❸➁➇➄✟➈r➉✵➊❺➉➋➈r➄➌❾▲➍➃➈➎❿❾❨➏❦❾❨➉✴➀❊❾➑➐➓➒➔⑤ ✈➓→ ➔✬➣➆✉❢✈✛↔ ④⑤❸↕❨➙ ➛❼➜✥➝❦➞❸➟➡➠❸➢➥➤✛➦✯➒➔⑤ ✈❈➧➩➨❯➫➜❇➭➫➜ ➨ ➜❨➠♣➝❦➟➡➠❸➯➳➲☞➜❨➵➡➵ ➧➺➸➜ ➫➻➧➯✛➜ ➨ ➔➒ ⑤ ✈ → ➼ ③▲✉ ➽➚➾❇➪⑨➶✬➹➘ ➾ ➪⑥➴✬➹➘ ➔✬➣➆✉✲↔ ④⑤ ↕✡➷✉ ➬➮➅➱➈✃➐➚❐❒➉✴❾❨❮✏❰✥➈r➉➋➄➌❾❨➏❒Ï✴➏❦❾❨➄Ð➁➃➄❒➈ÒÑ➃➉❼➀❊➁➇➉❼Ó❊❾▲❾❊➁➃➐❊➈r➂ÕÔ➓❾❦Ö➩➄Ð❾☞➉➮➍✛❾❊➍➳➄ÐÑ×➈r➏➑➏❦❾➌Ø✛Ù❢➂Ú➁➃➏❿Ø✛➏➑➈Ò➍➺➐ ➙ Û ÜÝ■▲❆✪❘✕❋✍❚✬Þ✯❲❳❉✬❄➑Þ✵❋àß❋❯❉✾á✪■✓â❈❘ ⑧❯❬❭❩ ãä❥✡å✥❤❿❣⑥❞❢❣❭❫✻❤ ♠✴♥♣♦rq✮s❼æ ç➭è➭❸➵Úé➱➟➡➠❸➯➳➟Ú➠♣➝➻➜❨➯➫❊➧➵➮➦➆➤➫❦ê ➤➇➦✾➲☞➤✛➠➨➜ ➫❦➸➃➧➝➻➟Ú➤➩➠➥➵➧➺ë ➝➻➤ ➧ ➲❨➜❨➵➡➵ ① ➷ ➷④ ➽②➾ ➪❒➶✕➹➘ ➾❇➪Ð➴✬➹➘ ➔ ➷✉☎✇íìïîrð✬➣➆➔✬➣➆✉✈❦ñ❳➹➘ ↔ ④ ↕❭↕ ì✽ð✬➣Ò➔✕➣Ò✉✈☞ò❿➹➘ ↔ ④ ↕❦↕➌ó ➨❦ô➯✛➯➩➜➨ ➝ ➨ ➔➒ ⑤ñ✲õ ✈ ì ➔➒ ⑤ ✈ ③④ ③▲✉☎✇ïì÷ö❇ø⑤ ✈❦ñ ➹➘ ì✽ø⑤ ✈☞ò ➹➘✛ù ú ➔➒ ⑤ñ✕õ ✈ ✇ ➔➒ ⑤ ✈ ì ③④ ③▲✉ ö❇ø⑤ ✈❦ñ ➹➘ ì✽ø⑤ ✈☞ò ➹➘➩ù û❈❾▲➀❊Ñ➃➉è➐➑➈Ò➍✛❾❨➏✏➅❸❾❨➏❦❾➳Ñ➃➉è➂ÕÔ➥❾❦Ö☞Ï✴➂Õ➈Ò➀☞➈r➄✖➐❨➀➻➅è❾☞ü×❾☞➐➻ý❯Ó☞Ù❢➄✟➈rü✍Ï➮➂þ➈Ò➀☞➈r➄✖➐❨➀❊➅❸❾❨ü➳❾☞➐▲➁➃➏❦❾✵➁➃➂þ➐☞Ñ✥Ï❸Ñ➃➐❊➐❊➈ÒÓ❨➂Ú❾ ➙ ➼
2.2 Numerical Flux function F+≡F(u Wi+r) nd F is a numerical flux function of l+r+ l arguments that sat isfies the following consistency condition We will sometimes omit the time superscript with the understanding that left and right hand sides are evaluated at the same time. Thus, the above fur function +是 F 2.3 Lax-Wendroff theorem SLIDE 5 If the solution of a conservative numerical scheme converges as Ac-0 with t fixed, then it converges to a weak solution of the conservation law shock capturing schemes are possible Note 1 The Lax- wendroff Theorem While the Lax-Wendroff theorem shows that if we converge to some solution as he grid is refined, then that solution will be a weak solution of the conser vation law, it does not guarantee that we will converge. In fact the consistency to our integral form of the conservation law is guaranteed if we employ a conservative numerical scheme as defined above. We know that in order to obtain convergence we require some notion of stability. Because we are dealing with a non-linear oblem the concepts of stability used until now are not applicable. At the end this lecture we will give sufficient conditions for a scheme to be non-linearly table and hence convergent The theorem also does not guarantee that the weak solution obtained sat isfies the entropy condition. If more than one weak solution exists for a given problem then different conservative numerical schemes may converge to different answers We will discuss entropy-satisfying schemes later in the lecture
ÿ✁⑨ÿ ✂☎✄✝✆✟✞✡✠☞☛✍✌✏✎✒✑✔✓✕✑✖✄✘✗✚✙✛✄✝✜✢✌✤✣✥☛✖✦✧✜ ★✤✩✫✪✭✬✯✮✱✰ ✲✡✳✍✴✶✵✷✔✸ ✲✺✹✼✻✽✳✿✾❁❀❃❂ ✽✻ ✳✿✾✤❀❄✴❆❅❇❂❉❈❉❈❊❈✿❂ ✽✻ ✳❋❂❉❈❊❈❉❈✿❂ ✽✻ ✳✍✴✯●❊❍ ■❇❏❁❑ ✲▼▲❖◆ ■✕P❘◗❚❙❱❯✫❲❨❳❬❩❪❭❴❫✁❵❚◗❆❛❝❜❞◗❆P❚❩❇❡✼❳❬❢✤P❱❣❇❤❥✐❁❦♠❧✝❦♦♥♣■❇qsr❋t❴✉✇✈❊❏❪① ◆ ①✍②❁■❨① ◆ ■❨① ▲③◆✖④✈◆ ①✍②❴✈ ❤⑤❣❋⑥③⑥③❣❨⑦▲❏❴r✕❩❪❢❁P❚⑧❊❳⑨⑧❉❡✼❯⑩P❆❩❋❶❸❷❉❣❋❏☞❑▲① ▲❣❪❏ ✲✕✹✽ ❂ ✽ ❂❊❈❉❈❉❈✛❂ ✽ ❂ ✽❍❥❹♦❺ ✹✽ ❍ ❻❽❼❿❾➁➀✭➂✭➂❪➃❉➄❇➅✇❼❉➆⑨➀✭➅✕❼✿➃✁➄❨➅➇➀✭➆❘➆⑤➈❴❼❿➆➉➀✭➅✇❼➊➃✛➋✼➌☞❼✿➍➎➃❉➏❉➍✛➀➌✤➆✡❾➁➀✭➆⑤➈✔➆✭➈☞❼❿➋✥➐✤➑❪❼✿➍➎➃✛➆➓➒❨➐✤➑❇➀✭➐✫➔✶➆⑤➈❴➒❇➆❘➂→❼❬➣➎➆✯➒❇➐✤➑ ➍✛➀❄➔↔➈⑩➆➊➈❴➒❇➐✤➑↕➃✛➀❬➑❋❼✿➃✇➒❇➍✍❼✕❼✿➙❨➒❨➂❄➋☞➒❨➆➓❼s➑➛➒❇➆➜➆✭➈☞❼♣➃❉➒❇➅✇❼➇➆➉➀✭➅✇❼✼➝❽➞✏➈⑩➋⑩➃➎➟❿➆⑤➈❴❼✕➒❪➠➎➄❨➙❨❼➜➡❿➋❋➢✢➣➎➋✥➐✏➏✿➆➉➀❬➄❨➐ ❼s➢✛➌✏➍✍❼✿➃s➃✛➀❬➄❇➐✱➀✭➅✝➌✤➂➤➀❬❼✿➃♣➆✭➈☞➒❨➆ ✲➦➥✳✍✴ ✵✷ ✸ ✲➨➧✼✻✽✳✿✾❁❀ ➥ ❂ ✽✻ ✳✿✾❁❀❄✴❆❅ ➥ ❂❉❈❊❈❉❈❉❂ ✽✻ ✳➥ ❂❊❈❉❈❉❈❊❂ ✽✻ ✳✍✴✡●❊➩ ➥ ❈ ÿ✁⑨➫ ➭♣✎❚✗❿➯❊➲➳✞✯✜✝➵✝✠❴✦➁➸➻➺➽➼✘✞✡✦❥✠☞✞✯✆ ★✤✩✫✪✭✬✯✮➽➾ ➚❤❆①s②❴✈ ◆❣❋⑥③t✥① ▲❣❋❏➛❣❇❤➁■✇❩❋❢✤P❆⑧✼❯✫❲↔➪❴❭✥❡❊❳❬➪❁❯➶❏✫t☞✉➇✈❊q▲ ❷❉■❋⑥ ◆ ❷➎②☞✈❉✉✇✈♣❷✿❣❪❏✫➹❪✈❉qsr❋✈ ◆ ■◆✝➘✔➴➬➷➱➮ ⑦▲①✍② ✃✒❐ ✃➁❒ ④❴❮✈❊❑❘❰⑩①s②❴✈❉❏ ▲①✝❩❋❢✤P✏➪❁❯✫❲❨Ï❁❯⑩⑧✇❡✼❢Ð❭✱ÑÒ❯⑩❭✥Ó✺⑧✼❢❁❫⑨◗✯❡✼❳❬❢✤PÔ❣❋❤✯①s②❴✈♣❷✿❣❪❏◆✈❊q✍➹❨■❇① ▲❣❋❏↕⑥❖■↔⑦➦Õ Ö ♥ × ⑧✼Ø❆❢✯❩❊ÓÙ❩❋❭✥Ú❚❡❊◗❆❲❨❳❬P❚Ï ◆ ❷➎②❴✈❊✉➇✈ ◆ ■❇qs✈✶Û✤❣◆s◆✍▲→Ü⑥③✈ Ö✶Ý Þ➽ß❘à✛á✱â ã➜ä✧á➶å➊æ⑩ç✯è❨éêá❇ë❥ì❘í❪ß✫îïã➜ä✧á⑩ß⑩í❪á❇ð ñ②▲ ⑥→✈✢①✍②☞✈✢ò✯■❮⑩ó➉ñ✈❉❏☞❑❴q✍❣❋ô➶①s②❴✈❉❣❪q✍✈❊✉ ◆②❴❣❨⑦◆ ①✍②☞■❇①❿❳❬❜❚⑦❿✈✢❷❉❣❋❏⑩➹❋✈❊q✍r❪✈✁①✍❣ ◆❣❋✉✇✈ ◆❣❪⑥→t✥① ▲❣❪❏↕■◆ ①✍②☞✈➜r❋q ▲❑ ▲❖◆ q✍✈④❏❴✈✼❑❘❰❇①s②❴✈❉❏✇①s②☞■❨① ◆❣❋⑥③t✥① ▲❣❋❏✇⑦▲ ⑥→⑥ Ü✈✝■õ⑦❿✈❊■❋ö ◆❣❋⑥③t✥① ▲❣❋❏✇❣❇❤✤①✍②☞✈✝❷✿❣❪❏◆✈❊q✍➹❨■❇① ▲❣❋❏ ⑥❖■↔⑦➦❰ ▲①✝❑✥❣⑩✈◆ ❏☞❣❇①➜r❋t☞■❋qs■❋❏✫①✍✈❉✈✝①✍②☞■❇①➜⑦➊✈õ⑦▲ ⑥→⑥✡❷❉❣❋❏⑩➹❋✈❊q✍r❪✈❋Õ ➚❏↕❤❬■❪❷✛①✁①✍②❴✈♣❷❉❣❋❏◆✍▲❖◆①s✈❉❏☞❷❉÷✇①✍❣✇❣❋t❴q ▲❏❪①s✈❉r❪qs■❋⑥❁❤⑤❣❪q✍✉ø❣❇❤✯①s②❴✈➦❷✿❣❪❏◆✈❊q✍➹❨■❇① ▲❣❋❏↕⑥❖■↔⑦ ▲❖◆ r❋t☞■❋qs■❋❏✫①✍✈❉✈✼❑ ▲❤❚⑦❿✈✶✈❉✉✇Û❴⑥③❣❨÷↕■✇❷✿❣❋❏◆✈❊q✍➹❨■❨① ▲➹❪✈ ❏⑩t❴✉✇✈❉q ▲❷❊■❇⑥ ◆ ❷➎②❴✈❊✉➇✈❥■◆ ❑✥✈④❏❴✈❊❑➦■Ü❣❨➹❪✈❋Õ ñ✈❥ö⑩❏❴❣❨⑦Ô①✍②☞■❇① ▲❏➦❣❪qs❑❴✈❉q❘①s❣➜❣Ü①➎■▲❏♣❷✿❣❋❏⑩➹❪✈❉qsr❋✈❉❏❁❷✿✈ ⑦❿✈➇qs✈❊ù✫t▲ qs✈ ◆❣❋✉✇✈✇❏❴❣❇① ▲❣❪❏Ô❣❋❤ ◆ ①s■Ü❴▲ ⑥ ▲①❃÷❋Õ✕ú❿✈❊❷❉■❋t◆ ✈✔⑦❿✈✕■❇qs✈➇❑✥✈✼■❇⑥ ▲❏☞r➬⑦▲①✍②Ð■➶❏❴❣❋❏ó ⑥ ▲❏☞✈❊■❇q Û❴qs❣Ü⑥③✈❉✉✟①s②❴✈➦❷✿❣❋❏❁❷✿✈❉Û❴① ◆ ❣❇❤ ◆ ①s■Ü❴▲ ⑥ ▲①❃÷✕t◆ ✈❊❑ût☞❏❪① ▲ ⑥✏❏❴❣❨⑦ü■❇qs✈✝❏❴❣❋①✁■❋Û❴Û❴⑥ ▲❷❊■Ü⑥③✈❋Õ➁ý❿①❿①✍②❴✈õ✈❉❏❁❑ ❣❇❤❚①✍②▲③◆ ⑥→✈✼❷✛①✍t☞q✍✈♣⑦❿✈➦⑦▲ ⑥→⑥❆r▲➹❋✈ ◆t✥þû❷▲✈❉❏✫①✝❷✿❣❪❏☞❑▲① ▲❣❋❏◆ ❤⑤❣❪q✢■ ◆ ❷➎②❴✈❉✉✇✈õ①s❣ Ü✈✔❏❴❣❋❏ó ⑥ ▲❏❴✈✼■❇qs⑥→÷ ◆ ①s■Ü⑥→✈➦■❇❏❁❑↕②❴✈❊❏☞❷✿✈♣❷❉❣❋❏⑩➹❋✈❊q✍r❪✈❉❏✫①❊Õ ÿ②❴✈Ò①✍②❴✈❊❣❋qs✈❉✉ ■❇⑥ ◆❣↕❑❴❣✫✈ ◆ ❏❴❣❋①✢r❪t☞■❇q➎■❇❏✫①✍✈❊✈õ①✍②☞■❇①✝①s②❴✈Ò⑦❿✈❊■❋ö ◆❣❋⑥③t✥① ▲❣❋❏❽❣Ü①s■▲❏☞✈❊❑ ◆ ■❨① ▲③◆✖④✈◆ ①✍②☞✈✧✈❉❏✫①sq✍❣❪Û✫÷✶❷✿❣❪❏☞❑▲① ▲❣❋❏✡Õ ➚❤☞✉✇❣❋qs✈✒①✍②☞■❋❏♣❣❋❏❴✈➊⑦➊✈✼■❇ö ◆❣❪⑥→t✥① ▲❣❪❏♣✈❮✥▲❖◆① ◆ ❤⑤❣❪q❆■✘r▲➹❪✈❉❏➦Û❴qs❣Ü⑥→✈❊✉➬❰ ①✍②☞✈❉❏Ò❑▲ô✏✈❉qs✈❉❏✫①❚❷❉❣❋❏◆ ✈❉qs➹❨■❨① ▲➹❋✈➁❏⑩t❴✉✇✈❊q▲ ❷❉■❋⑥ ◆ ❷➎②❴✈❉✉✇✈◆ ✉✕■↔÷✶❷❉❣❋❏⑩➹❋✈❊q✍r❪✈❆①s❣✢❑▲ô✏✈❉qs✈❉❏✫①❚■❋❏◆⑦❿✈❉q ◆ Õ ñ✈➦⑦▲ ⑥③⑥✯❑▲❖◆ ❷❉t◆s◆ ✈❉❏✫①✍qs❣❋Û⑩÷ó❞◆ ■❨① ▲③◆ ❤⑤÷▲❏☞r ◆ ❷➎②❴✈❉✉✇✈◆ ⑥❖■❨①s✈❉q ▲❏➬①✍②❴✈➦⑥③✈❊❷✿①✍t❴qs✈❋Õ Ý
Shock Capturing vs. Shock Fitting hocks when the shocks or di n the solution as regions of large gradients without having to give them any special treatment. If we use conservative schemes, the Lax-Wendroff theorem 's. will be to a weak solution We know tha reak solutions satisfy the jump conditions and therefore give the correct shock An alternative to shock capturing schemes are the so called shock fitting meth ods. In these methods, one needs to assume that a discontinuity will be present n the solution. The numerical algorithm iteratively determines the strength and speed of that discontinuity using the Rankine-Hugoniot jump relation. Shock fitting schemes will not be considered in this lectures. They are considered old and hardly used nowadays. The main disadvantage is that one requires a fair amount of knowledge about the solution before one