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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au af(u) 0 This equation has applications in oil reservoir simulation where one models the flow of oil and water through porous rock or sand. So varies between 0 and 1 u=0 represents a flow of pure oil, u=l represents pure water f(u)=2+a(1 a: constant w 1 Note 5 The Buckley-Leverett Equations For the most part, we consider equations where f(u) is convex(or a concave function of the unknown variable. In the convex(or concave)case, the solution of an initial discontinuos data distribution(Riemann problem) is always either a shock or a rarefaction(or expansion)wave. When f is not convex(nor concave the solution might involve both. The Buckley-Leverett equati example where this situation may occur 2 Smooth solutions 2.1 Total derivative SLIDE 10 Recall the primitive form of the conservation law +a(u) The total time variation of u(a, t), on an arbitrary curve a=a(t), in the a-t plane, is dt at dt 2.2 Characteristics SLIDE 11 If at=a(u) 0→ The curves r=c(t), such tha dt a(u)are called characteristics a )constant characteristics are straight lines
➆✢➇ ➆✢➈⑨➉ ➆✌➊❈➋s➇✌➌ ➆✢➍ ➎✉➏ ➐✑➑✴➒➔➓❧→❅➣☛↔❪↕❜➙④➒✲➛❜➜❯➑②↕✩➓❧↕➞➝♣➝✑➟➠➒✲➡❅↕✩➙✓➒✲➛✩➜❪➓✠➒✺➜➄➛✩➒✺➟✟➢➞→P➓❏→❏➢P➤➥➛❜➒✺➢➦➓❅➒✺➧⑨↔✆➟⑥↕❜➙④➒✲➛❜➜➩➨⑩➑②→❏➢✐→⑨➛✩➜✑→➫➧✘➛✡➭❝→☛➟➯➓➫➙s➑②→ ➲➛✩➨➳➛♠➵✠➛❜➒✺➟③↕❜➜✢➭✘➨✎↕❜➙➸→❏➢➦➙✺➑✆➢✐➛✩↔♣➺➥➑r➝❪➛✩➢➞➛✩↔✴➓➦➢✐➛✡➡❅➻➼➛✩➢➦➓❏↕✩➜✑➭♣➽❙➾✑➛ ➇ ➤➥↕❜➢P➒✲→P➓❧➚❅→❏➙④➨✎→❅→❏➜➄➪❀↕❜➜✢➭❀➶ ➹ ➇ ➎➳➏ ➢✐→➸➝✑➢✐→☛➓☛→❏➜❪➙④➓✠↕ ➲➛✩➨➘➛❂➵➴➝✢↔✆➢✐→⑨➛✩➒✺➟➠➷ ➇ ➎➮➬ ➢✐→➸➝✑➢✐→☛➓☛→❏➜❪➙④➓➴➝✢↔✆➢✐→✠➨✎↕✩➙➱→☛➢❏➽ ✃r❐ ➊❈➋s➇✌➌ ➎ ➇✑❒ ➇❒ ➉✬❮ ➋ ➬✰❰ ➇✑➌ ❒ ❮◆Ï❡Ð❏Ñ♣Ò❪Ó❂Ô➞Õ♣Ò✸Ô❙Ö×➬ Ø✽Ù✌ÚPÛ❃Ü Ý✰Þ✎Û➼ß✝à✌áPâ✟ã♠Û✡ä③å❂æ✰Û②ç✆Û❜è❝Û②Ú❏Ú➫é❧ê⑩à◆ë✟Ú❂ì➱Ù✆í③î ï Ñ❝ð❙Ô➞ñ②ò▼ó▼Ñ❝Ó❂Ô❿ô❪Õ♣ð❂Ô❞õ✌ö✧ò▼Ð☛Ñ❝Ò❪Ó✐÷✤ø✆ò✡ðùò❞ú✸û❪Õ✩Ô➞÷⑥Ñ❝Ò❪Ó❙ö✰ñ❪ò❏ð➞ò ➊❈➋✲➇✑➌ ÷✤Ó➦Ð❏Ñ♣Ò✴ü♣ò❏ý ➋ Ñ♣ð➫Õ❯Ð☛Ñ♣Ò◆Ð❏Õ➥ü♣ò ➌ þû②Ò❪Ð☛Ô✐÷✤Ñ♣Ò➼Ñþ Ô➞ñ②ò➦û②Ò②ÿ✴Ò②Ñ✩ö✰Ò ü✩Õ❜ð➞÷➔Õ✁✄✂ò✆☎✞✝➱Ò Ô➞ñ②ò➦Ð☛Ñ♣Ò✴ü❝ò☛ý ➋Ñ❝ð➴Ð❏Ñ♣Ò❪Ð✡Õ➥ü♣ò ➌ Ð❏Õ❝Ó❂ò❝õ♣Ô➞ñ②ò➫Ó❂Ñ✂û✆Ô➞÷⑥Ñ❝Ò