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 t仍t2)+∑t修ti conservation errors If the solution is smooth, the conser vation errors are O(A ) If the solution is not smooth, the conservation errors are O(1) 2.4 First Order Upwind 2.4.1 Linear Advection Equation SLIDE 7 0 a constant >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
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