actually computes it They are also very difficult to extend to multidimensions where one can have very complex interactions involving several shock systems and consequent ly no a-priori know ledge about the structure of the solution 2.3.1 Shock Capturing SLIDE 6 In the exact proble Here fo= f(u(ao, t)) and f,= f(u(aj, t)) conser vative numerical scheme satisfies an analogous discrete condition: N3] 4t∑(a2+1 + We see that due to the cancellation of all interior funes we are only left writh the boundary fiures. The form of these boun dary flutes will depend on the boundary conditions Note 3 Discrete Conservation The basic priciple underlying a conser vation law is that the total quantity of a ble i egion changes only due to Aux through the aries. We saw this in the last lecture when we derived some conservation law
✁✄✂✆☎✞✝✠✟ ✡☞☛✌✂✎✍✞✏✒✑✔✓✖✕✗☎✙✘✛✚✢✜✤✣✦✥★✧✪✩✖✫✬✡☞☛✭✂✎✍✞✏✯✮✰✜✱☎✲☎✳✜✤✣✦✥ ✴✄✵✷✶✹✸✢✺✔✻✳✼✛✸✽✻✌✸✔✶✳✾✿✼❀✵✲❁❂✵❃✾✙✸❅❄✪✻✹❆❈❇✹✵✙✶❉✶✳✼❈❊✪✾✿❋●✶■❍✰✼❈✵✙❏❑✻✹✼❈✵✗✶✳✼❈❊✪✾✿❋●✶■❊✖❇▼▲❈◆❖✶✹✾P❊✖❏✖✻✹◆◗❏●❆❈◆◗✻✳◆❖✵✙✶❉✸❘❄❈❄❙✵✙✸❅❇ ◆❖❏❚✻✳✼❈✵❯✶✳❊❘❱❖❆✪✻✹◆◗❊✖❏❚✸✖✶❲❇✹✵✲❳❘◆❖❊❘❏✛✶❲❊❅❨✷❱❖✸❘❇✳❳✖✵❩❳❘❇✿✸❘▲✪◆❖✵✲❏❬✻✿✶✔❍✰◆❭✻✹✼❈❊❘❆✪✻❪✼❀✸✢❫●◆◗❏❈❳❴✻✹❊❵❳❘◆❖❫❘✵❛✻✹✼❈✵✲❁❜✸❅❏●✺ ✶✳❄✛✵❝✾P◆❞✸❅❱▼✻✳❇✹✵✙✸❅✻✳❁❂✵✲❏❬✻❝❡❩❢❣❨❃❍❃✵❩❆❀✶✳✵❂✾✲❊❘❏❀✶✳✵✲❇✹❫✽✸✽✻✹◆◗❫✖✵❛✶✹✾✿✼❈✵✲❁❂✵❝✶✲❤☞✻✳✼❈✵❥✐✦✸✽❦●❧♠✴✄✵✲❏✛▲✪❇✳❊❘♥♦✻✳✼❈✵✙❊❘❇✹✵✲❁ ❳❘❆✛✸❅❇✿✸❅❏❬✻✳✵✙✵✙✶■✻✳✼✛✸✽✻❃✾✲❊❘❏●❫❘✵✙❇✳❳✖✵❘❤❅◆◗❨☞◆◗✻♣❊✪✾✲✾✲❆❈❇✿✶✲❤✖❍✰◆◗❱❖❱✛q❙✵r✻✳❊❑✸❑❍♣✵❝✸❅❋❪✶✳❊❘❱❖❆✪✻✹◆◗❊✖❏✎❡▼✴s✵r❋❬❏❀❊✽❍t✻✹✼❀✸✽✻ ❍❃✵✙✸❅❋❥✶✳❊❘❱❖❆✪✻✹◆◗❊✖❏❀✶✷✶✳✸❅✻✳◆❞✶✉❨✈✺❩✻✹✼❈✵❃✇✉❆❈❁❂❄①✾P❊❘❏✛▲✪◆❭✻✹◆◗❊✖❏❀✶✰✸❅❏✛▲❥✻✹✼❈✵✲❇✹✵P❨✈❊✖❇✳✵②❳❘◆❖❫❘✵r✻✳✼❈✵❲✾✲❊❘❇✹❇✳✵❝✾✞✻✰✶✱✼❈❊✪✾✿❋ ✶✳❄✛✵✙✵✙▲✆❡ ③❏✠✸❅❱◗✻✳✵✙❇✳❏❀✸❅✻✳◆❖❫❘✵④✻✳❊❂✶✱✼❈❊✪✾✿❋❥✾✙✸❅❄✪✻✹❆❈❇✳◆❖❏❈❳❛✶✳✾✿✼❈✵✙❁❂✵✙✶✷✸❅❇✹✵✗✻✹✼❈✵②✶✳❊❂✾✲✸❅❱❖❱❖✵✙▲❯✶✳✼❈❊✪✾✿❋❛⑤❈✻✱✻✹◆◗❏❀❳❛❁❂✵✲✻✳✼✪❧ ❊✪▲❈✶✙❡✦❢⑥❏❩✻✹✼❈✵✙✶✳✵④❁❂✵✲✻✳✼❈❊✪▲❈✶✙❤●❊❘❏❈✵✗❏❀✵✲✵✙▲❀✶✭✻✹❊❪✸❘✶✹✶✳❆❈❁❂✵✰✻✳✼❀✸❅✻❃✸✬▲❈◆❖✶✹✾P❊✖❏✖✻✹◆◗❏●❆❈◆◗✻✉✺❛❍✰◆◗❱❖❱✛q❙✵②❄❈❇✳✵❝✶✱✵✙❏❬✻ ◆❖❏❲✻✳✼❈✵❃✶✱❊✖❱◗❆❈✻✳◆❖❊❘❏✎❡■⑦✷✼❈✵✌❏●❆❈❁❂✵✲❇✹◆❞✾✲✸❅❱❬✸❅❱❖❳❘❊✖❇✳◆◗✻✳✼❈❁⑧◆◗✻✳✵✲❇✿✸✽✻✹◆◗❫✖✵✲❱❖✺②▲❈✵P✻✳✵✙❇✳❁❂◆❖❏❈✵✙✶✎✻✳✼❈✵✷✶✱✻✳❇✹✵✲❏❈❳❘✻✳✼✬✸❅❏✛▲ ✶✳❄✛✵✙✵✙▲s❊❅❨❃✻✳✼❀✸❅✻❲▲❈◆❖✶✹✾P❊✖❏✖✻✹◆◗❏●❆❈◆◗✻✉✺❵❆❀✶✳◆❖❏❈❳✠✻✳✼❈✵❥⑨✗✸❘❏❈❋●◆◗❏❀✵P❧⑥⑩✗❆❈❳✖❊❘❏❈◆❖❊❅✻❃✇✉❆❈❁❂❄❚❇✳✵✙❱❖✸❅✻✳◆❖❊❘❏✎❡❥❶●✼❈❊✪✾✿❋ ⑤❈✻✳✻✳◆❖❏❈❳❷✶✹✾✿✼❈✵✲❁❂✵✙✶❥❍✰◆◗❱❖❱④❏❈❊❘✻❯q❙✵❸✾P❊✖❏❀✶✱◆❞▲✪✵✙❇✳✵❝▲t◆❖❏❹✻✳✼❀◆❖✶❯❱◗✵❝✾✞✻✳❆❀❇✳✵❝✶✲❡❺⑦✷✼❈✵✙✺t✸❅❇✹✵❵✾P❊✖❏❀✶✳◆❖▲✪✵✙❇✳✵❝▲ ❊❘❱❞▲✄✸❅❏❀▲s✼❀✸❅❇✿▲✪❱❖✺❵❆❀✶✳✵✙▲s❏❈❊✽❍✷✸❘▲❈✸✢✺✪✶✙❡❑⑦✷✼❈✵❩❁❩✸❅◆❖❏✄▲❈◆❖✶✹✸❘▲✪❫✽✸❘❏✖✻✿✸❅❳✖✵❪◆❞✶②✻✳✼❀✸❅✻✬❊❘❏❈✵❂❇✹✵✙❻❬❆❈◆❖❇✳✵❝✶r✸ ❨✤✸❅◆❖❇✬✸❅❁❂❊✖❆❈❏❬✻✔❊❘❨❃❋●❏❈❊✽❍✰❱❖✵✙▲❈❳❘✵❂✸❅q❙❊❘❆✪✻❲✻✹✼❈✵❥✶✱❊✖❱◗❆❈✻✳◆❖❊❘❏✄q✛✵✲❨✈❊❘❇✹✵❩❊❘❏❈✵❩✸❘✾P✻✳❆❀✸❘❱◗❱❖✺❸✾✲❊❘❁❂❄❈❆✪✻✹✵✙✶❲◆◗✻✙❡ ⑦✷✼❈✵✙✺❸✸❅❇✹✵❩✸❅❱❞✶✱❊✠❫❘✵✙❇✳✺❸▲❈◆❭❼❥✾P❆❀❱❭✻❲✻✹❊①✵P❦●✻✳✵✙❏❀▲s✻✳❊❴❁❑❆❈❱◗✻✳◆❞▲✪◆❖❁❂✵✲❏❀✶✳◆◗❊✖❏❀✶❲❍✰✼❈✵✙❇✳✵❂❊✖❏❈✵❩✾✲✸❅❏✄✼❀✸✢❫✖✵ ❫❘✵✙❇✳✺❥✾✲❊❘❁❂❄❈❱❖✵P❦❯◆❖❏❬✻✳✵✙❇✹✸✖✾✞✻✳◆❖❊❘❏✛✶✷◆◗❏●❫❘❊✖❱◗❫●◆❖❏❈❳❂✶✱✵✙❫❘✵✲❇✿✸❅❱✆✶✳✼❈❊✪✾✿❋❯✶✳✺●✶✱✻✳✵✙❁❩✶✰✸❅❏❀▲❴✾P❊✖❏❀✶✳✵✙❻❬❆❈✵✲❏❬✻✹❱◗✺❩❏❈❊ ✸✽❧❣❄❈❇✹◆◗❊✖❇✳◆❙❋●❏❈❊✽❍✰❱❖✵✙▲✪❳✖✵✔✸❘q✛❊✖❆✪✻✷✻✳✼❈✵✬✶✱✻✳❇✹❆❀✾P✻✳❆❈❇✹✵✔❊❅❨■✻✳✼❈✵✬✶✳❊❘❱❖❆✪✻✳◆❖❊❘❏☞❡ ❽✎❾❖❿☞❾✤➀ ➁☞➂✦➃☞➄✙➅★➆❪➇✪➈■➉✙➊✦➋✢➌➎➍✦➏ ➐❙➑❬➒➔➓☞→✄➣ ❢⑥❏✠✻✳✼❈✵✔✵✲❦✪✸✖✾✞✻✗❄❈❇✹❊❘q❀❱◗✵✙❁✠↔ ↕ ↕❘➙ ➛❷➜✢➝ ➜❝➞➠➟ ↕❘➡✠➢❺➤✬➥➎➦✢➧④➤❚➦❅➨❀➩ ➫❑➭✲➯✳➭ ➦➧ ➢➲➦▼➥➟ ➥✤➡➧✖➳ ➙✱➩✱➩❪➵✽➸➻➺❪➦➨ ➢➲➦▼➥➟ ➥✈➡➨✎➳ ➙✱➩✳➩✲➼ ③ ✾✲❊❘❏❀✶✳✵✲❇✹❫✽✸✽✻✹◆◗❫✖✵❃❏●❆❈❁❂✵✙❇✳◆❞✾✲✸❘❱✛✶✹✾✿✼❈✵✲❁❂✵④✶✹✸✽✻✹◆❖✶✱⑤❀✵✙✶♣✸❅❏❥✸❘❏❀✸❅❱❖❊❘❳✖❊❘❆❀✶✭▲❈◆❖✶✹✾P❇✹✵P✻✹✵④✾P❊✖❏❀▲✪◆◗✻✳◆❖❊❘❏✎↔ ➽r➾ ➚➡ ➚➙ ➨ ➪ ➶✳➹➧ ➥✙➘➟➻➴❅➷✦➬ ➶ ➤➮➘➟➴ ➶ ➩➱➢ ➤ ➨ ➪ ➶✳➹➧r✃❝❐➶ ➷②❒❮ ➤ ❐ ➶P❰ ❒❮✖Ï ➢ ➤ ✃✲❐➨➷②❒❮ ➤ ❐ ❰ ❒❮ Ï Ð①➭✰Ñ✲➭✿➭✗Ò✈Ó ➵Ò ➺✽Ô➭④Ò❣Õ✬Ò➔Ó❀➭②Ö➵❅➸Ö✿➭P×➔×➵Ò♠Ø✤Õ➸ Õ✉Ù ➵×➔×❀Ø➸Ò❣➭✲➯✞Ø✤Õ✽➯☞ÚÔ❘Û➭PÑ❃Ü✌➭ ➵➯✹➭rÕ ➸×ÞÝ✬×❖➭✤Ù✿Ò▼Ü❉Ø➔Ò✈Ó❩Ò✈Ó❈➭ ßÕÔ✪➸➻➺❘➵➯✞Ý♣ÚÔ❘Û➭PÑ ➼❑àÓ❈➭▼ÙPÕ✽➯✞áâÕ✱Ù✗Ò✈Ó❈➭PÑP➭ ßÕÔ✪➸➻➺❘➵➯✞Ý❃ÚÔ❘Û➭PÑ✷Ü❉Ø➔×➔× ➺➭❣ã❀➭➸➻➺ Õ➸ Ò✈Ó❈➭ ßÕÔ✪➸➻➺❘➵➯✞Ý Ö✿Õ➸➻➺Ø➔Ò♠Ø✤Õ➸Ñ ➼ ✁✄✂✆☎✞✝①ä å✠✜✤✩❘✍❝✚✖✝❈☎✞✝❚✑✬✂✪✣■✩❅✝❅✚●✧❅✓✆☎✱✜✉✂●✣ ⑦✷✼❈✵❲q❀✸✖✶✱◆❞✾✔❄❈❇✹◆❖✾✲◆◗❄❀❱◗✵✔❆❈❏❀▲❈✵✲❇✹❱◗✺●◆❖❏❈❳❥✸❩✾P❊✖❏❀✶✱✵✙❇✳❫✽✸❅✻✳◆❖❊❘❏❯❱❞✸✢❍⑧◆❖✶✷✻✹✼❀✸✽✻✰✻✹✼❈✵❲✻✹❊❅✻✹✸❘❱☞❻❬❆❀✸❅❏❬✻✹◆❭✻✉✺❥❊❘❨▼✸ ✾P❊✖❏❀✶✳✵✲❇✹❫❘✵✙▲❵❫✢✸❘❇✳◆❞✸❅q❀❱◗✵❪◆◗❏❚✸❅❏●✺❴❇✳✵✙❳❘◆❖❊❘❏✄✾✿✼❀✸❅❏❈❳✖✵✙✶r❊❘❏❈❱❖✺❵▲✪❆❈✵❪✻✳❊✠æ❀❆❈❦①✻✳✼❈❇✹❊❘❆❈❳✖✼❵✻✹✼❈✵❂q✛❊✖❆❈❏❀▲●❧ ✸❅❇✹◆❖✵✙✶✙❡✰✴s✵❛✶✳✸✢❍⑧✻✳✼❀◆❖✶r◆◗❏❵✻✳✼❈✵❪❱❖✸✖✶✉✻②❱◗✵❝✾✞✻✳❆❀❇✳✵✬❍✰✼❀✵✲❏❵❍❃✵❪▲✪✵✙❇✳◆❖❫❘✵❝▲①✶✱❊✖❁❂✵❑✾P❊✖❏❀✶✱✵✙❇✳❫✽✸❅✻✳◆❖❊❘❏❴❱❖✸✢❍✗✶ ➾
(conservation of mass, cars,. The expression given in the slide is an analo- gous discrete form of this principle. This discrete conservation means that an shocks computed bN he conservative numerical scheme must be in the"correct ion. c non-conservative method can give aith the shock prop- ating at the arong speed. This cannot happen aith a conservative method since an incorrect shock speed aould lead to an incorrect and thus con tion aould not be preserved. The solution computed method might not accuratelNresolve the shock (it manbe smeared out), but ahen the grid is rested suo cientiN the discontinuitiNaill be located in the correct position or example, consider a non-conservative upaind scheme for Burgers'equation 1=t Bt12)D>0 1t1)仍 0 D t ER=( Note that for this definition of the numerical fut function the consistency con lition is clearly statified.eFD=①f「 △ta D=Dt△①D e& What about a 0? We can a rite
ç✤è✲é❘ê❀ë✳ì✲í✹î✽ï✽ð✹ñ◗é✖ê①é❅ò✭ó❩ï❘ë✹ë✲ô➻è✲ï❘í✹ë✙ô☞õ❖õ❖õ öPõ②÷✷ø❈ì❛ìPù✪ú❈í✹ì✙ë✹ë✳ñ◗é✖ê✠û✖ñ◗î✖ì✲ê❵ñ❖ê①ð✹ø❈ì❛ë✳ü❖ñ❖ý✪ì❪ñ❖ë②ï❅êsï❅ê❀ï❘ü◗é❘þ û❘é✖ÿ❀ë✰ý✪ñ❞ë✳è✲í✳ì✲ð✳ì✔ò✈é✖í✳óâé❘ò✦ð✹ø❈ñ❞ë✗ú❈í✹ñ◗ê✛èPñ❖ú❈ü◗ì✖õ✌÷✷ø❈ñ❞ë✗ý✪ñ❞ë✳è✲í✳ì✲ð✳ì✬è✲é❘ê❀ë✳ì✲í✹î✽ï✽ð✳ñ❖é❘ê✠ó❂ì✙ï❘ê❀ë❃ð✳ø❀ï❅ðrï❅ê✁ ë✳ø❈é✪è✄✂●ë▼è✲é❘ó❂ú❈ÿ✪ð✹ì✙ý✆☎✝✔ð✹ø❈ì✰èPé✖ê❀ë✱ì✙í✳î✽ï❅ð✳ñ❖î❘ì✌ê●ÿ❈ó❂ì✙í✳ñ❞è✲ï❘ü●ë✹è✿ø❈ì✙ó❛ì✷ó❪ÿ❀ë✉ð✞☎❙ì✷ñ◗ê❪ð✳ø❈ì✠✟✹èPé❘í✹í✹ì✙è✞ð✄✡ ü❖é●è✙ï✽ð✹ñ◗é✖ê✎õ☞☛ ê❈é❘ê✪þ⑥èPé✖ê❀ë✳ì✲í✹î✢ï❅ð✳ñ❖î❘ì❲ó❂ìPð✹ø❈é✪ý❵è✲ï❘ê❵û❘ñ❖î❘ì❪ï❯ë✳é❘ü❖ÿ✪ð✳ñ❖é❘ê✍✌✰ñ◗ð✳ø❵ð✳ø❈ì❂ë✱ø❀é●è✄✂✠ú❈í✹é❘ú✪þ ï❅û❬ï✽ð✹ñ◗ê❈û ï✽ð④ð✹ø❈ì✎✌✰í✹é❘ê❀û❯ë✱ú❙ì✲ì❝ý✆õ❲÷✷ø❈ñ❞ërè✙ï❅ê❈ê❈é❘ðrø❀ï❘ú❈ú❙ì✲ê✍✌✰ñ◗ð✳ø✄ï❯èPé✖ê❀ë✳ì✲í✹î✢ï❅ð✳ñ❖î❘ì❲ó❂ìPð✹ø❈é✪ý✆ô ë✳ñ◗ê❀è✲ì❥ï❅ê❚ñ❖ê❀èPé✖í✳í✹ì✙èPð❲ë✳ø❈é✪è✄✂së✳ú❙ì✲ì✙ý✏✌❃é❘ÿ❀ü❖ý✄ü❖ì✙ï❘ýsð✹é❵ï❅ê♦ñ❖ê❀èPé✖í✳í✹ì✙èPð✒✑✛ÿ✪ù✆ô■ï❘ê❀ýsð✳ø●ÿ❀ë❑è✲é❘ê✪þ ë✳ì✲í✹î✢ï❅ð✳ñ❖é❘ê✍✌❃é❘ÿ❀ü❖ý①ê❀é❅ð✓☎✛ì❛ú❈í✹ì✙ë✳ì✲í✹î❘ì✙ý✎õ②÷✷ø❈ì❂ë✳é❘ü❖ÿ✪ð✳ñ❖é❘ê❸è✲é❘ó❂ú❈ÿ✪ð✹ì✙ý✔✌✰ñ❭ð✹øsï✠èPé❘ê✛ë✱ì✙í✳î✽ï✽ð✹ñ◗î✖ì ó❂ìPð✹ø❈é✪ý❚ó❛ñ❖û❘ø❬ð✬ê❈é❘ð❪ï❘è✙èPÿ❈í✿ï✽ð✹ì✲ü✕❵í✹ì✙ë✳é❘ü❖î❘ì❛ð✳ø❈ì