Ñ þ Õ❜Ò⑨÷⑥Ò②÷⑥Ô✐÷➔Õ✂ ø②÷✤Ó➞Ð☛Ñ❝Ò❝Ô➞÷⑥Ò✴û②Ñ✸Ó❈ø②Õ✩Ô❅Õùø✆÷➔Ó❂Ô✐ð➞÷✁û②Ô✐÷✤Ñ♣Ò ➋✠✟÷✤ò❏ó✘Õ❜Ò❪Òrô②ð➞Ñ✁✡✂ò✡ó➌ ÷➔Ó✁Õ✂ö➴Õ☞☛✆Ó⑩ò❏÷⑥Ô✐ñ②ò✡ð❈Õ Ó✐ñ②Ñ✆Ð❅ÿ❿Ñ❝ð✁Õ❙ð❅Õ❜ð➞òþÕ❝ÐPÔ✐÷✤Ñ♣Ò ➋ Ñ♣ð❈ò☛ý✆ô❪Õ♣Ò❪Ó✐÷⑥Ñ❝Ò➌ öÕ➥ü♣ò✌☎✎✍✉ñ②ò✡Ò ➊ ÷➔Ó❈Ò②Ñ❜Ô✁Ð☛Ñ❝Ò✴ü♣ò☛ý ➋Ò②Ñ♣ð✁Ð☛Ñ❝Ò❪Ð❏Õ➥ü❝ò ➌ õ Ô✐ñ❪ò❃Ó❂Ñ✂û✆Ô➞÷⑥Ñ❝Ò❨ó⑨÷✑✏♣ñ✸Ô ÷✤Ò✴ü♣Ñ✂ü♣ò ✁ Ñ❜Ô✐ñ✎☎✓✒ñ②ò✕✔➴û❪Ð❅ÿ✂ò✖☛✘✗✚✙✄ò❏ü❝ò❏ð➞ò☛Ô❂Ô▼ò✡ú✸û❪Õ❜Ô✐÷✤Ñ♣Ò ÷✤Ó❀Õ➅Ó❂÷✤ó▼ô✂ò ò☛ý②Õ♣ó⑨ô✂ò➦ö✰ñ②ò✡ð✐ò❿Ô✐ñ❪÷✤Ó Ó✐÷⑥Ô✐û❪Õ❜Ô✐÷✤Ñ♣Ò❯ó▼Õ☞☛❀Ñ✴Ð✡Ð☛û②ð✛☎ ✜ ✢✤✣✦✥✧✥✩★✫✪✬✢✭✥✯✮✖✰✤★✫✱✲✥✴✳✶✵ ✷✹✸✻✺ ✼✴✽✫✾❀✿✫❁✯❂❄❃✎❅✡❆❈❇✞✿❉✾❀❆❊❇❋❃ ●■❍✘❏▲❑✎▼❖◆◗P ✟ ò❞Ð❏Õ✂✑✂ Ô✐ñ❪ò➫ô②ð✐÷✤ó▼÷Ô➞÷⑥ü❝ò þÑ♣ð➞ó Ñ þ Ô✐ñ②ò✠Ð❏Ñ♣Ò❪Ó✐ò❏ð➞ü✩Õ✩Ô➞÷⑥Ñ❝Ò ✂Õ➥ö ➆✢➇ ➆✢➈ ➉♦❮ ➋s➇✌➌ ➆✢➇ ➆✢➍ ➎➳➏ ✒ñ②ò➦Ô➞Ñ❜Ô➞Õ✂ Ô✐÷✤ó▼ò✠ü✩Õ❜ð➞÷✤Õ❜Ô✐÷✤Ñ♣Ò❯Ñþ ➇③➋✲➍✎❘✐➈❂➌ õ❪Ñ❝Ò✮Õ❜Ò✝Õ❜ð✁ ÷Ô➞ð➞Õ♣ð❙☛➼Ð☛û②ð➞ü♣ò ➍ ➎ ➍⑩➋✲➈❂➌ õ◆÷⑥Ò✝Ô➞ñ②ò ➍ ❰ ➈ ô ✂Õ❜Ò②ò❝õ✆÷✤Ó ❚ ❚➇ ➈ ➎ ➆✢➇ ➆✢➈ ➉ ❚ ❚➍ ➈ ➆✢➇ ➆✢➍ ✷✹✸❯✷ ❱❳❲❨✿❉❅✡✿✞❩■✾❀❃❬❅❭❆✻❪✌✾✄❆✻❩❫❪ ●■❍✘❏▲❑✎▼❖◆✄◆ ❴✲❵ ❚ ❚➍ ➈ ➎✉❮ ➋s➇✌➌❜❛ ❚ ❚➇ ➈ ➎➘➏ ❛ ➇ ➎ ➇■❝ ➋ Ð☛Ñ❝Ò❪Ó❂Ô➞Õ❜Ò✸Ô ➌ ✒ñ②ò➫Ð☛û❪ð✐ü❝ò✡Ó ➍ ➎ ➍⑩➋s➈❂➌ õ◆Ó✐û❪Ð❅ñ➼Ô✐ñ❪Õ❜Ô ❚ ❚➍ ➈❯➎➘❮ ➋s➇✌➌ Õ❜ð➞ò❿Ð✡Õ✂✑✂ò❞ø❡❞☞❢❉❣❀❤✐❣✄❞◗❥✛❦✘❤☞❧❯♠✖❥✛❧✠❞✌♠ ➇ Ð❏Ñ♣Ò❪Ó❂Ô➞Õ♣Ò❝Ô♦♥ ❮ ➋s➇✌➌ Ð☛Ñ❝Ò❪Ó❂Ô➞Õ❜Ò✸Ô♦♥ ❞☞❢❉❣❀❤✐❣✄❞◗❥✛❦✘❤☞❧❯♠✖❥✛❧✠❞✌♠✧❣❀❤✐❦✕♠♣❥✖❤✐❣✄❧✠q■❢r❥✧s✠❧❯t✉❦✈♠ ✇
The characteristics are straight lines in the x-t plane along which u is constant lfu(ao, 0)=0, the characteristic passing througha=ao, t=0, is the solution of the following initial value problem dr/dt= a(uo),a(0)=xo; i.e. a Note 6 Characteristics The slope of the characteristic lines is determined by the initial condition uo (a) except for the trivial case in which f is a linear function of u. In this latter case, he slope characteristics is constant i.e. the characteristics are parallel. We note that, for our problems, the characteristics are straight lines, even in the explicitly on a, the characteristics are no longer straight lines on tha.quations linear case, because f is determined by u only. Fo burce term or a flux fu If we solve a problem on a finite domain, the