Finite difference discretization of Hyperbolic equations Linear problems Lectures 8.9 and 10
✂✁☎✄✆✁✞✝✠✟☛✡☞✁✍✌✎✟✑✏✒✟✑✄✔✓✑✟✕✡✖✁☎✗✘✓✑✏✒✟✙✝✚✁☎✛✢✜✣✝✚✁✥✤✦✄✧✤✩★ ✪✬✫✮✭✯✟✰✏✲✱✯✤✴✳☎✁✥✓☛✵✷✶✹✸✎✜✣✝✚✁✥✤✴✄✆✗✢✺ ✻✼✁☎✄✔✟✣✜✽✏✿✾❀✏✒✤✦✱❁✳✥✟✑❂❃✗ ✻❄✟✑✓✙✝✚✸✔✏✒✟✑✗✖❅❁❆✮❇❈✜✽✄❁❉ ❊✰❋
1 First Order ave Equation SLIDE 1 The simplest first order partial differential equation in two variables(a, t)is the linear wave equation. Recall that all first order PDE's are of hyperbolic type INITIAL BOUNDARY VALUE PROBLEM (IBVP) 0,x∈(0,1) U is the wave speed which, for simplicity, we as sume to be constant Unlike the parabolic case, which involves second order spatial derivatives, the hyperbolic case only has a first order spatial derivative. We can intuition e the pect that the hyperbolic equation will require less boundary conditions than the parabolic case. Appropriate initial and boundary conditions for the above prob Initial condition Boundary condition u(0, t)=go(t) if U> a(1, t=gi() if U<O We note that the boundary conditions are specified always on u, not its deriva- tive, and that the side on which the boundary condition must be specified depen on the sign ofU. The reasons for this will become