ë✱ø❈é✪è✄✂✯ç✈ñ◗ð❑ó❩ï✖✔☎❙ì❯ë✱ó❂ì❝ï❅í✹ì✙ý✄é❘ÿ✪ð✞ö✞ô✗☎❈ÿ✪ð ✌✰ø❈ì✙ê❷ð✹ø❈ì❴û❘í✹ñ❖ý❷ñ❖ë❂í✹ì✙✘❀ê❀ì✙ý✯ë✱ÿ✛✚❥è✲ñ◗ì✙ê✖ð✹ü✜✖ô✭ð✹ø❈ì①ý✪ñ❖ë✹èPé✖ê❬ð✳ñ❖ê❬ÿ❀ñ❭ð✹ñ✜✢✌✰ñ❖ü❖ü✣☎✛ì❴ü❖é●è✙ï✽ð✹ì✙ýtñ◗êtð✳ø❈ì èPé✖í✳í✹ì✙èPð❃ú❙é✖ë✳ñ◗ð✳ñ❖é❘ê✎õ ✤é✖í✌ì✲ù✪ï❘ó❂ú❈ü◗ì✖ô❬è✲é❘ê❀ë✳ñ❞ý✪ì✲í✌ï✬ê❈é❘ê✪þ⑥èPé✖ê❀ë✳ì✲í✹î✢ï❅ð✳ñ❖î❘ì✰ÿ❈ú✁✌✰ñ❖ê❀ý❥ë✹è✿ø❈ì✲ó❂ì✰ò✈é✖í✦✥❃ÿ❈í✹û❘ì✙í✹ë★✧❅ì★✩❬ÿ❀ï❅ð✳ñ❖é❘ê✫✪ ✭✯✮✱✰✳✲ ✬ ✴ ✵ ✭✬ ✮ ✴✷✶ ✸✹✺✼✻✾✽ ✻✞✿ ✭✬ ✴✷❀ ✮ ✭✬ ✴✠✶ ✮ ✭✬ ✴✙❁✮ ✲✙❂ ✭✬ ✴❄❃❆❅ ✮ ✻✾✽ ✻✞✿ ✭✬ ✴✷❀ ✮ ✭✬ ✴✮✰✳✲ ✶ ✭✬ ✴✮ ❂ ✭✬ ✴❄❇❆❅ ✮ ❈✆❉ ❈✎❊●❋❍ ✴❏■ ✲ ❈✭✯✮✬ ✴ ✵ ❍ ▲◆▼P❖❘◗ ❑ ✶ ✭❙✮✬ ✴❚❀✭❙✮✬ ✴ ✶ ✭✯✮✬ ✴✙❁ ✲★❂❱❯ ❍ ▲◆▼✙❲❘◗ ❑ ✶ ✭❙✮✬ ✴❚❀✭❙✮✬ ✴ ✰✳✲ ✶ ✭✯✮✬ ✴ ❂ ✵ ✶❨❳★❩❋✰✒❬❭ ✶✏❩❁ ❭❫❪ ❬ ❯✢❴❛❵❘❜✗❝◆❞❛❡✖❢❤❣✛✐◆❥❦❵❙❜❧❞✝❡◆❡♠❵❘❡♠❝ ♥ò▼ð✳ø❈ì❛ë✱é✖ü◗ÿ❈ð✳ñ❖é❘ê❴ñ❖ërë✱ó❂é●é❅ð✹ø✎ô✛ð✳ø❈ì❑è✲é❘ê❀ë✳ì✲í✹î✽ï✽ð✹ñ◗é✖ê✠ì✙í✳í✹é❘í✿ë✷ï❅í✹ì♣♦❯ç❈✆❊öPõ ♥ò▼ð✳ø❈ì❛ë✱é✖ü◗ÿ✪ð✹ñ◗é✖ê❴ñ❖ë ê❈é❘ð✰ë✳ó❂é●é❅ð✳ø☞ô✪ð✳ø❈ì✬è✲é❘ê❀ë✳ì✲í✹î✽ï✽ð✹ñ◗é✖ê❯ì✲í✹í✹é❘í✿ë♣ï❘í✳ì✎♦❯çrq❝öPõ s✉t✇✈ ①✷②r③❘④❫⑤⑦⑥⑧③❘⑨❶⑩✫③❆❷❧❸❺❹❻②❽❼☞⑨ ❾✫❿✜➀✳❿❦➁ ➂ ❥❦❜✗❞✝❣❤❡➄➃➆➅✳❢❘❞✁❴❫✐★❥❦❵❙❜❧➇✒➈➊➉✳❣✛✐◆❥❦❵❙❜ ➋❙➌✝➍➏➎➑➐⑧➒ ➓ ✭ ➓❙❉ ❯→➔ ➓ ✭ ➓❘❊ ✵ ❅ ➔ èPé✖ê❀ë✉ð✿ï❅ê❬ð ❃➣❅ ✭❙✮✱✰✗✲ ✬ ✴ ✵ ✭✬ ✮ ✴✠✶ ❈✎❉ ❈✆❊ ❳★❩✎↔✯↕ ✴ ✰ ❭❬ ✶✏❩✎↔✯↕ ✴✙❁ ❭❬ ❪ ➙ì✲ð ❩✴ ↔✯↕ ✰ ❭✎➛ ❬ ➔ ✭✬ ✴ ❳◆❩✴✙❁ ↔➊↕ ❭❬ ✵ ➔ ✭✬ ✴✙❁ ✲ ❪ ➜✎➝✱➞➠➟➡➞➏➢➥➤♠➞❙➦✙➝♠➧➡➞✇➢✁➨➫➩➡➭❫➟➲➯➵➳❘➨➏➞➲➨❦➝✱➳➸➝r➦✓➞✇➢❤➟✎➳➥➺✛➻✷➟➼➧P➨❦➽➾➤✱➚❫➪❺➺❫➶❶➦✄➺✛➳✯➽✙➞➹➨❦➝♠➳✍➞✇➢❤➟♣➽✄➝♠➳➥➩✄➨➫➩P➞➘➟✙➳✯➽✙➴❚➽✄➝♠➳❘➷ ➭♠➨➏➞➹➨❦➝♠➳➬➨➫➩✆➽✙➚✕➟➾➤✱➧P➚➮➴✷➩P➞➘➤♠➞➹➨➯✦➟✄➭❫➱✃➨❦➱➏➟★➱ ❩ç✭ö ✵ ➔✭ ✵❨❐ ç✭ö ➱ ❒ ✭✯✮✱✰✳✲ ✬ ✴ ✵ ✭✬ ✮ ✴✠✶ ❈✆❉ ➔ ❈✆❊ ç✭✬ ✴ ✶ ✭✬ ✴✙❁ ✲ ö ➋❙➌✝➍➏➎➑➐➸❮ ❰ø❀ï❅ð④ï✱☎❙é❘ÿ✪ð ➔ ❇❆❅❛Ï ❰ì✬è✲ï❅ê❄✌✰í✹ñ❭ð✹ì❘ô Ð
a△t∫a-t a>0 艹1=仍-(奶1-奶-)+(1-2+奶 Note that by introducing the absolute value we are able to write a single eapres sion that takes into account the dependency of the difference stencil on the sign of sLIdE 9 In conservative form 0)-ad(0+1-0) >0 We see that altho ugh the first order upwind method was originally derived using finite difference, and characteristic interpolation arguments, it can also be in terpreted as a conservative finite volame scheme were we solve for cell solution averages rather than pointwise values We note that the upwind scheme written in this manner is precisely a FCTS scheme (forward in time centered in space)with the explicit addition of a second difference term. As we know from the linear analysis this term is required for 2.4.2 Nonlinear case SLIDE 10 In the nonlinear case au af(u) 0 =2(+1)
Ò❙Ó❫Ô✳Õ Ñ Ö × ÒÑ Ö❚ØÚÙ✛Û✆Ü Ó Û✎Ý Þ ÒÑ ÖÓ Ø ÒÑ Ö✙ßÓ Ñ Õ Ù➆à➣á ÒÖÓÔ✳Õ Ø ÒÑ ÖÓ Ù➆â➣á ã❫ä Ò❙Ó✱Ô✗Õ Ñ Ö × ÒÑ Ö✠Ø Ó Ù✁Û✆Ü åÛ✆Ý→æÒÑ ÖÓÔ✳Õ Ø ÒÑ Ö✙ßÓ Õ➼ç❱èêé Ù é Û✆Ü åÛ✎ÝêæÒÑ ÖÓÔ✳Õ Ø åÒÑ ÖÓ è ÒÑ Ö✙ßÓ Õ✙ç ë✎ì✱í➠îïí➏ð➥ñ♠í✦ò➼ó✠ô➏õ➥í➹ö❏ì★÷✱ø➥ù✙ô➏õ✁ú✷í✇ð❤î➡ñ❛òPû➼ì✱ü➮ø✛í➠î➡ý✖ñ✱ü➮ø➥îïþ❱î✎ñ♠ö❏î✆ñ❫ò➼ü✜î➡í➘ì♣þ✞öPô➏í➘î✎ñ✆ûPô➏õ✝ú❫ü✜î✎î➾ÿ✁❙ö❏î✙û✄✂ ûPô❦ì♠õ✍í✇ð❤ñ✱í➵í➘ñ✆☎❛îPû✓ô➏õ➥í➘ì❚ñ❫ù✄ù✄ì♠ø✛õ❘í➵í✇ð❤î➡÷❛î✝➥î✙õ✯÷❫î➼õ❙ù➼ó➆ì✟✞✓í✇ð❤î➡÷✱ô✠✃î➼ö❏î✙õ✯ù✄îïûPí➘î✙õ✯ù✙ô➏ü✫ì♠õ✔í➏ð➥î✓ûPôú✱õ ì✟✞ Ù☛✡ ☞✍✌✏✎✒✑✔✓✖✕ ✗✙✘✛✚✢✜✍✣✥✤✧✦✏★✪✩✬✫✮✭✆✯✰✩✱✦✳✲✝✜✱★✪✴✶✵ ÒÑ Ó✱Ô✗Õ Ö × ÒÑ Ö✠Ø Ó Û✎Ü Û✆Ý ✷✆✸✺✹✼✻Ó Ö Ô✾✽✿ Ø ✸✺✹✼✻Ó Ö✙ß ✿❁❀ ✽ ✸✺✹✼✻ Ö Ô ✿✽ ×❃❂ å Ù✍❄ÒÑ Ö Ô✗Õ è ÒÑ Ö✧❅ Ø å ❂ é Ù é ❄ÒÑ Ö Ô✳Õ Ø ÒÑ Ö✧❅ ✸✹✼✻ Ö Ô❆✽✿ × ÙÒÑ Ö ✸ Ù❚à❆á ✹✼✻ Ö Ô ✿✽ × ÙÒÑ Ö Ô✳Õ Ù❚â❆á ❇î☞û✙î✄î✒í✇ð❤ñ✱í❱ñ♠ü➮í✇ð❤ì✱ø❫ú◆ð➄í✇ð❤î❉❈❱ö✄û✄í➵ì✱ö❏÷❫î➼ö☞ø✧✯þ✞ô➏õ❙÷❋❊✷î✙í✇ð❤ì◆÷✆þ❱ñ✖ûïì♠öPôú❫ô➏õ❙ñ♠ü➏üó➆÷❫î➼öPô➏ý✖î✄÷♣ø✁ûPô➏õ✝ú ❈➵õ❘ô➏í➠î➄÷♠ô✠✃î➼ö❏î➼õ❙ù✄î✁●✓ñ♠õ✯÷✔ù✄ð❤ñ✱ö❏ñ❫ù➼í➠î➼öPô➫ûPí➲ô❦ù➄ô➏õ❘í➠î➼ö❍❤ì✱ü✜ñ♠í➹ô❦ì♠õ❧ñ♠ö➹ú✱ø✮❊✷î✙õ❘í❦û■●❶ô➏íïù✄ñ♠õ❻ñ✱üû➼ì➬ò✄î✷ô➏õ✱✂ í➘î✙ö❍✯ö❏î✙í➘î✄÷➆ñ♠û➡ñ➄ù✄ì♠õ❤û➼î➼öPý✖ñ✱í➲ô➏ý♠î❏❈➵õ❘ô➏í➠î➡ý✖ì✱ü➮ø✮❊✠î✎û➼ù✄ð❤î❑❊✠î✎þ❱î✙ö➾î➡þ❱î✎û➼ì♠ü➮ý♠î▲✞✙ì♠ö✆ù✄î➼ü➏ü✫û➼ì♠ü➮ø✛í➹ô❦ì♠õ ñ♠ý♠î➼ö❏ñ➼ú✝îPû➡ö➾ñ♠í✇ð❤î➼öïí➏ð➥ñ♠õ▼➥ì♠ô➏õ❘í➲þ✞ô➫û➼î✆ý♠ñ♠ü➮ø➥î✙û ✡ ❇î➆õ✯ì♠í➘î➄í➏ð➥ñ♠í✒í➏ð➥î❄ø✧❙þ✞ô➏õ❙÷✍û➼ù➾ð➥î✁❊✷î❄þ✞öPô➏í➹í➠î➼õ→ô➏õ➣í➏ð✛ô➫û◆❊✷ñ♠õ➥õ✯î✙ö✷ô➫û❁❙ö❏î✄ù➼ô➫û✙î➼ü➮ó✏ñP❖❘◗❚❙❱❯ û➼ù➾ð➥î✁❊✷î❳❲❨✞✙ì♠öPþ❱ñ♠ö➾÷✎ô➏õ➄í➲ô✒❊✷î✒ù➾î➼õ➥í➘î➼ö❏î✄÷ïô➏õ❚û❍➥ñ❫ù✄î✟❩➡þ✞ô➏í➏ð❚í➏ð➥î✓î❏ÿ✁❙üô❦ù✙ô➏í➵ñ❫÷❛÷♠ô➏í➹ô❦ì♠õ✍ì✟✞☞ñïû➼î✄ù✄ì♠õ❙÷ ÷♠ô✠☞î✙ö❏î➼õ❙ù✄î❚í➠î➼ö✄❊✡▼❬ û✆þ❱î✺☎✱õ❙ì✱þ❭✞✄ö➾ì❪❊ í✇ð❤î✷ü➮ô➏õ❙î✄ñ♠ö❚ñ✱õ❙ñ♠üó✖û✄ô➫û✷í✇ð✁ô➫û♣í➠î➼ö✄❊ ô➫û✠ö❏î■❫✙ø✛ô➏ö❏î✄÷❆✞✙ì✱ö ûPí➠ñ❛ò✙ô➏üô➏í➲ó ✡ ❴❱❵❜❛✥❵❝❴ ❞✜✍✣✥❡❢✯✰✣❣✦❤✫✮★❥✐❳✫✬✤✆✦ ☞✍✌✏✎✒✑✔✓❧❦♥♠ ✗✙✘P♦q♣✬r❆✘ã✘✬s✉t❜✘✬r✧✈ä❚✇✈✢①qr✢② ③ ③Ò Ü è ③☛④ ❄ ③ Ò ❅ Ý × á ♦q♣⑤r❆⑥⑤⑦✮⑧◆⑨✍r✇➼ã✢⑩r✆① ❶❸❷ ✸✺✹✼✻ Ö Ô ✿✽ ×❹❂ å ✷ Ñ④ Ö Ô✳Õ è Ñ④ Ö ❀ Ø ❂ å é ÑÙ Ö Ô ✿✽ é ❄ÒÑ Ö Ô✳Õ Ø ÒÑ Ö❑❅ ❺
i每1+1≠每 Here f, denotes f(i,). The above choice of a guarantees that one sided approx mation is obtained. i.e >0 F First Order Upwind Scheme The first order upwind scheme is conser vative, and for At sufficiently small, it can be shown to be convergent (later on in this lecture we will dis scuss for convergence). The Lax-Wendroff the hat it will converge to a weak solution We show below that the lar-Wen droff and Beam-Warming algorithms are also ervative schemes and therefore admit a finite volume interpretation 2.5 Lax-Wendroff SLIDE 1 =2(+)- (1+1-a aii is again defined as For the linea ("+1--)+(2+1-2+-) =a△/△t
❻❼✢❽q❾➀❿➁❳➂➄➃ ➅➆✙➇❢➈ ❿■➉ ➅➆✙➇ ➅➊➇❢➈ ❿■➉ ➅➊➇ ➋➍➌ ➎❻ ❽q❾❣➏✺➐➂ ➎❻ ❽ ➑✼➒❍➓➎❻ ❽✧➔ ➋➍➌ ➎❻ ❽q❾❣➏ ➂ ➎❻ ❽ →✺➣❑↔q➣ ➑❻❽➙↕➣❑➛✍➜♥➝✝➣✁➞ ➑❉➓➎❻ ❽ ➔✆➟❋➠✼➡ ➣✺➢✢➤■➜❪➥❪➣✺➦➡ ➜♥➧✰➦➨➣❳➜✟➩ ❻❼◆➫✢➭➢❪↔q➢♥➛⑤➝✙➣■➣✄➞❸➝➡ ➢♥➝▲➜❪➛✼➣➯➞✄➧↕➣↕ ➢➨➲➳➲✍↔q➜✆➵➳➸ ➧✒➺▼➢❪➝❍➧✰➜❪➛➻➧❝➞✺➜➳➤✁➝✙➢❪➧✒➛✼➣↕❪➼ ➧ ➟ ➣ ➟ ➽❽q❾ ❿➁ ➂ ➃ ➑❻❽ ❻❼❽q❾ ❿➁➚➾➶➪ ➑❻❽q❾✥➏ ❻❼❽q❾ ❿➁➚➹➶➪ ➘✖➴☛➷✄➬P➮ ➱❘✃✰❐✪❒➳➷➚❮❁❐➳❰❣➬♥❐ÐÏ❚Ñ▲Ò❆✃❢ÓÔ❰ÖÕØ×✁ÙÔ➬♥Ú➶➬ ÛÝÜ✬ÞPß⑤à■á✟â▼ã➳à➨ä✮Þ✆à➚å✬æ❤ç➋❜èä➶á➨é■Ü✬Þ❑ê➚Þ ➋ á◆é✁ãè áëÞ✆àqì❪í❪â ➋ì➳Þ✢îïíè ä ➌ã✢à◆ð❳ñ❳áqå✮ò➙é➋Þè âqó✉ôõáqê▼í♥ó✉ó❢î ➋â❳é❑íè✖ö Þ◆áqÜ✬ã❪çè âqã ö Þ◆é✁ãè ì✢Þ❑à➨÷✢Þè â ➓ ó✉í♥âqÞ❑à✺ãèÐ➋✉è âqÜ➋ á✾ó✉Þ✆é✄â➨å✬à➨Þ➙ç▲Þ▼ç➋ ó✉ó▲ä➋ áqé❑å⑤á➨á❆âqÜ✬Þ à➨Þ✆ø✏å➋ àqÞ✆ê❋Þè â➨á ➌ã✢à➙é❑ãè ì✢Þ✆àq÷➳Þè é❑Þ ➔✄ù ÛÝÜ✬Þ✳ú✥í❪û❤ü❍ý✖Þè ä✮àqã✢þ➶âqÜ✬Þ✆ã✢à➨Þ❑êÿâ➨Ü✬Þ❑à➨Þ➌ã➳àqÞ Þè áëå✬à➨Þ✆á âqÜ✱í❪â ➋â❚ç➋ ó✉ó❱é✁ãè ì➳Þ❑à➨÷✢Þ➀âqã▼í❋ç▲Þ✧í✁ áqã✢ó✉å✮â ➋ãè ù ✂✛➣❁➞➡ ➜☎✄ ➤■➣✝✆❜➜☎✄ ➝➡ ➢❪➝❏➝➡ ➣✟✞✥➢❑➵➳➸☎✂✛➣✁➛ ↕↔q➜✡✠ ➢♥➛↕☞☛➣■➢❪➺❋➸☎✂✛➢❪↔✄➺❋➧✒➛➫ ➢☎✆➫➜♥↔✄➧✒➝➡➺✺➞❋➢♥↔q➣➚➢☎✆❨➞❑➜ ➦■➜❪➛⑤➞✁➣❑↔✄➥❪➢❪➝❍➧✒➥✪➣✺➞❑➦➡ ➣✁➺▼➣✁➞❁➢♥➛↕ ➝➡ ➣✁↔q➣❢➩✁➜❪↔➨➣❋➢ ↕➺❋➧✒➝❚➢✍✌Ô➛✱➧✒➝✝➣❁➥✪➜✎✆➭➺▼➣❁➧✒➛✱➝✝➣❑↔❍➲✍↔q➣❑➝✙➢❪➝❍➧✰➜❪➛➟ ✏✒✑✔✓ ✕✗✖✙✘✛✚✢✜✤✣✦✥★✧★✩✫✪✭✬ ✮✰✯✲✱✴✳✦✵✷✶✫✶ ➽✹✸✰✺ ❽q❾ ❿➁ ➂ ✻ ✼ ✽ ❻➑❽q❾✥➏✿✾ ❻➑❽✝❀❂❁ ✻ ✼ ❻❼❄❃ ❽q❾ ❿➁ ð❳ñ ð❆❅ ➓➎❻ ❽q❾❣➏✛❁ ➎❻ ❽✆➔ ❻❼❽q❾ ❿➁ ➧❝➞❳➢➫➢♥➧✒➛ ↕➣❇✌ï➛✍➣↕ ➢❪➞ ❻❼❽q❾ ❿➁ ➂ ➃ ➅➆✟➇❢➈ ❿■➉ ➅➆✙➇ ➅➊➇❢➈ ❿■➉ ➅➊➇ ➧➩ ➎❻ ❽q❾✥➏ ➐➂ ➎❻ ❽ ➑✼➒❍➓➎❻ ❽ ➔ ➧➩ ➎❻ ❽q❾✥➏ ➂ ➎❻ ❽ ❈✬ã➳àÝâqÜ✬Þ✾ó ➋✉èÞ✆í✢à❘Þ✆ø✏å⑤í❪â ➋ãè ➎✰❉❻ ❾❣➏ ❽ ➂ ➎❻ ❽ ❁❋❊✼❍●➎❻ ❽q❾✥➏ ❉ ❁ ➎❻ ❽❉➉ ➏❏■ ✾❑❊❃ ✼▲●➎❻ ❽q❾✥➏ ❉ ❁ ✼➎❻ ❽❉ ✾ ➎❻ ❽❉➉ ➏❏■ ❊ ➂ ❼ð▼❅❖◆♥ð❳ñ P