number of boundary conditions to be prescribed, in the non-linear case, depends on the data itself. That is, in tho boundaries with incoming characteristics a boundary condition will be required Similarly, the solution at those boundaries with outgoing characteristics will be determined by the interior. We can see therefore that in the 1D case we can equire, two, one or no boundary condition For nonlinear conservation laws and arbitrary data, the characteristics may cross within finite time. This would suggest a multi-valued solution which does not make any will see that just at the point where th characteristics start crossing, the solution becomes discontinuous. At this point he differential primitive form of the equation, on which we are basing our solution procedure is no longer valid
①r②✄③♦④⑤②✡⑥✐⑦❙⑥✌④✲⑧❈③✲⑦⑩⑨❷❶⑩⑧❸⑨✠④⑩❶♦⑥◗⑦❙③✹❶⑩⑧❸⑦❙⑥✐⑨❺❹☞②✈⑧✉❻❼⑨▲❽■③✲❶✹⑨▲❽✧⑧▲②✡③✫❾➀❿✩➁✎➂r❻➃⑥✐❽r③❨⑥◗❻➃➄✐❽✈❹➆➅✉②❀⑨✠④⑤②✩➇✕⑨❷❶➈④➉➄◗❽✄❶⑩⑧✚⑥◗❽✡⑧❈➊ ➋➍➌➎➇❉➏✠❾✡➐✘➑❙➒✌➓✫➔→➇❭➐✆➣✉⑧➍②✄③↔④➉②✄⑥✐⑦⑤⑥✆④✖⑧✚③✖⑦⑩⑨❷❶⑩⑧❯⑨✠④➀➂✡⑥☞❶➉❶⑩⑨▲❽✘❹✴⑧➍②✈⑦⑤➄✐↕✆❹☞②✩❾✤➔→❾✡➐✌➣❭➁➎➔→➒✄➣✎⑨❷❶❨⑧➍②✄③❨❶✖➄✐❻❼↕❀⑧❸⑨✠➄✐❽ ➄❊➌✶⑧➍②✄③✯➌✲➄✐❻▲❻✑➄✐➅✫⑨▲❽✘❹➙⑨▲❽❭⑨▲⑧❯⑨✠⑥◗❻➜➛✐⑥✐❻❼↕✡③➝➂■⑦❙➄✌➞✲❻✑③✲➟➡➠✌❾r➢✐➠✌➁❡➔➥➤r➏➍➇■➐✛➓✲➑➦❾✉➏❯➒✌➓➧➔➨❾✡➐✌➩➝⑨✠➊▲③♣➊✓❾➫➔ ❾✡➐✹➭➯➤r➏➍➇■➐✛➓❊➁✖➊ ➲■➳✘➵▲➸✎➺❖➻✆➼ ➠✌❾ ➠✆➁ ➔→➤r➏➍➇➐ ➓ ➽ ❾❡➔➾❾➐ ➭➚➤r➏➍➇➐ ➓■➁ ➪♦➶ ➹❳➘❫➴⑩➷➮➬ ➱❐✃✞❒❀❮✐❒■❰✘➴✲➷✐❮☞Ï❯Ð✌➴❙Ï❈❰✛Ð Ñ✹Ò✄Ó➜Ô❙Õ➃Ö✌×❭Ó➈Ö✆Ø❭Ù⑤Ò✄Ó➜Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù❙Ó♣Ü❙Ý❷Ô✻Ù❙Ý❷Ú❋Õ✑Ý➃Þ✡Ó♣Ô✫Ý❷Ô➎ß❀Ó✖Ù❙Ó✖Ü⑤àáÝ➃Þ✡Ó♣ßáâ✈ã✩Ù❙Ò✡Ó➜Ý➃Þ✡Ý❺Ù⑤Ý✑Û✆Õ✡Ú✲Ö✌Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þá➇➐ ➏➍❾r➓⑩ä Ó✲å✄Ú✖Ó✖×❀Ù✫Ø➍Ö✌Ü✞Ù❙Ò✄Ó➈Ù⑤Ü❙Ý✑æ✈Ý✑Û✆Õ✄Ú✖Û✌Ô✻Ó✹Ý✑Þ➝ç➈Ò✄Ý✑Ú➉Ò✧è✧Ý✑Ô➎Û↔Õ✑Ý✑Þ✄Ó♣Û✆Ü✞Ø➍é✄Þ✡Ú⑩Ù⑤Ý➃Ö✌Þ➝Ö◗Ør➇✉ê❉ë❈Þ✴Ù❙Ò✄Ý❷Ô✫Õ❷Û✐Ù❙Ù❙Ó✖Ü➀Ú✖Û✆Ô❙Ó✆ä Ù❙Ò✡Ó➆Ô✻Õ✑Ö✆×■Ó✩Ú➉Ò✡Û◗Ü➉Û✆Ú✲Ù❙Ó✖Ü⑤Ý❷Ô❊Ù⑤Ý✑Ú♣Ô❋Ý❷Ô➜Ú✲Ö✌Þ✡Ô✻Ù⑤Û◗Þ✘Ù➈Ý❸ê Ó✌ê✞Ù❙Ò✄Ó➆Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖ÔìÛ◗Ü⑤Ó↔×✡Û◗Ü➉Û◗Õ✑Õ➃Ó♣Õ❯ê í❳ÓìÞ✡Ö◗Ù❙Ó✹Ù⑤Ò✡Û✐Ù✛ä✐Ø➍Ö✆Ü✫Ö✌é✄Ü✞×✄Ü⑤Ö✆â✄Õ✑Ó✖à➝Ô♣ä✐Ù❙Ò✄Ó➜Ú➉Ò❭Û◗Ü➉Û✆Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖Ô❉Û◗Ü⑤Ó✹Ô❊Ù⑤Ü⑤Û✆Ý➃î✌Ò✌Ù✞Õ✑Ý➃Þ✄Ó✛Ô✖ä✆Ó✖æ✆Ó♣Þ✯Ý✑Þ✴Ù❙Ò✄Ó