Hyperbolic Equations: Scalar One-Dimensional Conservation laws Lecture 11
✂✁☎✄✝✆✟✞✡✠✝☛✌☞✎✍✑✏✓✒✕✔✗✖✙✘✛✚✜✍✑☛✣✢✥✤✧✦✩★✪✏✫✘✬☞✭✘✮✞✰✯✱✢✲✆✴✳✡✵✶✍✑✷✸✆✹✢✥✤✺✍✑☛✣✢✙✘✬☞ ✻☛✣✢✥✤✜✆✟✞✽✼✗✘✛✚✜✍✑☛✣✢✿✾❀✘❂❁❃✤ ✾❄✆✟✏❅✚❆✖✥✞❇✆✸❈✌❈
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
umaC=LDER VL (m1) /ua=-0 The integral form is the form of the differential forms are derived. In contrast with the differential form, we note that the integral from is well defined even when the solution u and or the flur f are discontinuous. We show below, an example of derivation of the different forms of the conservation laws from physical principle. 1.2 Derivation Example 1.2.1Cor p(xt)°b4 ATE OF CHANGE OF MASS FLUX OF FLUID IASS INSIDEΩ THT OUGH aQ V·(pu) t v·(p)a=0 heIt all n,h wacc wtda Th cFtha differential forIt thaichatvat gc lar To derive the differential form of the conservation law, we hat P(a, t)and v(a, t) are differentiable function
ó✽ô❦ó✽ôâõ ö✑÷➄ø✭ù➢ú❽û⑦ü✉ý✫þ✜ÿ❽û✁ ✂☎✄✝✆✟✞✡✠☞☛ ✌✎✍✑✏✓✒✕✔✗✖✙✘✛✚✢✜✤✣✙✥✓✦★✧✤✖✙✍✪✩✫✜✬✔✭✏✯✮✱✰✳✲✴✓✵✷✶✸✴✓✹✻✺✡✼✯✽✾ ✿❁❀❃❂❅❄☎❆❄☎❇✤❈ ❄❊❉✷❋●❆❊❍ ❄✴❏■▲❑✝▼✳◆P❖ ❑ ❑ ❇ ✿❀ ❆ ❑✝▼✳◆✳◗ ✲❉✷❋●❆✹❍ ◗ ❉✷❋●❆✵ ❍ ✺ ❘❚❙❱❯❳❲✟❨✓❩❬❯❬❭✑❪✸❫✁❴✝❵❜❛✁❪❞❝❡❲✗❢❣❩●❙❱❯❤❝✤❛✁❢❞❩✐❭✝❯✛❨☎❯✛❪✸❫✁❴❁❵❜❛✁❪❞❝❥❛✕❵❣❩●❙❱❯✤❦❧❛✁❨❱❢✛❯✛❪❞♠★❫✬❩♥❲♦❛✬❨♣❴q❫✁r☞❵❧❪✸❛✁❝srt❙❁❲♦❦❧❙ ❩●❙❱❯✈✉✁❲✇①❯✛❪✸❯✛❨②❩③❲♦❫✁❴④❵❜❛✬❪❞❝⑤❢⑥❫✬❪✸❯✈✉✑❯✛❪❞❲✟♠★❯❧✉✑⑦⑨⑧❜❨⑩❦❧❛✁❨✓❩♥❪✸❫✁❢❞❩✡r✐❲✟❩●❙✫❩●❙❱❯✈✉✁❲✇①❯✛❪✸❯✛❨②❩③❲♦❫✁❴★❵❜❛✁❪❞❝❤❶tr❷❯①❨☎❛✬❩❬❯ ❩●❙❱❫✁❩✢❩●❙❱❯✫❲✟❨②❩❸❯③❭✑❪✸❫✬❴❁❵❧❪❹❛✁❝❥❲✗❢❳r❷❯❜❴✟❴❷✉✪❯♦❺❷❨☎❯❧✉❻❯✛♠★❯✛❨❃rt❙❱❯✛❨❃❩✟❙②❯❳❢✛❛✁❴❽❼✙❩③❲♦❛✁❨ ❆ ❫✬❨☎✉❞❾❚❛✁❪✤❩●❙❱❯✢❿➀❼✑➁ ❉ ❫✬❪✸❯✫✉✁❲✗❢✛❦❧❛✁❨✓❩♥❲✟❨✓❼②❛✁❼❁❢✛⑦➃➂✯❯⑤❢✸❙❱❛✬r➅➄❧❯✛❴q❛✁r✷❶➆❫✁❨➇❯❹➁✝❫✬❝➆➈❚❴q❯✫❛➉❵❳✉✑❯✛❪❞❲✟♠★❫✬❩♥❲♦❛✬❨➊❛➉❵⑤❩✟❙②❯✫✉✁❲✇①❯✛❪✸❯✛❨②❩ ❵❜❛✁❪❞❝❤❢⑤❛➉❵➋❩●❙❱❯❳❦❧❛✁❨②❢❜❯✛❪❞♠✁❫✁❩③❲♦❛✁❨➌❴✭❫✁r✷❢❷❵❧❪✸❛✬❝✳➈②❙❁➍★❢❞❲♦❦❧❫✁❴②➈☎❪❞❲✟❨☎❦✛❲➈☎❴✭❯❞❢✛⑦ ➎✐➏♥➐ ➑➓➒→➔✓➣❸↔✷↕t➙❱➣✕➛❷➜➞➝⑩➟✢↕✐➠➢➡⑥➤➉➒ ó✽ô✗➥❇ô❦ó ➦❯ÿ✽÷t➧✭ùrû➩➨✉ü✉ø④➫❦ÿ✽÷❴ÿ☎➭✢➯ü❱➧➩➧ ✂☎✄✝✆✟✞✡✠➌➲ ✌✎✍✑✏✓✒✕✔✗✖✙✘✛✚➆✜➵➳✑✍✑➸✭➺❱✩✤✘⑤✮➻✘✛✏②➼✛➸q✍✝✒✕✘➩✖➾➽❁➚✯✒✕➺②✚✕➪♦✜✪➼❜✘ ❄✮➻➼✛✍✑✏✝➶❹✜✑✔q✏②✔q✏❱➹✤➘✓➺❱✔✭✖✯✍✑➪✷✖❱✘✛✏②✒✸✔➴➶➉➚⑩➷ ❋♦➬ ✶ ❇✕❍ ✜✬✏✓✖➱➮❁✏❱✍✁✃✢✏⑩➳✑✘④➸q✍✙➼❜✔q➶➉➚➵❐ ❋♥➬ ✶ ❇✕❍ ✾➆❒✐❮✢❰PÏ✈ÐÑ✌➀Ò⑥❒①Ó✈ÔÕ❰➻Ï✈Ð ✰ Ö×❒✈Ø❱Ø✯Ð✷Ù✡Ú➆ÛÜÏ✈ÐÝÐ✻ÙtÚ➆✽➉Þ Ö×❒⑥Ø❱Ø⑩✽➉Ó✈Ø❁✽➉Þ✈❰P✮ ❮✢Ò⑥✾➆Ï✈Ú⑥ÔÕÒ ❄✮ ❄ ❄✓❇ ✿❁❀ ➷ ❑✝▼ ◆ ◗ ✿❱ß④❀ ➷✝❐✯à❜á ❑✝â ◆ ◗ ✿❀❳ã à ❋ ➷✝❐ ❍ ❑❁▼ ✂☎✄✝✆✟✞✡✠☞ä ✿❀Ñå❜❄➷❄☎❇❤❈ ã à ❋➷✝❐ ❍❬æ ❑❁▼✳◆➓❖ ç✍✪➸✭✖②✒➀➪●✍✑✚➆✜✬➸✭➸t✮✈è❱✒✸✍✤✃✎✘❣➼④✜✬✏⑩✃✢✚✸✔q➶✸✘ ❄➷❄☎❇❤❈ ã à ❋ ➷✝❐ ❍ ◆P❖ ❮ç✔✗✒✢✔✗✒➀➶ç✘❣é➫●ê✹ù➧û⑦ù➢÷➄ø④➫æü✉ý➀➭ÿ⑤û✁ ✍✑➪✡➶ç✘❣➼❜✍✪✏②✒✕✘④✚✸➳✁✜✬➶✸✔✭✍✑✏➱➸✗✜★✃➋ë ❘✓❛➌✉✑❯✛❪❞❲✟♠✁❯➱❩●❙❱❯⑩✉✁❲✇⑥❯❜❪✸❯✛❨②❩③❲♦❫✁❴✓❵❜❛✁❪❞❝ì❛➉❵✤❩●❙❱❯⑩❦❧❛✁❨❱❢✛❯✛❪❞♠★❫✬❩♥❲♦❛✬❨➊❴✭❫✁r✷❶⑥r❷❯❳❙❱❫✬♠★❯✯❫★❢❧❢❞❼✙❝✫❯❹✉×❩✟❙②❫✁❩ ➷ ❋♦➬ ✶ ❇✕❍ ❫✁❨❚✉ ❐ ❋♦➬ ✶ ❇✕❍ ❫✬❪✸❯❤✉✬❲✇①❯✛❪✸❯❜❨✓❩♥❲♦❫✪➄❜❴✭❯✎❵❧❼✙❨❚❦❜❩③❲♦❛✁❨②❢✛⑦ í
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)=fu2
î✐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❐ ❐
Burgers’ Equat. The actual equation studied by Burgers includes a viscous ter This is one of the simplest models that includes the nonlinear and vis cous ef fects of fluid dynamics. Again, when we include the viscous term, the equation becomes parabolic and does not admit discontinuous solutions An important aspect of the fux function, that will be used later, is that it is onvex;i.e f(u)==>0 1. 3. 3 Traffic Flow sLide 8 Let pla, t) denote the density of cars(vehicles/km)and u(a, t) the velocity t Ass that u is a function of where<ps Pmax and umax is some maximum speed(the speed limit?). N4 Note 4 rafic Flow Problem Typically on a highway, we wish to drive at some speed umax, but in heavy traffic we slow down. At some point, the highway reaches its maximum capacity of cars, Pmax, and our velocity is zero. The simplest model for this relationship between velocity and density is that given above. This function has been found to provide a fairly good model for actual traffic fows. For example, for the Lincoln tunnel a good fit to actual dat a was obt ained using the function f(e) which has a similar shape to our linear relation(see wD We point out that with either of the two relationships between car density and velocity, the Alux is a concave function of p; i.e. f"(p)<0 1.3.4 Buckley-Leverett Equation sLide 9 two phase(oil and water) fluid flow in porous medium. Let 0< (a, t)<I represent the saturation of water
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