2.6 Beam-Warming (-++3+1-3+-)+号计+计(+-+1一+一1 For the linear equation -立(-3+41+1-12)+(+2-213+1+)a<0 2.7 Entropy Solutions SLIDE 13 Do these schemes converge to the entropy satisfying solution? EXAMPLE: Consider a non-physical solution to Burgers'equation 1x<0 either 1 2.7.1 Example SLIDE 14 First order upwind =2(1+1+1) +t(u+1-a a,+1 or ij+1-ij is zero Vj Be ther f;=fi F+i=2 F一F The entropy-violating solution is preserved 5 Entropy Satisfying Solutions It turns out that the first order upwind, the Lax Wendroff and the beam Warming schemes allow for entropy violating solutions. These sche distinguish between shocks and expansions To determine in advance if a general numerical scheme will only produce en tropy satisfying solutions is very difficult. One possible approach is to derive an
◗✒❘✔❙ ❚❱❯❳❲✭❨❑❩✢❬✤❲✙❭❪❨❴❫❛❵✍❜ ❝✰❞✲❡✴❢✦❣✷❤❥✐ ❦✭❧♥♠♦q♣✭rs✉t ✈✇②①❇③⑤④⑥ ♦✔♣✲⑦⑨⑧☞⑩ ④⑥ ♦✔♣ ✈ ⑧☞⑩ ④⑥ ♦ ③❶④⑥ ♦❸❷ ✈✡❹ ③ ④❺ ⑦ ♦q♣✙rs☞❻♥❼ ✇❻✫❽ ① ④❾♦q♣✲⑦ ③ ④❾♦q♣ ✈ ⑧ ④❾♦ ③ ④❾♦❸❷ ✈❸❹ ③ ❿➁➀➁➂ rs ✇➃①❇③❂④⑥ ♦q♣✲⑦⑨⑧➄⑩ ④⑥ ♦q♣ ✈ ③ ⑩ ④⑥ ♦✰⑧ ④⑥ ♦➅❷ ✈✡❹ ⑧➇➆♦✔♣ rs ❺ ⑦ ♦q♣ rs❱❻✫❽ ✇❻♥❼ ① ④❾♦q♣✲⑦ ③ ④❾♦✔♣ ✈ ③ ④❾♦⑨⑧ ④❾♦➅❷ ✈❸❹ ➈❏➉❛➊➌➋➍▼➎❍➏➉❛➊➌➋➍✲➐♥➑ ➏ ➉❛➊➒➋➍⑨➑ ➓✫➔➣→✒↔❛↕✫➙✹➛➝➜➝➞✫➙✢➟❥→➠➙✢➡✲➢♥➟☎↔➤➜➥➔➣➞ ➧✝➨❸➩☎➫ ➦ ➭ ➯ ➧➦ ➭➳➲▼➵ ➨ ➸✹➺➼➻➧➦ ➭✦➲✛➽ ➨ ➧➦ ➭❇➾➨ ➫✎➚ ➧➦ ➭❇➾✝➪➹➶ ➨ ➚ ➵ ➪ ➸➘➺➧➦ ➭✙➲ ➨ ➸➧➦ ➭❇➾➨ ➫❥➚ ➧➦ ➭❇➾✝➪➹➶ ➨ ➴✙➷➒➬ ➧➦ ➨❸➩☎➫ ➭ ➯ ➧✝➨➦ ➭➳➲ ➵ ➸ ➺➝➲➮➻➧✝➨➦ ➭ ➚ ➽➧✝➨➦ ➭➩☎➫ ➲ ➧✝➨➦ ➭➩ ➪q➶ ➚ ➵ ➪➸ ➺➧✢➨➦ ➭➩ ➪➮➲ ➸➧✢➨➦ ➭➩☎➫❥➚ ➧✝➨➦ ➭❏➶➱➴✙✃➒➬ ❐✛❒➹❮ ❰ÐÏ②Ñ❄Ò♥Ó✿Ô✛Õ×Ö✛ÓÙØ❛Ú➠Ñ✫Û❸Ó②Ï★Ü Ý✰Þ✲ß✴à✦á✷â➣ã äæå❱ç✢è✙é✲êëéÐêëìëè✙é❄íîé❄ê❆ì➣å❪ï⑨ð✰é✲ñóò✰éÐçëå❱ç✢è✙éôé❄ï⑨ç✝ñ☎å♥õ❖ö÷êëø➮ç✢ù✔ê✢úqö❳ù❇ï✙ò❱ê✢å✰û❇ü➳ç✢ù✔å❪ï✿ý þ✿ÿ✁✄✂✆☎✞✝✦þ✄✟ ✠☛✡✌☞✎✍✑✏✓✒✕✔✗✖✙✘✚☞✛✡✜☞✕✢✤✣✛✥✧✦✧✍★✏✓✩✗✘✫✪✬✍★✡✌✪✮✭✕✯★✏✮✡✌☞✰✯★✡✲✱✳✭✛✖✵✴✌✔✶✖✵✍✶✷✧✔✶✸✹✭✺✘✻✯✵✏✼✡✜☞ ✟ ✽✿✾❁❀❃❂✑❄✑❅☛❆❈❇❊❉ ❀●❋■❍ ❏ ❉ ❀●❑■❍ ✏▼▲ ✔✌▲❖◆✽◗P❘ ✏✓✍✙✔✗✏✼✯★✥✛✔✶✖ ❉ ✡✌✖ ❏ ❉❚❙ ❯ ❘ ❆❊❱❲❨❳✕❩ ❬✬❭❫❪✬❭❵❴ ❛❝❜ø✫íõû❇é Ý✰Þ✲ß✴à✦á✷â✻❞ ❡✏✼✖❢✍✑✯❣✡✜✖✵✒✛✔✗✖❣✭✛✣✧❤✙✏✼☞✎✒ ✟ ✐❦❥♠❧ ❘★♥♣♦q ❆ ❉rts ◆❯ ❘★♥ ❱✈✉ ◆❯ ❘✶✇ ❏ ❉r✈① ◆②❘★♥③♦q ① ✾ ◆✽❘★♥ ❱ ❏ ◆✽❘ ❅ ④✧✏✮☞✺✩⑤✔❣✔✶✏⑥✯✵✥✛✔✗✖✄◆② ❘★♥ ♦q ✡✌✖⑦◆✽❘★♥ ❱ ❏ ◆✽❘ ✏✓✍✿⑧✗✔✗✖✵✡ ❳✛❩⑩⑨❷❶❢❸❢❹✻❺✧❻✗❶✁❶⑤❼❫❽❵❾✕❶✗❿ ◆❯ ❘ ❆ ◆❯ ❘★♥ ❱➁➀➃➂ ❿ ◆✽❘ ❆ ◆✽❘★♥ ❱✌➄ ❙ ✐❥♠❧ ❘★♥ ♦q ❆❊❱❲❨❳✕❩ ❙ ✐❥♠❧ ❘★♥ ♦q ❏ ✐❥◗❧ ❘⑤➅ ♦q ❆➆❍ ❳✕❩ ❙ ◆✽P♥ ❱ ❘ ❆ ◆✽◗P❘ ➇è✙éôé❄ï⑨ç✝ñ☎å❪õ❖ö◗➈❛ð❳ù❇å✰û❇ø✫ç✢ù❇ï✙ò êëå❪û✔ü✦çëù❇å✰ï✷ù✔ê❆õ✙ñ☎é❄ê✢é✲ñóð❪é✧➉ ➊✁➋ ➌t➍✬➎➐➏✆➑ ➒❝➓➔➎★→✜➍✌➣✬↔➙↕➜➛➜➎✑➝❵➞➠➟❁↔❃➝❵➓❃➡■↕➃➍✕➢★➤❃➎✑➝➥➍✧➓➃➞ ➦✯➧✯★✭✛✖✵☞✺✍➧✡✜✭✕✯➧✯★✥✺✘✫✯➨✯★✥✛✔t➩✺✖❢✍➥✯➨✡✜✖✵✒✕✔✶✖➧✭✛✣✧❤✙✏✼☞✺✒➜➫❝✯✵✥✛✔ ✝✘✫➭✹✢✤➯t✔✗☞✺✒✕✖✵✡✫➲➳✘✌☞✺✒➵✯✵✥✛✔➸✱✳✔✶✘✌➺❦✢ ➯➻✘✫✖✵➺➼✏✮☞✛✴➽✍✵✩❢✥✛✔✗➺✚✔➁✍✄✘✫✪✮✪✮✡✻❤➚➾❁✡✜✖✄✔✗☞✹✯✵✖★✡✜✣✹✦✰➪✧✏✼✡✜✪✮✘✫✯★✏✮☞✛✴➶✍★✡✌✪✮✭✕✯★✏✮✡✌☞✺✍✶▲✙➹❣✥✛✔✶✍★✔⑩✍✵✩❢✥✛✔✗➺✚✔➁✍✄✩✗✘✌☞✛☞✛✡✌✯ ✒✕✏✓✍➥✯✵✏✼☞✺✴✌✭✛✏✓✍✑✥➴➘♠✔⑤✯➥❤✳✔✗✔✗☞➨✍★✥✛✡✕✩❢➷✕✍✳✘✌☞✺✒➴✔⑤➭✕✣✺✘✌☞✺✍✑✏✮✡✌☞✎✍✗▲ ➹❃✡●✒✕✔⑤✯✵✔✗✖✵➺✚✏✼☞✛✔➶✏✼☞➸✘✜✒✕➪✻✘✫☞✺✩✗✔➽✏⑥➾✄✘➧✴✜✔✗☞✛✔✶✖✵✘✌✪✈☞✧✭✛➺✚✔✗✖✵✏✮✩✶✘✫✪✳✍★✩❢✥✛✔✶➺✚✔✲❤✙✏✼✪✮✪✞✡✜☞✛✪✮✦➬✣✛✖✵✡✕✒✕✭✺✩✗✔✲✔✗☞✕✢ ✯★✖✵✡✌✣✧✦➼✍★✘✫✯★✏✓✍➥➾❁✦✧✏✮☞✛✴⑦✍✑✡✜✪✼✭✛✯★✏✮✡✌☞✺✍☛✏✮✍☛➪✌✔✗✖✵✦❦✒✛✏⑥➮➽✩⑤✭✺✪⑥✯➁▲☛➱✄☞✺✔❷✣✎✡✹✍★✍★✏✮➘✛✪✼✔❝✘✫✣✛✣✛✖✵✡✜✘✜✩❢✥❦✏✓✍➔✯✵✡⑩✒✕✔✶✖★✏✮➪✌✔❷✘✌☞ ✃
