1.3 Examples 1.3.1 Linear Advection Equation Model convection of a concentration p(a, t) 0 at a t tant Ad vection-Diffusion Equation Consider the flux of a chemical past some point in a stream. If there is no diffusion in the fow, the concentration profile will convect downstream with a velocity a, and is described by the linear advection equation. In practice molecular diffusion and tur bulence will cause the concentration profile to change With the simple one-dimensional model we cannot model turbulence the effect of molecular diffusion Can be included by determining the diffusive Aux. This flux is described by Fourier's Law of heat conduction(the diffusion of a chemical concentration is similar to diffusion of heat diffusive flux =-Dop Combining this with the advective flux, ap, we obtain the advection-diffusion equation Note that for the advection-diffusion equation, the flux function now depends af as well as p. The advection-diffusion equation is a parabolic equation, while the linear advection equation is hyperbolic. This means that the advection diffusion equation always has smooth solutions, even if the initial data is dis continuous, while the linear advection equation admits discontinuities. We will onsider some solutions of the linear advection equation later in the lecture. 1.3.2 Inviscid Burgers?Equation SLIDE 7 Flux function f(u)=fu2î✐ï♥ð ñ⑩ò✢ó✷ô➢õ✈ö➉÷✡ø ù☎ú✭û✡ú♦ù ü✎ý♦þ✻ÿ✁￾✄✂✆☎✞✝✠✟✓ÿ☛✡✌☞④ý✎✍☎þ✑✏✓✒✕✔✠￾✖☞➩ý✎✍☎þ ✗ ✘✁✙✛✚✢✜✤✣ ✥✧✦✖★✖✩✫✪✭✬✮✦✌✯☛✰✱✩✫✬✳✲✵✴✶✦✱✯✆✦✌✷✹✸✺✬✮✦✱✯✻✬✮✩✫✯✁✲✽✼✾✸✿✲✽✴❀✦✌✯✆❁❃❂✎❄✢❅✽❆❈❇✳❉ ❊❁ ❊❆●❋ ❊❁☛❍ ❊❄❏■ ❊❁ ❊❆●❋ ❍ ❊❁ ❊❄❑■✑▲ ❍✧❉▼✬✮✦✱✯✻◆❖✲✾✸P✯✁✲ ◗❙❘ ❚✤❯✕❱✳❲❨❳ ❩❭❬❫❪✄❲✁❴☛❱❈❵❖❯☛❛❫❜❖❝❨❵❡❞❙❢❤❣✐❵❥❯✖❛❧❦♥♠✠❢❃♦✕❱❈❵❖❯☛❛ ♣✦✌✯❤◆❈✴q★✖✩r✼s✲✵t✄✩✈✉✻✇✖①②✦✌✷③✸✤✬✾t✄✩✫④s✴q✬r✸✌✪⑥⑤✻✸✌◆❈✲⑦◆❈✦✱④✺✩❨⑤❃✦✌✴❀✯✁✲⑧✴✶✯✑✸⑨◆❖✲✵✼✽✩✐✸P④✈⑩❧❶❷✷❙✲✵t✄✩r✼✵✩✈✴❀◆✺✯✄✦ ★✖✴✶❸❃✇❤◆❈✴❀✦✌✯❧✴✶✯❑✲✽t✻✩✈✉✻✦✿❹❻❺▼✲✵t✄✩✧✬✮✦✌✯❤✬✮✩r✯✁✲✵✼✵✸P✲✽✴❀✦✌✯②⑤✻✼✽✦✌❼✻✪✶✩❨❹⑥✴❀✪✶✪❽✬r✦✌✯☛✰✌✩✐✬✳✲⑧★✖✦✿❹⑥✯✻◆❈✲✽✼✵✩✫✸✌④❾❹⑥✴❿✲✵t ✸➀✰✌✩r✪❀✦✖✬✮✴✶✲❖➁②❍❃❺➂✸✌✯✻★②✴q◆⑧★✖✩✐◆✽✬r✼✽✴❀➃❤✩✐★➄➃☛➁➅✲✵t✄✩✧✪✶✴❀✯✄✩✐✸P✼⑧✸✌★✖✰✱✩✫✬✮✲✽✴❀✦✌✯②✩✐➆✁✇✻✸✿✲✵✴✶✦✱✯✭⑩❑❶❥✯❧⑤✄✼✾✸✌✬✮✲✽✴q✬✮✩✌❺ ④✺✦✌✪❀✩✫✬r✇✄✪❀✸✌✼✭★✖✴✶❸❃✇✻◆✽✴❀✦✌✯♥✸✌✯✻★❙✲✽✇✄✼✵➃✄✇✄✪❀✩r✯✻✬r✩▼❹⑥✴❀✪✶✪✁✬r✸P✇❤◆❈✩➇✲✵t✄✩➈✬r✦✌✯✻✬r✩r✯✁✲✽✼✾✸✿✲✵✴✶✦✱✯❙⑤✄✼✵✦P❼❤✪✶✩❫✲✵✦❽✬✾t✻✸P✯✻➉✌✩✌⑩ ➊✑✴✶✲✽t❭✲✵t✄✩✺◆❈✴❀④✺⑤✄✪✶✩●✦✌✯✄✩r➋❷★✄✴✶④✺✩r✯❤◆❈✴❀✦✌✯✻✸✌✪✠④✺✦✖★✖✩✫✪✠❹➌✩●✬✫✸P✯✄✯✄✦✌✲❙④✺✦✖★✖✩r✪✹✲✵✇✄✼✽➃✻✇✄✪✶✩✫✯✻✬✮✩✱❺➍t✄✦✿❹➂✩✫✰✌✩✫✼ ✲✽t✻✩✞✩✮❸➍✩✫✬✮✲●✦✌✷➎④s✦✱✪✶✩✐✬✮✇✄✪q✸P✼s★✖✴✶❸❃✇✻◆✽✴❀✦✌✯②✬✫✸P✯➅➃❃✩✞✴❀✯✻✬✮✪❀✇✻★✖✩✐★➄➃☛➁✤★✄✩✮✲✽✩✫✼✽④✺✴❀✯✄✴❀✯✄➉❭✲✵t✄✩✆★✖✴✶❸➍✇✻◆❈✴❀✰✌✩ ✉✻✇✖①✕⑩❻➏➐t✄✴q◆➎✉❤✇✖①❭✴❀◆✓★✖✩✐◆✽✬r✼✽✴❀➃❤✩✐★✧➃☛➁❭➑✄✦✱✇✄✼✽✴❀✩r✼✐➒ ◆❽➓✢✸➔❹→✦✌✷▼t✄✩✫✸P✲✓✬r✦✌✯✻★✖✇❤✬✳✲✽✴❀✦✌✯❧❂✛✲✵t✄✩s★✄✴❿❸➍✇✻◆✽✴✶✦✱✯ ✦P✷❫✸✺✬✾t✄✩r④✺✴q✬r✸P✪✭✬r✦✌✯✻✬r✩r✯✁✲✽✼✾✸✿✲✵✴✶✦✱✯✆✴q◆⑥◆✽✴✶④✺✴❀✪❀✸✌✼➐✲✽✦⑧★✖✴✶❸❃✇✻◆✽✴❀✦✌✯✈✦P✷❫t✄✩✫✸P✲✾❇✮❉ ★✖✴❿❸➍✇✻◆✽✴✶✰✱✩③✉✻✇✖① ■→➣➎↔ ❊❁ ❊❄➎↕ ♣✦✌④●➃✄✴❀✯✄✴✶✯✻➉❙✲✵t✄✴q◆➇❹⑥✴❿✲✵ts✲✽t✄✩➎✸✱★✖✰✌✩✐✬✳✲✽✴❀✰✌✩➌✉✻✇✄①➍❺✁❍✁❁❤❺P❹➌✩⑥✦✌➃✄✲✵✸P✴❀✯●✲✽t✄✩ ￾✄✝✠✟✓ÿ☛✡✌☞④ý✎✍☎þ✢➙✵✝✻ý❡➛➈✔✹➜④ý➝✍✓þ ÿ☛✒✕✔✠￾✄☞④ý✎✍☎þ ❉ ❊❁ ❊❆●❋ ❊ ❊❄ ➞ ❍✁❁ ➣⑨↔ ❊❁ ❊❄➈➟➠■➡▲ ◗ ✦✌✲✽✩➂✲✽t✻✸P✲✠✷❡✦✱✼✹✲✽t✄✩➐✸✌★✖✰✱✩✫✬✮✲✽✴❀✦✌✯✖➋❥★✖✴✶❸❃✇✻◆✽✴❀✦✌✯➢✩✐➆✱✇❤✸✿✲✽✴❀✦✌✯✢❺➔✲✽t✄✩➌✉✻✇✖①❻✷❡✇✻✯✻✬✳✲✵✴✶✦✱✯♥✯✄✦✿❹②★✄✩r⑤❃✩r✯✻★✄◆✹✦✱✯ ➤✐➥ ➤✐➦ ✸✌◆➐❹➌✩r✪❀✪✭✸✌◆➐❁❃⑩▼➏➐t✄✩➢✸✌★✄✰✌✩✫✬✮✲✽✴❀✦✌✯✖➋❥★✖✴✶❸❃✇❤◆❈✴❀✦✌✯✆✩✫➆✁✇✻✸✿✲✵✴✶✦✱✯✆✴q◆➎✸♥➧✻➨✿➩✽➨✱➫✵➭P➯➳➲✎➵❽✩✫➆✁✇✻✸✿✲✵✴✶✦✱✯✭❺✖❹⑥t✄✴❀✪✶✩ ✲✽t✻✩❭✪✶✴❀✯✄✩✐✸P✼✆✸✱★✖✰✌✩✐✬✳✲✵✴✶✦✱✯❧✩✐➆✁✇✻✸✿✲✵✴✶✦✱✯❧✴q◆✧➸☛➺✵➧✻➻✮➩r➫✾➭✿➯❿➲✎➵✮⑩➼➏➐t✄✴❀◆✞④✺✩✫✸P✯❤◆✺✲✽t✻✸P✲✞✲✵t✄✩➽✸✌★✄✰✌✩✫✬✮✲✽✴❀✦✌✯✖➋ ★✖✴✶❸❃✇❤◆❈✴❀✦✌✯⑨✩✫➆✁✇✻✸P✲✽✴❀✦✌✯⑨✸P✪❀❹➌✸➔➁✖◆✓t✻✸✱◆❻◆❈④✺✦☛✦P✲✵t✤◆✽✦✌✪❀✇✖✲✽✴❀✦✌✯❤◆r❺✠✩r✰✌✩✫✯➀✴✶✷➌✲✽t✄✩⑧✴❀✯✄✴✶✲✽✴q✸P✪➈★✄✸P✲✵✸✈✴q◆❻★✖✴q◆❖➋ ✬✮✦✱✯✁✲✽✴❀✯✁✇✻✦✌✇✻◆✫❺☛❹⑥t✄✴✶✪❀✩③✲✽t✻✩❻✪✶✴❀✯✄✩✫✸✌✼➎✸✌★✖✰✱✩✫✬✮✲✽✴❀✦✌✯❨✩✫➆✁✇✻✸✿✲✵✴✶✦✱✯❨✸✱★✖④✺✴❿✲✾◆⑥★✖✴q◆✽✬r✦✌✯✁✲✽✴❀✯☛✇✄✴❿✲✵✴✶✩✐◆r⑩➇➊✤✩❻❹⑥✴❀✪✶✪ ✬✮✦✱✯✻◆✽✴❀★✖✩✫✼⑥◆❈✦✱④s✩❻◆✽✦✌✪❀✇✖✲✽✴❀✦✌✯❤◆➐✦P✷✹✲✽t✄✩❻✪❀✴✶✯✻✩✫✸P✼➎✸✱★✖✰✌✩✐✬✳✲✵✴✶✦✱✯✞✩✫➆✁✇✻✸P✲✽✴❀✦✌✯❨✪❀✸P✲✽✩✫✼⑥✴✶✯❨✲✽t✻✩❻✪✶✩✐✬✳✲✽✇✻✼✽✩✱⑩ ù☎ú✭û✡úq➾ ➚❜þ➍✟→ý➝➜✫✡✪ý✎✝✑➪s✔✢✂✿➶☎ÿ✁✂➔➜✱➹➇✏❙✒✕✔✹￾✖☞➩ý✎✍☎þ ✗ ✘✁✙✛✚✢✜➅➘ ➑✹✪❀✇✖①✞✷❡✇✻✯✻✬✳✲✵✴✶✦✱✯✧➴❫❂✎➷➍❇ ■➮➬➱ ➷➱ ♣✦✌✯❤◆❈✩✫✼✽✰✿✸✿✲✵✴✶✦✱✯✞✪q✸➔❹✃❉ ❊➷ ❊❆s❋ ❊ ➱➬ ➷➱ ❊❄ ■ ❊➷ ❊❆✺❋ ➷ ❊➷ ❊❄❑■➡▲ ◗❽❐ ❐