apparent when we look at the form of the solution bele 1.1 Solution SLIDE 2 Let u(a, t), be the solution to the above equation. Assuming that u is differen tiable we can write dt U→|x=Ut+ Fi ↓ d=0→【(a21=()=( In other words, if we restrict the variations of a and t, to be on a characteristic line, then u must be a constant. We note that this constant can be different for different characteristics, i. e. different hence u(a, t)=(S). Alternatively, we ify that( Dt) particular funct n will be determined by initial and boundary conditios. For
● ❍❏■▲❑◆▼✒❖◗P☞❑◆❘❚❙✣❑✿❯❱❳❲❨❙✕❩✷❬✮❭✔❱✽❖◆■✥❪✹❫ ❴ ❵❜❛❞❝❢❡❤❣ ✐❦❥♠❧♦♥☎♣❞q✢rts✉❧✥♥☎✈①✇✙②✍♥☎✈✙③④②⑥⑤⑦❧✥②✙r♠⑧⑨②☎✈⑩♣❶⑧⑨s✒⑤④♣❷✢❧▲②⑥❧▲❸①✈❹♣❶⑧④s✒❧✍❺✥❻①⑧⑨✈⑩♣❶③⑨❸❁♣❞❸✔✈❹❼❽③✴❾④⑧④②☎♣❶⑧⑦❿✥s➀❧☎♥✦➁➃➂✚➄➆➅➆➇✽♣➈♥✢✈➃❥♠❧ s➉♣❞❸❦❧✞⑧⑨②✦❼❽⑧④❾④❧➊❧✞❺▲❻①⑧④✈❹♣❶③④❸❦➋➍➌✢❧✍➎✍⑧④s❞s✒✈➃❥♠⑧⑨✈➍⑧⑨s❞s➏✇✙②✍♥☎✈✰③⑨②⑥⑤➏❧▲②✽➐◆➑✩➒➊➓♥➔⑧④②✞❧❨③➆→♦❥↔➣✞r①❧✥②▲❿✍③⑨s➉♣❶➎✹✈⑩➣✞r①❧↕➋ ➙➜➛✣➙➜➝➍➙➜➞♦➟❄➠❳➡✽➢♦➛✢➤✣➞♦➥✚➦➨➧✘➞♦➟❢➢✢➩➭➫✑➥✣➡✽➠✰➟✚➩❽➯✕➁➃➙➜➠✰➧♦➫✰➇ ➲t➳ ➲➅➸➵❤➺ ➲➻➳ ➲➂❤➼❀➽ ➄➾➂❚➚❄➁ ➽ ➄▲➪➶➇ ➺ ♣➈♥✦✈❞❥①❧✴❼❽⑧⑨❾➶❧➔♥❹r①❧✞❧✍⑤➹❼✚❥↔♣❶➎✍❥❜➘t→✥③⑨②✩♥☎♣❞q✣r❦s➉♣❶➎✥♣❞✈❹➣➶➘➍❼❽❧❨⑧④♥✞♥☎❻➴q➸❧✦✈➷③✮❿✍❧✹➎✞③⑨❸♠♥☎✈➷⑧④❸①✈➷➋ ➬✚❸➻s➉♣✉➮⑦❧❁✈❞❥①❧❨r♠⑧⑨②⑥⑧➏❿✍③⑨s➉♣❶➎✎➎✍⑧➶♥▲❧✥➘✴❼✚❥↔♣❶➎✍❥➱♣❞❸①❾④③④s✃❾➶❧✥♥✮♥▲❧✍➎✍③④❸t⑤❄③⑨②⑥⑤➏❧▲②✆♥⑩r①⑧④✈❹♣❶⑧④s✣⑤⑦❧✥②☎♣❞❾④⑧④✈❹♣❞❾➶❧✥♥✍➘❨✈➃❥♠❧ ❥↔➣✞r①❧▲②▲❿✞③⑨s➉♣❶➎✮➎✍⑧④♥✥❧✮③④❸①s✃➣➊❥♠⑧④♥➊⑧✩✇❽②✍♥✍✈✣③⑨②⑥⑤➏❧▲②➔♥❹r①⑧④✈❹♣❶⑧④s✙⑤➏❧▲②☎♣❞❾④⑧④✈❹♣❞❾➶❧↕➋❒❐❚❧➸➎✍⑧④❸❮♣❞❸①✈❹❻➴♣❞✈❹♣❞❾➶❧▲s➉➣❚❧⑥❰⑦Ï r①❧✞➎▲✈✣✈➃❥♠⑧⑨✈✣✈❞❥①❧➔❥↔➣✞r①❧▲②▲❿✞③⑨s➉♣❶➎❁❧✞❺▲❻①⑧④✈❹♣❶③④❸❄❼◆♣❞s❞s❽②⑥❧✍❺✥❻➴♣❞②⑥❧➸s✉❧☎♥✍♥➸❿✞③⑨❻➴❸t⑤⑦⑧④②☎➣Ð➎✍③④❸t⑤⑨♣❞✈⑩♣❶③⑨❸♠♥➊✈❞❥①⑧④❸❄✈➃❥♠❧ r①⑧④②⑥⑧⑦❿✞③⑨s➉♣❶➎➸➎✍⑧➶♥▲❧↕➋✢Ñ✑r⑦rt②⑥③✍rt②☎♣❶⑧④✈➷❧➊♣❞❸➻♣❞✈⑩♣❶⑧⑨s✘⑧④❸❦⑤✆❿✍③⑨❻➴❸t⑤➏⑧⑨②☎➣❁➎✍③④❸❦⑤④♣❞✈❹♣❶③④❸①♥❽→✥③④②✴✈➃❥♠❧➊⑧➏❿✍③⑨❾➶❧✣rt②✞③➏❿▲Ï s✉❧▲qÒ⑧⑨②⑥❧➔✈❞❥①❧❽→✥③⑨s❞s✉③④❼◆♣❞❸↔Ó⑦Ô ➙➷Õ♠Ö✉×⑥Ö➈Ø⑨Ù❢Ú▲Û➏Õ①Ü➴Ö✉×⑥Ö➀Û➏Õ❢Ý ➳ ➁❶➂❢➄ ➽ ➇ ➼ ➳➻Þ ➁❶➂t➇ ➠✰Û➏ß♠Õ①Ü①Ø⑨à✞á➹Ú✥Û➏Õ➻Ü➴Ö✃×✞Ö✉Û⑦Õ①âãÝ❈ä ➳ ➁ ➽ ➄➆➅➆➇ ➼➭åÞ ➁❶➅➆➇ Ö✉æ ➺èç➱➽ ➳ ➁➆➪➏➄➆➅➆➇ ➼➭å❜é ➁❶➅➆➇ Ö✉æ ➺èê➱➽ ❐❚❧✴❸❦③④✈➷❧✦✈➃❥♠⑧④✈✰✈➃❥♠❧✹❿✞③⑨❻➴❸t⑤⑦⑧④②☎➣✆➎✍③④❸❦⑤④♣❞✈❹♣❶③④❸①♥❨⑧④②⑥❧✴♥❹r♠❧✍➎▲♣✇✑❧✍⑤✆⑧⑨s➉❼❽⑧④➣➶♥❨③⑨❸ ➳ ➘✙❸t③⑨✈✑♣❞✈⑩♥➔⑤➏❧▲②☎♣❞❾➶⑧⑨Ï ✈❹♣❞❾➶❧✥➘✚⑧④❸❦⑤✦✈➃❥♠⑧④✈❢✈❞❥①❧➍♥☎♣❶⑤➏❧✽③④❸➹❼✚❥➴♣❶➎✞❥➸✈❞❥①❧✢❿✍③④❻➴❸❦⑤➏⑧⑨②☎➣✦➎✍③⑨❸t⑤④♣❞✈❹♣❶③④❸✆q❨❻↔♥☎✈✚❿✍❧✰♥❹r①❧✍➎✥♣✇✰❧✞⑤❨⑤➏❧➷r♠❧▲❸t⑤④♥ ③④❸✔✈➃❥♠❧✽♥☎♣✃Ó➏❸Ð③➜→ ➺ ➋✹✐❦❥①❧✩②⑥❧✍⑧➶♥▲③④❸①♥✘→✥③⑨②♦✈➃❥↔♣➈♥✽❼◆♣❞s❞s✲❿✍❧✍➎✍③④q➸❧✴⑧✞r⑦r♠⑧⑨②⑥❧▲❸①✈✙❼✚❥①❧✥❸❚❼❽❧✽s✉③↕③ã➮➹⑧④✈✙✈➃❥♠❧ →✥③④②☎që③➜→✴✈❞❥①❧✦♥▲③④s✃❻➴✈⑩♣❶③⑨❸✼❿✍❧✥s➀③④❼❽➋ ì◆í➆ì î✰ï✑ð➆ñ❳ò♠ó➆ï❽ô ❴ ❵❜❛❞❝❢❡✼õ ö✚❧✥✈ ➳ ➁➃➂✚➄➆➅➆➇✍➘✣❿✍❧✹✈❞❥①❧✹♥▲③④s➉❻➴✈❹♣❶③④❸❮✈➷③❁✈❞❥①❧➸⑧➏❿✍③④❾④❧➹❧✍❺▲❻①⑧④✈❹♣❶③④❸t➋✹Ñ✽♥✍♥☎❻➴q❨♣❞❸↔Ó✆✈❞❥①⑧④✈ ➳ ♣➈♥➊⑤⑨♣❷✢❧▲②⑥❧✥❸➻Ï ✈❹♣❶⑧➏❿▲s✉❧❨❼❽❧❨➎✍⑧④❸✎❼◆②☎♣❞✈÷❧↕Ô ø➳ ➼ ➲➻➳ ➲➅ ø➅ ➵ ➲t➳ ➲➂ ø➂ ➼úù ➲t➳ ➲➅ ➵ ø➂ ø➅ ➲➻➳ ➲➂✑û ø➅ ü▲ý ø➂ ø➅ ➼✷➺ þ ➂ ➼✷➺➅ ➵❒ÿ ✂✁Ø➏à✞Ø⑦Ú☎×☎✄▲à✞Ö➀â➆×⑥Ö➈Ú▲â ✆ ø➳ ➼❏➽✞✝◗þ ➳ ➁➃➂❢➄⑥➅➆➇ ➼✠✟ ➁ ÿ ➇ ➼✠✟ ➁❶➂☛✡ ➺ ➅➆➇ ☞✄ãÕ✌✄▲à✍Ø⑨Ù✒â⑥Û➏Ù➀ß➴×✞Ö✉Û⑦Õ ✍☎❸Ð③⑨✈❞❥①❧✥②✽❼❽③④②⑥⑤④♥✍➘✠♣→♦❼❽❧✩②⑥❧✥♥☎✈⑩②☎♣❶➎▲✈◆✈➃❥♠❧✩❾④⑧④②☎♣❶⑧⑨✈⑩♣❶③⑨❸♠♥✴③➜→✑➂✂⑧④❸t⑤✽➅✍➘✠✈➷③✹❿✍❧✴③⑨❸Ð⑧➸➎✞❥①⑧④②⑥⑧⑦➎✥✈➷❧✥②☎♣➈♥☎✈❹♣❶➎ s➉♣❞❸❦❧☎➘✙✈➃❥♠❧▲❸ ➳ q✹❻↔♥☎✈✑❿✍❧❨⑧➹➎✍③⑨❸♠♥☎✈÷⑧⑨❸①✈➷➋➹❐❚❧✦❸❦③④✈➷❧✦✈➃❥♠⑧⑨✈❽✈➃❥↔♣➈♥✴➎✞③⑨❸♠♥☎✈➷⑧④❸①✈➍➎✍⑧④❸✯❿✍❧➔⑤⑨♣❷✢❧▲②⑥❧✥❸➻✈➻→✥③⑨② ⑤④♣❷♦❧✥②⑥❧▲❸①✈❽➎✍❥♠⑧⑨②⑥⑧➏➎▲✈➷❧✥②☎♣➈♥☎✈❹♣❶➎☎♥✍➘❽♣❶➋❞❧↕➋✩⑤④♣❷✢❧▲②⑥❧▲❸①✈ ÿ✏✎ ❥♠❧▲❸t➎✍❧ ➳ ➁➃➂✚➄➆➅➆➇ ➼✠✟ ➁ÿ ➇▲➋❳Ñ♦s➉✈➷❧▲②☎❸t⑧④✈❹♣❞❾④❧✥s➉➣➶➘◆❼❽❧ ➎✍⑧④❸➱❾➶❧▲②☎♣→✍➣✎✈➃❥♠⑧⑨✈ ✟ ➁➃➂✑✡ ➺ ➅➆➇✍➘✩♣➈♥❁⑧Ð♥✥③⑨s➉❻➴✈❹♣❶③④❸ ✈÷③✼③④❻➴②❁❧✞❺▲❻①⑧④✈❹♣❶③④❸✔→✥③④②✔⑧④②▲❿▲♣❞✈⑩②✞⑧④②☎➣ ✟ ➋ ✐❦❥♠❧ r①⑧④②☎✈❹♣❶➎✥❻➴s✉⑧⑨②✚→✍❻➴❸❦➎✥✈❹♣❶③④❸ ✟ ❼◆♣❞s❞s✒❿✍❧✦⑤➏❧▲✈÷❧▲②☎q❨♣❞❸❦❧✍⑤✹❿▲➣➔♣❞❸➻♣❞✈⑩♣❶⑧⑨s❦⑧⑨❸t⑤➊❿✍③④❻➴❸❦⑤➏⑧④②☎➣➊➎✍③④❸t⑤⑨♣❞✈⑩♣❶③⑨❸♠♥▲➋✓✒❢③⑨② ➪
example u(c, t)=(c-Ut), u(a, t)= sin(r-Ut), or u(a, t)=e-dt are solutions of the linear wave equation 1.11U>0 SLIdE 3 (x,t) ∫u°(x-U),ifr-Ut>0 go(t-a/0), if -Ut<0 1.1.2U<0 Ut<1 (t-r/0), if a-Ut> We note