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 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
3.3 First Order Upwind Scheme 0 sugg 仍-C(-奶-1) <n< N 0≤n≤N Courant number C=U△/A The Courant number is a non-dimensional number thatthat plays a centrai role in the numerical solution of hyperbolic equations. If we imagine particles traveling at speed u, we can think of C, as the distance, meas ured in grid points, that a particle will move in an increment of time At 3.3.1 Interpretation SLIDE 1 Aa 了+1 uP≈Ca-1+(1-C) Note 1 Exact nodal solution for c For C= 1, the sche In this case, the grid is such that the same characteristic line goes through(a;, tn+)and (ai-1, tr) The interpolation is then exact, and the numerical scheme reproduces the exact solution with no erre SLIDE 1 nfr0<n≤N
ý❈þ✼ý ÿ✁✄✂✆☎✞✝✠✟✡✂✆☛✌☞✍✂✏✎✒✑✔✓✕✗✖✘☛✚✙✜✛✣✢✤☞✦✥✧☞ ★ ✩✫✪✭✬✦✮✕✯✞✰ ✱✳✲✵✴✷✶✸✱✳✹✻✺✽✼✿✾✗❀❂❁❃❁✞❄❅✾✄❆❇✾❉❈❊❈❋❈ ✱✵❍❏■▲❑ ● ▼ ◆ ✱● ▼❍ ❖◗P ✴✷✶ ✱● ▼❍ ◆ ✱● ▼❊❘❍ ❑ ❖◗❙ ✺✠✼ ❚ ✱✳❍✞■▲❑ ● ▼ ✺ ✱● ▼❍ ◆❱❯❳❲✱● ▼❍ ◆ ✱● ▼❊❘❍ ❑❩❨ ❬❪❭ ❫✡❴✁❫ ❵ ✼ ❫✕❛❜❫ ❝ ✱● ❞❍ ✺ ✱● ❍❡ ✼ ❫❢❛❜❫✷❝ ❣✐❤✳❥▲❦♠❧✣♥✵♦♣♥q❥▲rts✈✉✫❦ ❯ ✺✇✶❖◗P②①❏❖◗❙ ③✵④❂⑤✷⑥⑧⑦⑩⑨❷❶❹❸⑩❺✣❻❳❺✆⑨❷❼❳❽❇⑤❋❶❿❾➁➀t❸✕❺✳⑦⑩❺✆➂✗➃⑩❾✭❼✁⑤❊❺✣➀➄❾➅⑦⑩❺✳❸❏➆❉❺✣⑨❷❼✁❽❹⑤❋❶➇❻➈④❂❸⑩❻✁❻➈④❂❸❏❻✘➉✳➆➊❸❏➋♠➀❜❸✏➌❹⑤❋❺✣❻➍❶②❸❏➆ ❶②⑦❏➆➊⑤✁❾✭❺✕❻✭④✣⑤✁❺✣⑨❷❼✁⑤❋❶➄❾➅➌❹❸❏➆➎➀❋⑦⑩➆➏⑨❷❻➍❾➅⑦⑩❺✏⑦✄➐✐④➑➋❹➉✣⑤❋❶❋❽❹⑦❏➆➏❾➅➌➒⑤❇➓❊⑨✣❸❏❻➔❾➅⑦❏❺❂➀❋→↔➣➅➐❳↕✈⑤✁❾✭❼✁❸❋➙✞❾✭❺✳⑤✻➉❂❸❏❶➄❻➔❾➅➌❋➆➊⑤❊➀ ❻➍❶②❸⑩➛⑩⑤❊➆➜❾✭❺✫➙➝❸⑩❻▲➀➔➉✣⑤❇⑤❇➃ ✶◗➞ ↕✈⑤✸➌❇❸⑩❺♣❻➈④➑❾✭❺❷➟✐⑦✗➐ ❯➞ ❸⑩➀✜❻➈④❂⑤✸➃⑩❾➁➀➄❻➠❸⑩❺✵➌❹⑤ ➞ ❼✁⑤❇❸♠➀➄⑨❷❶②⑤❇➃➡❾✭❺➝➙❏❶➄❾➅➃➢➉❂⑦❏❾✭❺✣❻➔➀ ➞ ❻➈④❂❸⑩❻✜❸❉➉✣❸⑩❶➄❻➍❾➅➌❋➆➊⑤◗↕➤❾✭➆✭➆✦❼✁⑦⑩➛⑩⑤✻❾✭❺❱❸⑩❺➥❾✭❺✵➌❊❶❹⑤❊❼✁⑤❊❺✆❻✔⑦✗➐➡❻➔❾✭❼✁⑤ ❖◗P → ➦✦➧➨➦✦➧➅➩ ➫♥✵♦❅✉✫❦❅➭➯❦⑩✉✫♦❩❧❷♦❅➲➅❤✳♥ ★ ✩✫✪✭✬✦✮✕✯✞➳ P ✱❍❏■▲❑ ▼ ✺✏✱✵➵ ➸➻➺ ✉❿➼✈➲➔♥▲✉➑❧❷❦ ➫♥✵♦❅✉✫❦❅➭➽❤✆➾➔❧❷♦❩➲➔❤✆♥ ❽❇⑤❊❻➍↕✈⑤❇⑤❊❺❿❻➈④❂⑤➢➉✣⑦⑩❾✭❺✣❻➔➀ ❴ ◆ ❭❃➚✣❴ ➪ ❭ ✱➵❱➶ ❯❱➹➘✈➴➷✞➬➽➮ ✴ ❲ ❭ ◆❱❯❨ ➘✈➴➹ ➷ ➱❜✃q❐➄❒➥❮ ❰✘Ï✦Ð✳Ñ✫❐➡Ò✈✃✍Ó✍Ð❷Ô➽Õ✞✃❷Ô②Ö▲❐✗×➠✃❷Ò✏Ø✄✃❷Ù ❯ ✺ ❭ Ú❂Û❃Ü ❯ ✺ ❭❃Ý ❆②Þ❂❄❱✾❹ß❇Þ❂❄❋à❳❄ Ü❄❩á❂❀✣ß❊❄❅✾♣❆ Û ✱✵❍❏■▲❑ ▼ ✺â✱▼❊❘❍ ❑ ❈✧ã➠ä✽❆②Þ❂å➁✾➒ß❋æ❃✾✗❄ Ý ❆②Þ❂❄❜❁Ü å➨á✠å➨✾ ✾②❀✣ß❇Þ✡❆②Þ✆æ⑩❆➝❆②Þ❂❄➇✾❹æ❏à❳❄➻ß❇Þ✣æÜæ❃ß➄❆❹❄Ü å➨✾✗❆②å➁ß✁ç➨å➊ä❂❄è❁Û ❄❅✾✻❆❹ÞÜ②Û❀❂❁✞Þ ❲❙ ▼ ➚ P❍❏■▲❑ ❨ æ✞ä✣á ❲❙ ▼❊❘ ❑ ➚ P❍ ❨ ❈ éÞ❂❄✌å➨ä✫❆②❄Ü②ê✳Ûç➁æ⑩❆❹åÛä➝å➁✾➤❆❹Þ❂❄❋ä↔❄❊ë❂æ✞ß❊❆ Ý æ✞ä✣á➝❆②Þ✣❄✤ä➑❀❂à❳❄Ü å➨ß❩æ❏ç✣✾❹ß❇Þ❂❄❩à➝❄ Ü❄ê❂Ü②Û á❷❀✆ß❊❄❩✾➤❆②Þ❂❄✘❄❊ë❂æ✞ß❊❆ ✾Û ç➨❀❷❆②åÛäèì➢å➜❆❹Þ➒äÛ ❄Ü❹Ü②Û❃Ü ❈ ➦✦➧➨➦✦➧➁í î✘ï➭➯➾➔➲➅ð❃➲➈♦↔ñ✍❤✳➾➅❥▲♦❩➲➔❤✆♥ ★ ✩✫✪✭✬✦✮✕✯⑩ò ó❾✭➛♠⑤❋❺ ✱● ❞ ❲ ✺✒✱❞ ❨ ↕✈⑤◗➌❇❸⑩❺❱➌❇⑦⑩❼✤➉✳⑨❷❻➠⑤ ✱● ❍ ➐❊⑦⑩❶ ✼ ❫✷❛t❫✏❝ ô