Þ✄Ö✌Þ❀ï❸Õ✑Ý✑Þ✄Ó♣Û✆Ü❐Ú♣Û✆Ô❙Ó✆ärâ❭Ó✛Ú✖Û✆é✡Ô✻Ó➝èðÝ❷Ô↔ß❀Ó✲Ù⑤Ó✖Ü⑤àáÝ➃Þ✄Ó✛ß✕â✈ã✕➇ñÖ✆Þ✡Õ➃ã✌êóò✄Ö✌Ü↔Ô❙ã❀Ô❊Ù⑤Ó✖à➝Ô♦Ö◗Ø➀Ó✛ô✘é✡Û✐Ù⑤Ý➃Ö✌Þ✡Ô✖ä Ö✆Ü➎Ø➍Ö✆Ü❋Ô⑤Ú✖Û✆Õ✑Û✆Ü✫Ó♣ô✘é✡Û✐Ù⑤Ý➃Ö✌Þ✡Ô✫ç➈Ý➃Ù❙Ò✧Ó♣Ý❺Ù⑤Ò✄Ó✖ÜìÛ✩Ô✻Ö✌é✄Ü➉Ú✲Ó✹Ù❙Ó♣Ü❙àõÖ✆Ü➀Û❐ö✡é❀å✴Ø➍é✄Þ✡Ú⑩Ù⑤Ý➃Ö✌ÞáÙ❙Ò✡Û◗Ùìß❀Ó♣×❭Ó♣Þ✡ß✄Ô Ó✲å❀×✄Õ✑Ý❷Ú✲Ý➃Ù❙Õ✑ã✧Ö✆Þ✶❾✎ä✄Ù❙Ò✄Ó➆Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖ÔìÛ◗Ü⑤Ó↔Þ✄ÖáÕ➃Ö✌Þ✄î✆Ó♣Ü➜Ô❊Ù⑤Ü⑤Û✆Ý➃î✌Ò✌Ù✹Õ✑Ý➃Þ✄Ó✛Ô✖ê ë✚Ø❬çìÓ♦Ô❙Ö✆Õ✑æ✆Ó❨Û✩×✡Ü❙Ö✌â✄Õ➃Ó♣à÷Ö✆Þ✧Û✩ø❭Þ✄Ý❺Ù⑤Ó♦ß❀Ö✌à➝Û◗Ý✑Þ❬ä✆Ù⑤Ò✄Ó❐Þ✘é✡àóâ■Ó✖ÜìÖ◗Ø❫â❭Ö✌é✄Þ✡ß✄Û✆Ü❙ã➝Ú✖Ö✆Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þ❭Ô✫Ù❙Ö â■Ó➎×✄Ü⑤Ó♣Ô⑤Ú✲Ü⑤Ý➃â■Ó♣ß❫ä☞Ý✑Þ↔Ù❙Ò✄Ó➀Þ✄Ö✆Þ✄ï❸Õ✑Ý➃Þ✡Ó♣Û◗Ü✉Ú✖Û✆Ô❙Ó✆ä☞ß❀Ó♣×❭Ó♣Þ✡ß✄Ô✉Ö✆Þ↔Ù❙Ò✡Ó➀ß✄Û◗Ù⑤Û➈Ý➃Ù⑤Ô❙Ó✖Õ➃Ø❊ê❉Ñ✹Ò❭Û✐Ù✎Ý❷Ô✖ä☞Ý✑Þ↔Ù❙Ò✄Ö✘Ô✻Ó â■Ö✆é✄Þ✡ß✡Û◗Ü⑤Ý➃Ó✛Ô✞ç➈Ý❺Ù⑤Ò✴Ý✑Þ✡Ú✖Ö✆àáÝ➃Þ✡î↔Ú➉Ò❭Û◗Ü➉Û✆Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖Ô❉Û↔â■Ö✆é✄Þ✡ß✡Û◗Ü⑤ãóÚ✲Ö✌Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þ✴ç➈Ý➃Õ✑Õ✄â■Ó➈Ü❙Ó✛ô✌é✡Ý➃Ü⑤Ó♣ß❫ê ùÝ✑àáÝ➃Õ❷Û◗Ü⑤Õ➃ã✌ä✆Ù⑤Ò✄Ó↔Ô✻Ö✌Õ➃é❀Ù⑤Ý➃Ö✌Þ✤Û◗Ù❋Ù❙Ò✡Ö✌Ô❙Ó❨â■Ö✆é✄Þ✡ß✡Û◗Ü⑤Ý➃Ó✛Ô❋ç➈Ý➃Ù❙Ò✤Ö✆é❀Ù⑤î✆Ö✆Ý✑Þ✄îóÚ➉Ò❭Û◗Ü➉Û✆Ú⑩Ù⑤Ó✖Ü⑤Ý✑Ô✻Ù❙Ý❷Ú✖Ô❋ç➈Ý➃Õ✑Õrâ❭Ó ß❀Ó✖Ù❙Ó✖Ü⑤àáÝ➃Þ✡Ó♣ß❳â✈ã➮Ù⑤Ò✄Ó➝Ý➃Þ✘Ù⑤Ó✖Ü⑤Ý➃Ö✌Ü♣ê✤íúÓ✧Ú♣Û◗ÞðÔ✻Ó♣Ó➝Ù❙Ò✡Ó✖Ü⑤Ó✲Ø➍Ö✆Ü⑤Ó✴Ù❙Ò✡Û◗ÙóÝ✑ÞñÙ❙Ò✄Ó➧û✛üýÚ✖Û✌Ô✻ÓáçìÓ✧Ú✖Û✆Þ Ü⑤Ó♣ô✘é✄Ý✑Ü❙Ó✌ä✌Ù❊çìÖ✡ä❀Ö✌Þ✄Ó↔Ö✆Ü➈Þ✄Öáâ■Ö✆é✄Þ❭ß✄Û◗Ü⑤ã✭Ú✲Ö✌Þ✡ß❀Ý➃Ù❙Ý✑Ö✆Þ❬ê ò✄Ö✌Ü✤Þ✡Ö✆Þ✄Õ✑Ý➃Þ✡Ó♣Û◗Ü✶Ú✲Ö✌Þ✡Ô❙Ó✖Ü⑤æ☞Û◗Ù❙Ý✑Ö✆Þ→Õ❷Û☞ç➜Ô❡Û◗Þ✡ß✓Û◗Ü⑤â✄Ý❺Ù⑤Ü⑤Û✆Ü❙ã➾ß✄Û◗Ù⑤Û✄ä❨Ù❙Ò✄ÓñÚ➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù❙Ó♣Ü❙Ý❷Ô✻Ù❙Ý❷Ú✖Ô✧à➝Û☞ã Ú✲Ü⑤Ö✌Ô⑤Ô➎ç➈Ý➃Ù❙Ò✡Ý➃Þ✤ø✡Þ✄Ý➃Ù❙Ó❐Ù❙Ý✑àáÓ✆ê➎Ñ✹Ò✄Ý❷Ôìç➀Ö✌é✄Õ❷ß✭Ô❙é✄î✆î✌Ó♣Ô✻ÙìÛ➆à✯é✄Õ➃Ù❙Ý➃ï❸æ✐Û◗Õ✑é✄Ó✛ß✤Ô❙Ö✆Õ✑é❀Ù❙Ý✑Ö✆Þ✭ç➈Ò✄Ý✑Ú➉Ò❡ß❀Ö✈Ó♣Ô Þ✄Ö✆Ù✯à➝Û◗þ✌Ó✤Û◗Þ✈ãñÔ✻Ó♣Þ✡Ô❙Ó✭×✄Ò✈ã❀Ô✻Ý❷Ú✖Û✆Õ➃Õ✑ã✆êúíúÓ❡ç➈Ý✑Õ➃Õ➈Ô❙Ó✖Ó✧Ù⑤Ò✡Û✐Ù♦ÿ❊é✡Ô✻ÙáÛ◗Ù✯Ù⑤Ò✄Ó✭×■Ö✆Ý✑Þ✘Ù✴ç➈Ò✄Ó♣Ü❙Ó✭Ù❙Ò✄Ó Ú➉Ò✡Û✆Ü⑤Û✌Ú⑩Ù❙Ó♣Ü❙Ý❷Ô✻Ù❙Ý❷Ú✖Ô✎Ô✻Ù⑤Û✆Ü✻Ù➎Ú✖Ü❙Ö✘Ô❙Ô❙Ý➃Þ✡î✡ä✛Ù❙Ò✄Ó➈Ô❙Ö✆Õ✑é❀Ù❙Ý✑Ö✆Þ✯â❭Ó✛Ú✲Ö✌à✴Ó✛Ô✫ß❀Ý❷Ô❙Ú✖Ö✆Þ✘Ù❙Ý✑Þ✈é✄Ö✆é✡Ô♣ê✁ìÙ✞Ù⑤Ò✄Ý✑Ô✞×■Ö✆Ý✑Þ✘Ù♣ä Ù❙Ò✡Ó❳ß❀Ý✄✂rÓ✖Ü⑤Ó✖Þ✘Ù❙Ý❷Û◗Õ↔×✄Ü⑤Ý➃àáÝ➃Ù❙Ý✑æ✆Ó➮Ø➍Ö✆Ü⑤à Ö◗Ø✩Ù❙Ò✡Ó❳Ó✛ô✘é✡Û✐Ù⑤Ý➃Ö✌Þ❬ä➜Ö✆Þ✓ç➈Ò✄Ý❷Ú➉Ò çìÓúÛ◗Ü⑤Ó➮â❭Û✆Ô❙Ý➃Þ✄î❖Ö✆é✄Ü Ô❙Ö✆Õ✑é❀Ù❙Ý✑Ö✆Þ❡×✄Ü⑤Ö✈Ú✖Ó♣ß❀é✡Ü❙Ó❐Ý✑Ô➜Þ✄ÖáÕ➃Ö✌Þ✄î✆Ó♣Ü➈æ☞Û✆Õ➃Ý❷ß❫ê ➶