that the regularity of the solution is determined by the initial and bound ary data. For the moment we will assume that the solution u(, t is smooth The non-smooth case, including the dis continuous case will be considered in the 1.2 Stability SLIDE 5 In the remainder of this course we will only be considering p-norms. In order to simplify the notation lllp will denote the p-norm of a function(usually defined over [0,1] and llzllp will denote the p-norm of a vector
✔☎✕✗✖✙✘✛✚✢✜✣✔✥✤✧✦✩★✫✪✭✬✯✮✑✰✱✦✩★✳✲✵✴✶✬✯✮✸✷✙✹✺✤✻✦✼★✽✪✯✬✯✮✑✰✿✾✯❀❂❁✫✦✼★❃✲❄✴✺✬✯✮❅✹❇❆❉❈☛✤✻✦✼★✽✪✯✬✯✮✑✰✿❊●❋■❍❑❏✢▲▼✖❉❈✭✔ ◆❖❆❉✜◗P❙❘❯❚✩❆❉❱✞◆❲❆✸❳✺❘✼❨✌✔❲✜◗❚❩❱❬✔❅✖❉❈❲❭❪✖❉❫❉✔❴✔❅❵❖P✞✖❉❘❯❚✩❆❉❱❬❛ ❜❬❝✩❜❬❝✩❜ ✴❡❞❣❢ ❤❬✐✗❥❩❦✫❧✳♠ ✤✻✦✼★✽✪✯✬✯✮♥✰♣♦ ✤❬qr✦✩★☛✲s✴✶✬✯✮t✪ ❀✣✉✈★✇✲s✴✶✬✂❞❣❢ ① q ✦✼✬✧✲②★❑③■✴④✮⑤✪ ❀✣✉✈★✇✲s✴✶✬✂⑥❣❢ ❜❬❝✩❜❬❝⑧⑦ ✴❡⑥❣❢ ❤❬✐✗❥❩❦✫❧▼⑨ ✤✻✦✼★✽✪✯✬✯✮♥✰ ♦ ✤❬qr✦✩★☛✲s✴✶✬✯✮t✪ ❀✣✉✈★✇✲s✴✶✬✂⑥❄⑩ ①✗❶✙✦✼✬✧✲②★❑③■✴④✮⑤✪ ❀✣✉✈★✇✲s✴✶✬✂❞❄⑩ ❷✔❸❱❬❆❉❘❹✔❸❘✼❨✌✖❉❘❑❘✼❨✌✔✓❈✭✔❻❺✙P❙✜✣✖✙❈t❚❩❘❽❼✺❆✸❳✂❘❩❨✞✔❸◆❖❆❉✜◗P❙❘❯❚✩❆❉❱❾❚⑧◆✓❿■✔❖❘❻✔❖❈t✘❴❚❩❱✢✔❅❿④➀⑤❼✶❘❩❨✞✔❸❚❩❱✞❚❩❘❯❚✩✖❉✜✞✖✙❱❬❿✺➀❅❆❉P❙❱❬❿✙➁ ✖❉❈t❼▼❿r✖❉❘❹✖■❛➃➂✫❆✙❈✇❘❩❨✞✔✇✘✇❆❉✘✇✔⑤❱➄❘➅❭❪✔❇❭➆❚❩✜❩✜❸✖❉◆☎◆tP❙✘✇✔✇❘✼❨✌✖✙❘➅❘✼❨✌✔☛◆⑤❆✙✜◗P❙❘❯❚✩❆❉❱▼✤✻✦✼★✽✪✯✬✯✮❾❚⑧◆❾◆t✘✇❆➇❆✙❘❩❨✞❛ ➈❨✌✔➉❱❬❆❉❱➄➁❻◆t✘❾❆●❆❉❘✼❨➋➊☎✖❉◆❖✔t✹♥❚❩❱✢➊⑤✜◗P✞❿✙❚❩❱✗❺❾❘✼❨✌✔❲❿❉❚⑧◆❖➊❅❆❉❱✞❘❯❚❩❱✞P✞❆✙P✏◆✺➊❅✖➌◆❖✔➉❭➆❚❩✜❩✜✫➀☎✔❲➊☎❆✙❱✌◆t❚✩❿■✔❖❈✭✔❅❿❾❚❩❱➍❘✼❨✌✔ ❱❬✔☎✕r❘♥✜✣✔❅➊❖❘❽P❙❈✭✔⑤◆❖❛ ➎➆➏❽➐ ➑✂➒❙➓✻➔➣→✭↔✯→✸➒✙↕ ❤❬✐✗❥❩❦✫❧✳➙ ➛✷■✦✭➜❢✌✪➇⑩⑤➝⑧✮❹➞❯❁✞➟■➠☎➡ ➢❱✥❘❩❨✞✔➣❈✭✔⑤✘✇✖❉❚❩❱✢❿■✔❖❈✶❆✯❳✛❘✼❨✏❚⑧◆➣➊❅❆❉P❙❈❅◆❖✔➣❭❪✔➣❭➆❚❩✜❩✜✢❆✙❱✞✜◗❼❾➀☎✔✺➊❅❆❉❱✞◆t❚✩❿■✔❖❈t❚❩❱✗❺❸➤✫➁❹❱✢❆❉❈t✘❴◆❖❛ ➢❱➃❆❉❈✭❿r✔⑤❈➣❘❻❆ ◆t❚❩✘✛✚✢✜◗❚❳❅❼❾❘❩❨✞✔✺❱❬❆✙❘❻✖✙❘❽❚✩❆✙❱▼➥t➦❑➥☎➧❇❭➆❚❩✜❩✜✫❿r✔⑤❱✢❆❉❘❹✔✶❘❩❨✞✔✻➤✫➁❹❱✢❆❉❈t✘➨❆✸❳✺✖➩❳❅P❙❱❬➊❖❘❯❚✩❆❉❱➭➫❻P✏◆tP✞✖❉✜❩✜◗❼✇❿■✔❽➯♥❱✢✔☎❿ ❆❉❫❉✔❖❈➉➜❢✞✪❖⑩⑤➝❂➲☛✖❉❱❬❿✥➥t➦ ➥☎➧➍❭➆❚❩✜❩✜✧❿r✔⑤❱✢❆❉❘❹✔➉❘✼❨✌✔❪➤✫➁❹❱✢❆❉❈t✘➳❆✯❳➉✖☛❫❉✔❅➊⑤❘❹❆❉❈❖❛ ➵
lul2(t)=(/u2(a,t)da l=-U(x2(1,t)-2(0,t) This gives us an erpression for the time variation of the L norm,( or 2-norm). of the solution. We note that this variation only depends on the value of the olution at the boundaries 2 Model Problem To further simplify the presentation and analysis of the different schemes we will consider a problem writh periodic boundary conditio Initial condition (x,0)=°(x) Periodic Boundary conditions: u(0, t)=u(1, t) l=0→|l|2(t)=|°= constant 2.1 Exa 2.1.1 Periodic Solution(U >O) SLIDE 7 t=T t= 21