entropy function for our discrete scheme and prove a corresponding entropy cell inequality of the form U(+1)-U(),H一 0 where Hi+i is the numerical entropy fux associated with U Verifying an entropy cell inequality is not easy to do in general. There are certain lasses of schemes, as we ich can be shown to guarantee entropy satisfying solutions. It is often much simpler to verify whether a scheme belong to one such classes 3 Entropy Satisfying Schemes 3.1 Monotone schemes SLIDE 15 If a scheme can be written in the form H(i-1,0 ≥08= ,J,…,j+T then the scheme is monotone and tropy satisfying e at most first order accurate Convergence in the presence of Discontinuities For discontinuous solutions we seek convergence in weaker norms that those we have used for problems with smooth solutions. Most of the convergence results available for finite volume methods for conservation laws measure convergence thep=l norm. This norm can be seen to be weaker than the p=2 and p=oo in the sense that it is possible to find a scheme that will converge in the p=1 norm, but will not converge in the other norms. In fact, shock apturing methods are not conver gent in the p= oo norm when the solution discont inuous Moreover, the so called "first order schemes"i.e. schemes that have a trunca on error O(Ar, At), often converge at an even lower rate in the presence of discontinuities [L]. Note that our definition of truncation error is based on the ssumption that the exact solution is smooth and therefore is not applicable the discontinuous case
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3.1.1 Godunov's method SLIDE 1( The best know monotone scheme is Godunov's method j+1 Assume piecewise constant solution over each cell. Compute interface flux by solving interface(Riemann) problem exactly SLIDE 17 min f(a) uj U∈[v, Then △t The above eapression 2ves the eract 6ut for the Riemann pro bem and is vad for any sca-tr conservation -w with conver as we- as non conver 6ures. Fn addition it 2ives the correct 6u correspondin2 to the weak 5o-stion satisfyin2 nik'y- entro e point out that this 6ut is on-y vaid for a short time. Fn fact it is va-id 2enerated from the so-itiot one riemann pro hem start interactin2 writh the waves 2enerated by ne 2hborin2 inter faces. The fact that the so tion is eract for short times is due to the particu-ar form of the so-itions of the Riemann prob-em(i. e a-thou2h the 2emera n of the equations is a function of a and t the so tion to the rier m can be expressed as a function of a sinDe varibl-e name-y c/t3 this ty is known as simi-arity3 see /-] for detai- SLIDE 1 Applie to ur ers'e uation 2+1j,0+1s>u 2(u+1+t)
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