2.3 Examples 2.3.1 Linear Advection Equation SLIDE 13 p(a, t)=po(a-at) +at 2.3.2 Burgers' Equation SLIDE 14 Recall f(u)=2u2, so a(u) at ar 0 Solution: u(, t)=uo(a-ut) The solution is constant along the characteristic lines defined by a -ut=To We note that the above solution is defined implicitly(e.g. the definition of the function requires the function itself and therefore it is often not very useful We can verify however by direct differentiation that it is in fact a solation of the partial diffe Consider the initial data <0 0<x<1 16
☎✝✆✟✞ ✠☛✡✌☞✎✍✑✏✓✒✕✔✗✖ ✘✚✙✜✛✗✙✣✢ ✤✦✥✣✧✩★✫✪✭✬✯✮✱✰✁✲✳★✵✴✷✶✸✥✣✹✺✧✼✻✓✽✿✾✁✪❀✶❁✥✣✹✺✧ ❂✺❃✫❄❆❅✗❇❉❈✷❊ ❋✵●✷❍✜■❀❏▲❑✄●◆▼ ❖✺P✣◗✗❘❚❙❱❯❳❲❨❖✫❩✫P✣◗❭❬❫❪◆❙❱❯ ❴✦❵❜❛✷❝▲❛◆❞❡❏❚❢✸❝❚❑❤❣❱❏❚❑❤❞✐❍✜❑✄▼❜❢✸❣ ◗✯❲❥◗❩❧❦ ❪✫❙ ✘✚✙✜✛✗✙❤✘ ♠♥✾✗✬♣♦✺★✫✬rq◆st✻✉✽✿✾✩✪❀✶❁✥✣✹✺✧ ❂✺❃✫❄❆❅✗❇❉❈♣✈ ✇❢❁❞①❛②❍✜❍✗③ P⑤④✿❯⑥❲⑧⑦⑨ ④⑨②⑩ ❣❚● ❪✺P✣④❶❯⑥❲✼④ ❷ ④ ❷ ❙ ❦ ④ ❷ ④ ❷ ◗ ❲❥❸ ❋❀●✷❍✜■❀❏❚❑✜●✷▼❺❹ ④✩P✣◗✗❘❚❙❱❯❳❲❨④✺❩◆P✣◗❭❬❻④✺❙❱❯ ❼❵✭❢❽❣❚●✷❍✜■❀❏❚❑✜●✷▼☛❑✜❣✐❞❾●✷▼✳❣✕❏❿❛②▼✫❏➀❛②❍✜●✷▼✭➁➂❏❚❵✭❢➃❞❿❵✳❛②❝❿❛✷❞❡❏▲❢①❝▲❑✜❣❱❏❚❑❤❞✉❍✄❑✜▼✭❢✸❣✌➄✭❢❾➅❜▼✭❢❁➄☛➆✫➇ ◗❭❬❺④✳❙t❲✼◗✳❩✷➈ ➉➋➊♥➌✺➍♣➎➏➊♥➎⑤➐✭➑♣➎✌➎⑤➐✭➊➒➑◆➓❿➍♣➔♣➊♥→①➍♣➣↕↔❀➎➛➙✣➍♣➌❫➙❤→➝➜◆➊✣➞❳➌✺➊❿➜✯➙❆➟➀➠❶➣↕➙✣➡①➙❆➎✟➣↕➢❫➤✕➊❁➥➧➦◆➥➝➎⑤➐✭➊➒➜◆➊✣➞❳➌❜➙❆➎➛➙✣➍♣➌➨➍❱➩➫➎⑤➐✭➊ ➩❿↔❀➌✺➡①➎➛➙✣➍♣➌➯➭❚➊❿➲❾↔❀➙❆➭❚➊❾→❭➎⑤➐✭➊✓➩❿↔❀➌✺➡①➎✟➙✣➍②➌❉➙❆➎✟→①➊❾➣➩①➳❻➑②➌✺➜❻➎⑤➐✭➊①➭❚➊✣➩❾➍②➭❚➊☛➙❆➎➵➙❤→✯➍✕➩❿➎➏➊❾➌➯➌✺➍②➎➵➔r➊①➭❡➢➸↔✵→①➊✟➩❿↔❀➣✄➥ ➉➋➊✯➡❿➑♣➌❉➔r➊①➭❡➙➩❿➢☛➐❜➍♣➺❳➊①➔r➊①➭☛➓①➢➸➜♣➙❆➭❚➊❿➡①➎➵➜②➙➻✐➊①➭❚➊❾➌✳➎✟➙✣➑②➎✟➙✣➍②➌❉➎⑤➐✭➑②➎✉➙❆➎✓➙❤→➒➙❆➌✯➩❾➑◆➡❾➎➃➑➼→①➍②➣↕↔❀➎✟➙✣➍②➌➯➍❱➩ ➎⑤➐✭➊✌➠✭➑②➭❡➎✟➙✣➑②➣✩➜♣➙➻✉➊❾➭❚➊①➌❜➎➛➙✣➑♣➣✎➊❿➲❾↔❜➑②➎✟➙✣➍②➌✭➽⑥➙✣➥❆➊❁➥✄➾ ④✺➚➫❲✼④✺➪❩ ❬❺④❶➪❩ ④✺➚✷❙ ➶ ④✺➚➃❲ ④❩➪ ➹ ❦ ④❩➪ ❙ ④✺➘✎❲➴❬✌④❩➪ ④❭❬❺④❩➪ ④✺➘➛❙ ➶ ④✺➘✎❲ ❬✌④❩➪ ④ ➹ ❦ ④❩➪ ❙❀➷ ❂✺❃✫❄❆❅✗❇❉❈✷➬ ❴✦●✷▼✳❣❱❑❤➄❀❢①❝❧❏▲❵✭❢❽❑✄▼❜❑➮❏▲❑✜❛✷❍✗➄✭❛♣❏❿❛ ④✎P⑤◗✗❘▲❸◆❯⑥❲ ➱✃❐ ➹ ◗➋❒❨❸ ➹ ❬❻◗ ❸➝❮➯◗➋❮ ➹ ❸ ◗➋❰ ➹ ❂✺❃✫❄❆❅✗❇❉❈✷Ï Ð