➸t➺➆➸⑤➻■➼✼➽✯➾♥➚➶➪➄➹➴➘ ➷ ➺ ➻ ➼✼➬✫➮✭➽✯➾✓➱r➬✞✃➋❐❒ ➹❮➘ ➷ ➺②➪❪❰➺ ❰ ➽ÐÏ❮Ñ ❰ ➺ ❰ ➬ ✃Ò➱r➬➋➚ÔÓ ➱ ➱r➽ ➸⑤➺✻➸ ➻ ➻ ➚ÖÕ Ñ ➼✩➺➻ ➼✸×■➮✭➽✯➾✧Õ②➺➻ ➼❽Ó✌➮✭➽✯➾✯➾ Ø✢Ù✏Ú⑧Û✓Ü✙Ú❩Ý❉ÞtÛ➣ß✏Û➣à❉á✑Þ✭â⑤ã✢ä✭ÞtÛ❅ÛtÚ✩å❉á④æ⑤å✙ä➣ç✼Ù✌Þ✶ç❯Ú❩è❾Þ✶Ý❉à❉ätÚ✩à✙ç❽Ú✩å✙á▼å✸æ➅ç❩Ù✞Þ➩é➻ á❬å✙ätè❲ê❸ë❹å✙ä❸ì✌í❹á❬å❉ätè➣î➌ê å✸æ☛ç❩Ù✞Þ☛Û❖å✙ï◗ß❙ç❽Ú✩å✙á❬ðòñ➃Þ❇á❬å✙ç❻Þ✥ç✼Ù✌à❉ç✶ç✼Ù✏Ú⑧Û☛Ý❉à❉ätÚ✩à❉ç❯Ú✩å❉á✠å✙á✞ï◗ó✳ô■Þ❹ã✌Þ❖á❬ô❉Û➋å❉á➴ç❩Ù✞Þ❇Ý❉à❉ïõß✞Þ➋å✯æ☛ç✼Ù✌Þ Û❖å❉ï◗ß❙ç❯Ú✩å❉á❃à❉ç✂ç❩Ù✞Þ❴ö❅å❉ß❙á✢ô■à✙ätÚ✩ÞtÛ❖ð ÷ øù☛ú➍û➅ü✈ý❡þ➆ù❴ÿ➍ü❖û✁ ✂☎✄✝✆✟✞✡✠☞☛ Ø➄å❴æ❅ß❙ätç✼Ù✌Þ❖ä✥Û❅Ú❩è➅ã❬ïõÚæ❅ó✳ç✼Ù✌Þ❲ã✢ä✭Þ⑤Û⑤Þ❖á✞ç❹à❉ç❯Ú✩å❉á à✙á❬ô➭à❉á✢à❉ï◗ó➌ÛtÚ⑧Û➍å✸æ✥ç❩Ù✞Þ✑ô❉Ú✌➅Þ❖ä✭Þ❖á✞ç✺Û✎✍❅Ù✌Þ❖è❾Þ⑤Û✑✏❪Þ ✏➆Ú❩ï❩ï✒✍❅å✙á✌ÛtÚ✩ô■Þ❖ä❴à✶ã✢ä✭å■ö❖ï✣Þ❖è✓✏➆Ú❩ç✼Ù❾ã✌Þ❖ätÚ✩å➇ô✙Ú✔✍✇ö❅å❉ß❙á❬ôrà❉ätó✕✍❅å❉á✢ô❉Ú❩ç❯Ú✩å❉á✞Û⑤ð ❰ ➺ ❰ ➽✇Ï➴Ñ ❰ ➺ ❰ ➬ ➚✠Ó✌➮ ➬✗✖➭➼✩Ó✞➮❖×➌➾ ✘✚✙✜✛✣✢✤✛✦✥★✧✡✩✎✪✫✙✭✬✜✛✮✢✯✛✣✪✰✙✲✱ ➺✻➼✼➬✽➮✭Ó✗➾♥➚Ô➺➷ ➼✼➬✢➾ ✳✁✴✝✵✷✶✔✸✡✹✺✶✼✻✾✽✿✪✰❀✜✙✭✬✜✥✫❁✤❂✕✩✎✪✫✙✭✬✜✛✮✢✯✛✣✪✰✙✭❃✎✱ ➺✻➼✩Ó✞➮✯➽✯➾♥➚Ô➺✧➼✯×■➮✯➽✯➾ ➱ ➱r➽ ➸t➺➆➸ ➻ ➻ ➚ÔÓ ❄ ➸t➺➆➸ ➻ ➼✩➽✯➾♥➚❡➸⑤➺➷ ➸ ➻ ➚ ✩✎✪✫✙✭❃❅✢✯✥✫✙✝✢ ❆❈❇❅❉ ❊●❋■❍❑❏▼▲❖◆◗P ❘✲❙✔❚☎❙✔❚ ✳❯✴✰✵✷✶✼✸✲✹✺✶✼✻❲❱✲✸☎❳✔❨✺❩❬✶✼✸❪❭❴❫Ñ❛❵ Ó✝❜ ✂☎✄✝✆✟✞✡✠❞❝ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 x ➽➆➚✠Ó ➽➆➚❢❡ ➽➆➚ ì❡ ❣
3 Finite difference solution 3.1 Discretization DE Discretize(0, 1) into J equal intervals A r d (0, T) into N equal intervals N ≈u≡(xj,t"),fon ≤ SLiDE 9 N t △ 10 NOTATION E IR