U>0. known values s tote anpI wniqut S,t hs-sa 33 ut t-Hio Fotr e can E nSgl hang 1 33N JOz Pct sua sz ht mh△mht/r h tt
known values unknown values õ▲ö➒÷✁ø⑩ù❹ú②û➊ü↔û➨ý➑þ✞ÿ❋ú✁②û✂ý ☎✄ ✆✞✝✠✟☛✡✌☞✎✍✏☞ ø❏ý✑ û✒ ✓ õ ✡✕✔✖✓✗✝ ☎✄ ✆✙✘✛✚ ✜ ✢ ☎✄ ✜✤✣✦✥★✧ ✆ ☎✄ ✜✩✣ ✆ ☎✄ ✜✫✪✆ ✚✫✬ ✭✯✮✰✭✯✮✰✭ ✱✳✲✍✎✴✵✡✶✟✸✷ö ✴✵✹ ✺✼✻✾✽❀✿✯❁❃❂❅❄ ❆ ÿ ❇ø❏ý ❈ ú❹û➜ù❹ÿ ☎✄ ✆ ✢ ❉✄ ☎✄ ✆ ✪ ✚ ✢ ❉✄ ✆ ☎✄ ❊ ☎✄ ❊●❋ ☎❊ ❍■ ■ ■ ■ ■ ■ ■❏ ❑▼▲❖◆◗P❙❘ ❚ ❚ ❯❱❯❱❯ P P ❑✌▲❖◆❲P❙❘ ❚ ❳ ❳ ❳ ❚ ❚ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❚ ❳ ❳ ❳ ❳ ❳ ❳ P ❑✌▲❖◆❲P❙❘ ❚ ❚ ❯❱❯❱❯ ❚ P ❑▼▲❖◆❲P❙❘ ❨❱❩❩ ❩ ❩ ❩ ❩ ❩ ❬ ❭ ❪❴❫ ❵ ❜❛ ✭✯✮✰✭✯✮❞❝ ❡✟✲✹❣❢❖❤✕✝ ✺✼✻✾✽❀✿✯❁❃❂❅✐ ☎❦❥✖❧❣☎♥♠ ✢♣♦ qsr ✢ t t ♦✠♦ ✥ ✢ q✈✉ qsr ✢✇♦②① ③ ④ ✢ t⑥⑤ ⑦ ✢⑨⑧❅♦❅♦ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 1.2 x EXACT t=1 t=0.25 t=0.5 t=0.75 ⑩
r Ca n t mi s mn d m 2.1 p ey citic Thk finit. il kIknpk ai3Pliths col, a u ch d -PI al I initiai pPm itiN i nial FX w2X2EcS2 Gxc xum 2 tExt an F in; i.2. tG2 poixtSi42 2FBEiX Ln