dt SLIDE 17 For t1 0→x=x0→ For t1
Ñ✺Ò✫Ó❆Ô✗Õ❉Ö✫× Ø✭Ù◆ÚÜÛ➫ÝßÞ Ø✭Ù◆Ú✌à➋á❉â✉ã äà ä â➋åçæ✑è à å â✿éêà❜ë è ì✎íà✁î❱â❱ï åðæ Ø✭Ù◆Ú✌â✦ñ❉àòñ æ ã äà ä â➋åçæ✝ó à❜ë è à åôí✕æ✌ó à❜ë♣ï➏â✗éêà❜ë ì✩íà✗î❚â❱ï åçæ✝ó à❜ë å æ✌ó à æ✌ó â Ø✭Ù◆Ú✌à➋õ æ ã äà ä â å❥ö÷è à å àë è ì✎íà✁î❱â❱ï åøö ù✁ú♣û✓â✉ñüàêñ æ✷ý❧þ❳ÿ✁û✄✂✆☎✝✂①ú✟✞✡✠ ÿ☞☛ú♣û✌☎✎✍ ÿ✑✏✍✓✒②û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✗✕ ✏ ✞✡✕✙✘ ÿ ✂✛✚✢✜✣✘✤☎✙✍✥✕✖✂ ✏✒✦✂ ÿ ☎✎✍ ÿ✛✧ ✒♣ûÿ ★ ÿ✗✘ ÿ★✪✩ ✧✬✫✄✭✫✯✮ å❥ì ✚✱✰✲✕✙✘✏❿ÿ✉ì ✕✖✂ ✏ú✳✘✓✂✆☎✴✒✳✘✵☎☞✒✟✞✄ú✳✘✷✶ ÿ✒ ✏✍ ✏✍✵✒♣û✔✒ ✏ ☎ÿ û✆✕✖✂✄☎✸✕ ✏❾ý❧þ❳ÿ✺✹✘✺úþ ☎✙✍✵✒✟☎ ú✟✘ ÿ✒ ✏✍✪✞✡✕✙✘ ÿ✻✫✄✭✫✯✮ å❥ìë✷✚✺✼✝✘✏❿ÿ➵þ❳ÿ ✍✓✒✟✠ ÿ ★ ÿ ☎ ÿ û✆✽✾✕✙✘ ÿ★ ☎✎✍ ÿ✢✏✍✵✒♣û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✸✕ ✏ ✞✿✕✙✘ÿ ✂ ýtþ❳ÿ✱❀✂ ÿ ☎✎✍ ÿ❁☛✒ ✏ ☎❁☎✎✍✓✒✳☎ ì ✕✖✂ ✏ú✳✘✓✂✆☎✴✒✳✘✵☎✝✒✟✞✄ú✳✘✷✶ ÿ✒ ✏✍❂✞✡✕✙✘ ÿ ☎➏ú ★ ÿ ☎ÿ û✆✽❃✕✙✘ÿ ☎✎✍ ÿ ú✟✠ ÿ û✯✒✟✞✙✞❄✂①ú✳✞❀ ☎✗✕✣ú✳✘✲✚ Ñ✺Ò✫Ó❆Ô✗Õ❉Ö❆❅ Ø✭Ù◆ÚÜÛ❈❇ Þ ì✩íà✗î❱â❱ï å ❉ æ à➋ñ æ ö à❋❊ æ ● ☎✩â å æ ☎✎✍ ÿ ✂①ú✟✞❀ ☎✸✕✣ú✟✘ ★ ÿ ✠ ÿ ✞✜ú✯❍■✂✑✒ ★✕✖✂ ✏ú✳✘✵☎✸✕✙✘❀ ✕✙☎✧ ✚❑❏▲✍✥✕✖✂ ✏ú♣û❡ûÿ ✂✸❍✭ú✳✘★✂✢☎ú✻☎✎✍ ÿ ☎✗✕✙✽ÿ ✒✟☎ þ ✍▼✕ ✏✍◆☎✙✍ ÿ✾✏✍✓✒②û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✗✕ ✏ ✂ û✄✂✄☎ ✏ û❚ú✟✂✄✂✣✚ ❖✝P✓◗❙❘✭Ú▲Ù✥❚✣◗❱❯✓❲✭Ú✔◗❈❳✭Ú✯◗❱❨❆❩▼❬✝❯❀Ù✟❭☞❪❑❫⑤Ù◆Ú✌â❁❊ æ ❴
3 Discontinuous solution 3.1 Shock formation 19 SHOCK PA When the characteristics cross, the function u(a, t) has an infinite slope. A discontinuity or shock forms, and the differential equation is no longer valid Note 7 Vanishing viscosity approach After the characteristics have crossed, there are some points a where more than one characteristic leads back to t=0. This would imply that the solution is multi-valued at such a point, which in most cases is not physically realisable The correct physical behaviour can be determined by recalling that the inviscid Burgers'equation was a simplified version of a viscous equation with a term ea on the right hand side. If the initial data is smooth and e is very small, then this term is negligible compared to the lefthand-side terms, and