vector of approximate values at time n 2mE IR vector of exact values at time n lu(ai, t ) 3.2 Approximation For example∴(forU>0) SLiDE 11 du (x,t)-(x-1,t)- (x1,tn+1)-(x;,t")v △t Forward in Time Backward(Upwind) in Space
❤ ✐❢❥❬❦✗❥♠❧✒♥♣♦q❥sr☞♥✁t✉♥✁❦✇✈■♥②①✕③✾④✎⑤✑❧❑❥✎③⑥❦ ⑦❈⑧❅⑨ ⑩❢❶✤❷✝❸❺❹✭❻❽❼✜❶◗❾❽❿✒❼✜❶◗➀➂➁ ➃☎➄✝➅✟➆✡➇☞➈ ➉❯➊➌➋✯➍s➎✯➏s➐✯➊✣➑❬➏➓➒✼➔✜→❬➣↕↔➙➊✣➛✝➐✯➜✇➝➞➏❬➟✝➠✭➡✫➢❽➊✣➛✝➐✤➏❬➎✤➤✷➡✫➢➌➋➦➥⑥➧ ➥⑥➧●➨ ➣ ➝ → ➧➫➩❯➨➯➭✝➥⑥➧ ➡★➛❪➲➳➒✼➔✜→❅➵❖↔➂➊➌➛✝➐✤➜➺➸q➏❬➟✝➠✭➡✫➢❽➊✣➛✝➐✤➏❬➎✤➤✷➡✫➢➌➋➦➥⑥➻ ➥⑥➻✉➨ ➵ ➸ → ➻◗➼➓➨❢➽✡➥⑥➻ ➚➾ ➩✕➪ ➼ ➚➩✕➶ ➼ ➚ ➒✔➧➩ →❅➻➼ ↔➹→ ➘➴➜✫➎➬➷ ➔ ➮❞➭➺➮ ➝ ➔ ➮➬➽☞➮ ➸ ➃☎➄✝➅✟➆✡➇☞➱ ➃☎➄✝➅✟➆✡➇➯✃★❐ ❒❖❮❖❰✒Ï✉❰❈Ð✤❮❖❒ÒÑ Ó ➾Ô➩➼ ➡✫Õ✜Õ✜➎✯➜×ÖØ➊➌Ù➺➡✷➐✯➊✣➜✰➛Ú➐✤➜ Ô ➒➴➧➩ →❅➻➼ ↔ ➶ Ô➩➼ Ó ➾Ô ➼✕Û ÐÜ✁Ý✑➤✫➏❬➍s➐✤➜✰➎■➜★➘✒➡✫Õ✜Õ✜➎✯➜×Ö➫➊✣Ù➺➡✷➐✯➏❯➤✷➡✫➢✣➠✜➏↕➋■➡✷➐■➐✯➊✣Ù➓➏Þ➽✉ß ➾Ô ➼ ➨áà ➾Ô➩❑â ➼ ➩✤ã✒ä Ý Ó Ô ➼ Û ÐÜ✁Ý✑➤✫➏❬➍s➐✤➜✰➎■➜★➘✺➏✎Ö✜➡✫➍➹➐■➤✷➡★➢➌➠✜➏↕➋➦➡✷➐❈➐✤➊➌Ù➓➏Þ➽✉ß Ô ➼ ➨áà Ô ➒➴➧➩ →✤➻➼ ↔ â ➩✤ã✺ä Ý ⑦❈⑧✼å æèç✁ç❯❹✜➀✺é■❶✤ê▼❿✺❼✜❶❅➀➂➁ ➃☎➄✝➅✟➆✡➇➯✃✜✃ ë✜➜✰➎■➏sÖ✜➡★Ù➓Õ✜➢➌➏➓ì✎ì❬ìí➒✟➘➴➜✫➎❯î❛ï➬➔✰↔ ð Ô ð➧òññ ñ ñ ➼ ➩ ➪ Ô ➒➴➧➩ →❅➻➼ ↔❑ó Ô ➒✔➧➩sô✲ä →✤➻➼ ↔ ➥⑥➧ ➨ Ô➩➼ ó Ô➩sô✲ä ➼ ➥ò➧ ð Ô ð➻õññ ñ ñ ➼ ➩ ➪ Ô ➒➴➧➩ →❅➻➼★ö ä ↔❑ó Ô ➒✔➧➩ →✤➻➼ ↔ ➥⑥➻ ➨ Ô ➼★ö ä ➩ ó Ô➩➼ ➥⑥➻ ë✭➜✫➎✯÷✿➡✫➎✯➲➺➊➌➛✗❰❈➊✣Ù➓➏Òø❈➡✫➍♠ùØ÷❈➡★➎♠➲❲➒✔ú✁ÕØ÷■➊✣➛✭➲❪↔❈➊✣➛❲ûØÕ❪➡✫➍s➏ ü