i4 22E FoEtG2 Xox-p2Eodic cx2 tG2 xdou 2 d2fixitiox must b2 xdxpt2 xccoFixWa to ixcad2 bouxdxE coxditiox4 1/2 F|△ F√△ Wk phPPsk PuI nFIs with thk Aa pIks uitipiipatiPn tP s akk sulk that, as Aa, orv; F via, rtfl -FI sPs k 3ivkn unFtim viartfl tkn s tP a INstant iin -apt, thk intk3Iai P-thk squalk: P-viartfl PkI ior, 11. This is, in kssknpk, an pIPxis ation tP thk pPntinuPus p F 2 nPIs P-a unFtiPn. In PuI paltipuial ifs→0 PI h h s pinks that Friel→o,hchd I-wk wkIk tP nFt inpiu k thk Aa pIkaptfl, Pui nFIs wPul aptuaiiy bk thk P-an inglkasin3 nus bkI P-pFintwisk KIIPIs, an. hknpk nFt a wkly 3PP. s kasulk P-thk appulapy 5 Ca n sistm ay 5.1 p ey citic Thk,让 kIknpk sphks k Llyf F is col uut lt with kIkntiai hquatin Ci Fo
❶ ❷❹❸s❺★❻❽❼❿❾➁➀✈❼➂❺➄➃➅❼ ➆❿➇❴➈ ➉✇➊✖➋➍➌➂➎➐➏➑➎❴➒➓➌ ➔✼→✾➣❀↔✯↕❃➙✾➛ ➜➞➝➠➟➍➡②➢➠➤❞➥➦➟➨➧➑➤❞➩♥➟✎➫❱➟✎➢②➭✎➟➍➯❅➲❞➳✠➵❅➫❱➤➸➥❱➝➠➺➼➻✠➽❦➾♥➚✼➪✠➶✵➹✼➪➴➘➷➤➸➬ ➲✰➤❞➺ ➮●➱✼✃❴➮➂❐☛❒❰❮ Ï➮➂❐✛ÐÒÑ Ó➮➂➱✈Ð✸Ô Õ②Ö×ØsÙÒ×Ø Õ❿Ú♣Û➠Ü Ý➷Þ❃ßàÞ✸á ➬✶➵❅➫❿â➠➾♥ã❣➤✰➢➠➤❞➥➦➤✒➯✙➲✯➭✎➵❅➢②➧➑➤❞➥➦➤✰➵❅➢ ×✼ä✠å✶æ♥çéè êéë➄ì✠í✎ë✼í✎î➦ï✙ð✯ñ➓í✈ò➦ó➠ï✵ð❀ð➁ïôò✁òéõ➑ö✤í➨÷✶ó➠ï✙÷ Ö×ä Ú ×ä✙ø➓ù✕ú í ú ÷✶ó➠í❿û②ü ùë②÷▼ñùò✎í✤í✫îéî❱ü✵î ùë