the solution of the almost identical to that of the inviscid equation. However, as the discontinuity begins to form, amf becomes very large, and the viscous term becomes important. This term keeps the solution smooth for all time(recall the equation is now parabolic), and determines the correct physical nature of th hown in the figure below This behaviour is evident in the equations governing fluid flow. The Euler equations, which ignore the viscous terms, are hyperbolic and admit discontin lous solutions. Conversely, the Navier-Stokes equations are parabolic, and the viscosity ensures that the solution is always smooth D Exercise 1 (from [LvI) Show that the viscous Burgers'equation has a trav elling wave solution of the form u(a, t)=(as tt) by deriving an ODE for and verifying that this OdE has solutions of the form f (uh s ui )[1 s tanh((uh s ui )y/4e1
❵ ❛❝❜✣❞❢❡☞❣✾❤❃✐❥❜✛❤❧❦❂❣❃❦❂❞♥♠✻❣✾♦✛❦✪✐❥❜✣❣❃❤ ♣✝qsr t✝✉✇✈☞①✓②④③✁✈⑥⑤✵⑦⑨⑧❢⑩✥❶s✈❸❷ ❹✲❺✷❻✙❼❾❽➀❿❆➁ SHOCK PATH ➂➄➃✓➅❱➆➀➇✯➃✓➅➉➈✄➃✵➊❆➋✯➊➌➈✆➇✯➅✛➋✯➍➏➎s➇✔➍✖➈✛➎❃➈✣➋✯➐➌➎✯➎✛➑❸➇✔➃✓➅◆➒✎➓✓➆✵➈✆➇✯➍➔➐➌➆➣→❢↔➙↕❾➛s➜s➝❃➃■➊❆➎❧➊❆➆➀➍➏➆✥➞✵➆✓➍➔➇✔➅✬➎s➟➏➐❆➠✲➅❆➡➤➢ ➥➍✖➎✔➈✛➐❆➆✷➇✔➍➏➆▼➓✓➍➔➇➧➦◆➐❆➋❙➨➫➩➯➭❄➲➫➳➀➒✎➐❆➋✯➵✑➎✛➑❾➊❆➆➥ ➇✔➃✓➅ ➥➍➔➸▲➅✛➋✯➅✛➆✷➇✔➍✖➊✳➟❥➅➫➺➌➓■➊✟➇✔➍➏➐❆➆➻➍✖➎❙➆✓➐❂➟➔➐➌➆✓➼❆➅❱➋✱➽✦➊❆➟➔➍➥ ➡ ➾❈➚ ➪✇➶ ➹➻➘➷➴✆➬✤➮ ➱❸✃✥❐❢❒➙❮➫❰❢❒➙❐➯ÏÐ➱✱❒➙❮❆Ñ❆➘✥❮❱❒✔➴➙ÒÔÓ✌Õ❢Õ❥Ö➌➘✓✃✲Ñ✣❰ ➢☞➒✙➇✯➅✛➋❁➇✔➃✓➅×➈✄➃✵➊✳➋✄➊❆➈✣➇✔➅✛➋✯➍✖➎➧➇✯➍➏➈❱➎Ø➃✵➊✦➽❆➅✇➈✛➋✔➐✷➎✔➎✔➅➥ ➑✟➇✔➃✓➅❱➋✔➅×➊✳➋✯➅✇➎s➐➌➵✌➅☞➠■➐➌➍➔➆✷➇✯➎❁↕✻Ù☞➃✵➅✛➋✯➅☞➵❃➐➌➋✔➅✁➇✔➃✵➊❆➆ ➐❆➆✵➅❧➈✄➃✵➊✳➋✄➊❆➈✣➇✔➅✛➋✯➍✖➎➧➇✯➍➏➈✾➟➔➅➫➊➥➎✺Ú✵➊❆➈✄Û❋➇✯➐❋➜❙ÜÞÝ✓➡✻ß✝➃✵➍➏➎✢Ù❁➐➌➓✓➟➥ ➍➏➵✌➠✓➟➔➦➉➇✔➃✵➊✳➇✺➇✔➃✵➅❧➎s➐➌➟➔➓✥➇✯➍➔➐➌➆à➍➏➎ ➵✢➓✵➟✿➇✯➍✿á✴➽✟➊✳➟➏➓✓➅➥ ➊✟➇✺➎✔➓✵➈✄➃✤➊❑➠■➐➌➍➔➆✷➇❱➑➯Ù☞➃✓➍✖➈✄➃➻➍➏➆à➵✌➐➌➎s➇✢➈✛➊❆➎✔➅❱➎❈➍✖➎❙➆✓➐❆➇✺➠✓➃▼➦▼➎✔➍✖➈✛➊✳➟➏➟➏➦◆➋✔➅➫➊✳➟➏➍➏➎✯➊✳Ú✵➟➔➅➌➡ ß✝➃✓➅❈➈✣➐❆➋✯➋✯➅❱➈✆➇⑥➠✵➃✷➦✥➎✔➍➏➈❱➊✳➟■Ú■➅❱➃✵➊✦➽▼➍➔➐➌➓✓➋❁➈❱➊✳➆✑Ú✲➅ ➥➅✛➇✔➅❱➋✔➵✌➍➏➆✓➅➥ Ú▼➦❃➋✔➅➫➈✛➊❆➟➔➟➏➍➔➆✵➼❙➇✔➃✵➊✳➇❸➇✯➃✓➅✇➍➏➆▼➽✷➍✖➎✯➈✣➍➥ âã➓✓➋✯➼❆➅✛➋✄➎❱ä✣➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆❙Ù✝➊❆➎❾➊×➎s➍➏➵✌➠✓➟➏➍✿➞✵➅➥ ➽❆➅❱➋✯➎✔➍➏➐❆➆✱➐✳➒✵➊✇➽✷➍✖➎✯➈✣➐❆➓■➎❄➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆❙Ù☞➍➔➇✔➃✢➊✁➇✔➅❱➋✔➵æå✥ç➫èsé ç➫ê è ➐❆➆❃➇✯➃✓➅✁➋✯➍➔➼➌➃➌➇✯➃✵➊✳➆➥ ➎✔➍➥➅❆➡Øë✴➒▲➇✔➃✓➅✇➍➔➆✓➍➔➇✔➍✖➊✳➟ ➥➊✟➇✄➊❙➍✖➎⑥➎s➵✌➐▼➐✳➇✯➃❧➊✳➆➥ å❁➍➏➎❸➽❆➅❱➋✔➦✢➎✔➵✑➊✳➟➏➟✗➑❆➇✔➃✓➅❱➆✑➇✔➃✵➍➏➎ ➇✔➅❱➋✔➵ì➍✖➎☞➆✓➅✛➼➌➟➔➍➏➼❆➍➏Ú✓➟➏➅✢➈✣➐➌➵✌➠✵➊✳➋✯➅➥ ➇✔➐✑➇✯➃✓➅✺➟➏➅✣➒✙➇✯➃✵➊✳➆➥á✴➎✔➍➥➅❙➇✔➅❱➋✔➵✑➎❱➑■➊✳➆➥ ➇✔➃✓➅✢➎✔➐❆➟➏➓✥➇✯➍➔➐➌➆❋➐✳➒Ø➇✔➃✓➅ ➽▼➍➏➎✯➈✣➐➌➓✵➎➯➅❱➺✷➓✵➊✟➇✯➍➔➐➌➆✢➍✖➎❥➊✳➟➏➵✌➐➌➎s➇❢➍➥➅✛➆✷➇✯➍➏➈❱➊✳➟✷➇✔➐×➇✔➃✵➊✳➇❥➐✳➒✵➇✯➃✓➅ã➍➔➆▼➽▼➍➏➎✯➈✣➍➥ ➅❱➺✷➓✵➊✟➇✯➍➔➐➌➆❄➡➯í✇➐✟Ù❁➅❱➽❆➅❱➋❱➑✟➊❆➎ ➇✔➃✵➅ ➥➍➏➎✯➈✣➐➌➆➌➇✯➍➔➆▼➓✓➍➔➇➧➦✑Ú✲➅✛➼❆➍➏➆✵➎ã➇✯➐✾➒✎➐➌➋✔➵❂➑ ç➫èsé ç➫ê è Ú■➅➫➈✣➐➌➵❃➅➫➎ã➽❆➅✛➋✯➦✑➟✖➊✳➋✯➼❆➅❆➑✥➊❆➆➥ ➇✯➃✓➅✱➽▼➍➏➎✯➈✣➐➌➓✵➎❁➇✔➅✛➋✯➵ Ú✲➅❱➈✣➐➌➵✌➅❱➎Ø➍➏➵❃➠✲➐❆➋✔➇✯➊❆➆✷➇❱➡Øß✝➃✓➍➏➎❥➇✔➅✛➋✯➵îÛ➌➅✛➅❱➠✵➎Ø➇✔➃✵➅✁➎s➐➌➟➔➓✥➇✯➍➔➐➌➆✑➎s➵✌➐▼➐✳➇✯➃✾➒✎➐➌➋⑥➊✳➟➏➟✥➇✔➍➏➵✌➅✺↔✎➋✯➅❱➈❱➊✳➟➏➟✥➇✔➃✓➅ ➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆➻➍➏➎✱➆✵➐✟Ùï➠✵➊❆➋✯➊❆Ú■➐➌➟➔➍✖➈❱➝✣➑❾➊✳➆➥✬➥➅✛➇✔➅❱➋✔➵✌➍➏➆✓➅❱➎✱➇✯➃✓➅❧➈✣➐➌➋✔➋✯➅❱➈✣➇❈➠✵➃✷➦✥➎✔➍➏➈❱➊✳➟❥➆✵➊✳➇✔➓✓➋✯➅✌➐✳➒ã➇✔➃✓➅ ➎✔➦▼➎s➇✔➅❱➵ð➊❆➎☞➎✔➃✓➐✟Ù☞➆❑➍➔➆❑➇✯➃✓➅✱➞✵➼❆➓✵➋✔➅✱Ú✲➅✛➟➏➐✟Ù❙➡ limiting solution ε -> 0 ß✝➃✓➍✖➎✻Ú✲➅✛➃■➊✦➽✷➍➏➐❆➓✵➋✻➍✖➎✪➅❱➽▼➍➥➅❱➆✷➇✪➍➏➆ñ➇✔➃✵➅✬➅➫➺➌➓■➊✟➇✔➍➏➐❆➆■➎❧➼❆➐✟➽➌➅✛➋✯➆✓➍➏➆✓➼àò✵➓✓➍➥ ò✵➐✟Ù❙➡óß✝➃✓➅à➪⑥➓✓➟➏➅✛➋ ➅❱➺✷➓✵➊✳➇✔➍➏➐❆➆✵➎❱➑✥Ù☞➃✓➍✖➈✄➃✪➍➏➼❆➆✵➐❆➋✯➅×➇✯➃✓➅❙➽▼➍➏➎✯➈✣➐➌➓✵➎ã➇✔➅❱➋✔➵✑➎❱➑✓➊✳➋✯➅✱➃✷➦▼➠✲➅✛➋✯Ú■➐➌➟➔➍✖➈✱➊✳➆➥ ➊➥➵✌➍✿➇ ➥➍✖➎✯➈✣➐❆➆✷➇✯➍➔➆✥á ➓✓➐➌➓✵➎✁➎✔➐❆➟➏➓✥➇✔➍➏➐❆➆✵➎❱➡✁ô❁➐❆➆▼➽❆➅❱➋✯➎✔➅✛➟➏➦❆➑▼➇✯➃✓➅✾➾×➊✦➽✷➍➏➅✛➋✔á➧õ✷➇✔➐➌Û❆➅❱➎ã➅➫➺➌➓■➊✟➇✔➍➏➐❆➆■➎✁➊✳➋✯➅❙➠✵➊✳➋✄➊✳Ú✲➐❆➟➏➍✖➈✳➑✵➊❆➆➥ ➇✔➃✓➅ ➽▼➍➏➎✯➈✣➐✷➎s➍➔➇➧➦✑➅✛➆■➎s➓✓➋✯➅❱➎ã➇✯➃✵➊✟➇☞➇✯➃✓➅✺➎✔➐❆➟➏➓✥➇✔➍➏➐❆➆❑➍➏➎☞➊❆➟➔Ù✝➊✦➦✥➎✝➎s➵✌➐▼➐✳➇✯➃❄➡ ö❂÷✇ø➬✳Ö➌Ñ➫❒✗❮✳➬➻ù◆↔✙➒✎➋✯➐❆➵ûúü➌ý✁þ✎➝☞õ▼➃✓➐✟ÙÐ➇✔➃✵➊✳➇☞➇✔➃✓➅✢➽✷➍✖➎✯➈✣➐❆➓■➎☞â❁➓✵➋✔➼➌➅✛➋✄➎✛ä✥➅➫➺➌➓■➊✟➇✔➍➏➐❆➆❑➃✵➊➌➎✁➊✾➇✔➋✄➊✦➽➌á ➅✛➟➏➟➏➍➔➆✓➼✌Ù✝➊✦➽❆➅✱➎s➐➌➟➔➓✓➇✔➍➏➐❆➆❑➐✳➒❾➇✯➃✓➅✱➒✎➐❆➋✯➵ð→➷ÿ✆↔➙↕❾➛s➜s➝❁Ü✁✢↔➙↕✄✂✆☎✛➜s➝ãÚ▼➦ ➥➅✛➋✯➍➔➽▼➍➏➆✓➼✑➊✳➆✞✝✠✟×➪➀➒✎➐❆➋✡ ➊✳➆➥ ➽➌➅✛➋✯➍✿➒✎➦▼➍➏➆✓➼❃➇✔➃✵➊✳➇☞➇✔➃✓➍✖➎☛✝✠✟×➪➣➃■➊❆➎☞➎✔➐❆➟➏➓✥➇✔➍➏➐❆➆✵➎✝➐❆➒❾➇✯➃✓➅✱➒✎➐❆➋✯➵ ✢↔✌☞✓➝❸Ü➄→✎✍✑✏ ➶ ✒ ↔✎→✓✍✔✂➻→✓✕▲➝✛ú✿➶☛✂➉➇✄➊✳➆✓➃❾↔✔↔✎→✓✍✔✂➻→✓✕▲➝✖☞✘✗✚✙➌å✆➝ þ ✛