Ö×ä ùòþýÿí✎î➦ü ú ✛ü✵î◗÷✶ó➠í❲ë♥ü✵ë✂✁✰û②í✎îù ü☎✄ù✝✆✞✆ïôò✎í✞÷✶ó➠í ï✠✟✁ü☛✡ôí☞✄✠í✝✌➓ëù ÷ ù ü✵ë✸ö❽õ➴ò✁÷✍✟✁í ï✠✄✠ï❱û✼÷í✎✄ ï ✆✎✆ü✵î✏✄ùë✾ì❅ð✒✑Ò÷ü ùë✆ ðõ✂✄✠í✓✟✁ü✵õ➑ë✔✄❅ï✙î✕✑ ✆ü✵ë✖✄ù÷ ù ü✙ë➠ò ú Õ✕✗ Õ❿Ú ✘✙✛✚æ ✢✜✣ ✤✦✥✛✧ ✗✩★✤✫✪✬ ✧✦✭ ★ Ú✯✮✚ æ Õ✰✗ Õ ★ ✱✳✲ ✴✶✵✸✷✕✹✻✺ ✴✶✵✽✼✿✾❁❀✰❂❃✵✽❄❅❀✠✹ ❆Ò➟ ➭✁➝➠➵➴➵✫❇➦➟✦➵✠❈➠➫ ➢②➵❅➫❱➺❊❉➅➤❞➥➦➝ ➥➦➝②➟ ✚ ✚ æ●❋➫❱➟✎➺✓❈➠➲➸➥❱➤❋➲❞➤✒➭✎➯✙➥➦➤✰➵❅➢ ➥➦➵♣➺✤➯✠❍❅➟■❇✦❈➠➫➦➟✦➥❱➝②➯✵➥❑❏✈➯✠❇ æ▼▲ Û➠Ü◆✗✤ Ú✢✗ å✶æ✤ Ü◆❖Ø ç ➬✶➵❅➫P❇➦➵❅➺★➟➄➳❅➤❘◗❅➟ÿ➢ ➬❙❈➠➢②➭✫➥➦➤✰➵❅➢ ✗ å✶æ Ü✦❖Ø ç ➥➦➟ÿ➢②➧❚❇★➥❱➵❣➯Ò➭✎➵❅➢✂❇❴➥❱➯❅➢✾➥ å ➤✰➢✩➬✕➯✠➭é➥❑❏✠➥❱➝➠➟➂➤✰➢✾➥➦➟ÿ➳❅➫✁➯✙➲✼➵✙➬✖➥➦➝②➟✳❇✏❯✩❈②➯✙➫❱➟❿➵✙➬ ✗ å✶æ Ü◆❖Ø ç ➵❱◗❅➟ÿ➫ å Û②Ü✎Ý ç❴çéè ➜➞➝➠➤❲❇❙➤❘❇☎❏➴➤❞➢◗➟☎❇✏❇❴➟ÿ➢②➭✫➟✫❏✾➯❅➢ ➯❋✂❋➫➦➵✿❳➑➤✰➺✤➯✵➥➦➤✰➵❅➢ ➥➦➵✞➥➦➝②➟✩➭✫➵❅➢✾➥❱➤❞➢✽❈➠➵✫❈✂❇❩❨ Ú ✲ ➢➠➵❅➫❱➺ ➵✙➬➞➯✞➬❙❈➠➢②➭✫➥➦➤✰➵❅➢ èP❬➢Ò➵✠❈②➫ ❋➯✙➫➦➥➦➤✒➭✰❈➠➲✒➯✙➫ ➭✎➯✫❇❴➟ Õ❦Ö×Ø Ù✦×Ø Õ ▲ Û ➬✶➵❅➫ ÝsÞ✇ß❣Þ⑨á❏❦➤❞➺❋➲✰➤❞➟❑❇➅➥❱➝②➯✵➥❪❭ Ö×✤Ø ÙÒ×✤Ø ❭ ▲ Û ➬✶➵✠➫ ÝsÞ✇ß❣Þ♣á ➯✙➢❦➧ Ý➷Þ❴❫✤Þ❛❵è ❬➬✸❉➟❜❉❙➟✎➫❱➟➞➥➦➵➷➢②➵✙➥➓➤✰➢②➭✫➲❘❈②➧➑➟❿➥➦➝➠➟ ✚ æP❋➫➦➟✎➬✕➯❅➭é➥❱➵❅➫❑❏❅➵✫❈➠➫ ➢➠➵✠➫➦➺❝❉➵✫❈➠➲✰➧★➯✠➭é➥✦❈❦➯✙➲✰➲❡❞✓❢❦➟➅➥❱➝➠➟❣❇✦❈➠➺ ➵✙➬☛➯❅➢❽➤✰➢②➭✎➫➦➟✏➯✠❇➦➤❞➢➠➳➷➢✽❈➠➺✓❢❦➟ÿ➫ ➵✙➬ ❋ ➵❅➤✰➢✾➥❅❉➅➤❘❇➦➟❿➟✎➫❱➫❱➵❅➫✎❇❤❏✙➯❅➢②➧★➝➠➟✎➢❦➭✫➟❿➢➠➵❅➥ ➯✐◗❅➟ÿ➫✦❞s➳❅➵➴➵➑➧❽➺❽➟✏➯✠❇✦❈➠➫❱➟ ➵✙➬✛➥➦➝➠➟➨➯✠➭✎➭❤❈➠➫❱➯✠➭✰❞è ❥ ❷❹❸s❺❧❦♥♠✰❦♣♦✗❼➂❺➄➃❩q r➞➇❴➈ ➉✇➊✖➋➍➌➂➎➐➏➑➎❴➒➓➌ ➔✼→✾➣❀↔✯↕❃➙✠s ➜➞➝➠➟➷➧➑➤❞➩✼➟ÿ➫➦➟ÿ➢②➭✫➟t❇❱➭✁➝➠➟ÿ➺❽➟ ✉Ö Ö×Ø Ú♣Û ❏ ➤❲❇➅➻❅➽✼➾✛➘❑✈✕➘☎✇ÿ➪➴➾✔✇✐❉➅➤➸➥❱➝✞➥➦➝➠➟➨➧➑➤❞➩♥➟✎➫❱➟✎➢✾➥➦➤✒➯✙➲☛➟❑❯✩❈②➯✵➥❱➤❞➵✠➢ ✉× Ú✇Û ①✰②✫③ ④
For all smooth functions v 1≤j≤ ≤ when△x,△t→0. First Order Upwind Scheme SLIDE 1 {n+-Sn"} Lu at (C2);≡ ut+U02)1+=(utt+U-vza)"+ 6 Truncation error into diffe 1<j<J (C)-(C)=可 1<n<N nsistencv令 O(△x,△t) nl njs hj randa n jrrrJnsp( jr( hinA, 4f jn3 rt ih< hq riDipj3hirj 3 qj h q xq 6, itn hrih b+ 23(At-t Cj3crj 3 dn q..lt n
⑤❚⑥✠⑦⑨⑧✂⑩✝⑩♥❶◆❷❸⑥✽⑥☛❹✏❺❼❻❙❽✂❾✂❿✕❹✏➀❡⑥✫❾✂❶❜➁ ➂➄➃➅ ➁ ➆❚➇➉➈⑨➊ ➂➅ ➁✽➇◆➆➈✻➋➍➌❚➎ ❻❙⑥✠⑦➐➏➒➑✍➓→➔❪➓✯➣ ➑✍➓↕↔✶➓➛➙ ➜❺❚➝❤❾☞➞✓➟ ➎ ➞➡➠ ➋➍➌✂➢ ➤❩➥➧➦ ➨❪➩❅➫➯➭✠➲➵➳→➫➯➸⑨➺➻➫➛➼➾➽➪➚➶➩◆➹➘➸✢➴❩➷✂➬❣➺♣➮➱➺ ✃✖❐✩❒❰❮♣Ï➶Ð✠Ñ Ò➘➀ÔÓ✔➝❤⑦✏➝❤❾✂❿❤➝✍⑥✠Õ✖➝❤⑦✎Ö❱❹✏⑥✠⑦ ➅➃➁ ➆❸× ➑➞✓➠✽Ø ➁ ➆☛Ù➄Ú ➊ Û➃ ➁ ➆✖Ü Ò➘➀ÔÓ✔➝❤⑦✏➝❤❾✩❹✦➀❲Ö☛Ý✸⑥✫Õ➯➝☎⑦✏Ö☛❹✦⑥✫⑦ ➅➁ßÞáà➁ à ➠➡â➐ã à ➁ à ➟ ✃✖❐✩❒❰❮♣Ï✶äæå ➂♣➃➅➁ ➆ ➇ç➈èÞ ➁➈➆☛Ù➄Ú ➊✶➁➈➆ ➞✓➠ â➐ã ➁➈➆ ➊✶➁➈✰é➆ Ú ➞✓➟ × ➂➁❱ê â➐ã➁☛ë✠➇◆➆➈ â ➞✓➠ ì ➂➁❱ê❰ê◆➇◆➆➈ â➐ã ➞✓➟ ì ➂➁☛ë❑ë✫➇❅➆ â➵í❤í❤í ➂➅➁✽➇ ➈➆ Þ ➂➁❱ê â➐ã➁☛ë✠➇ ➈➆ ➂➅➃ ➁ ➆ ➇ç➈⑨➊ ➂➅ ➁æ➇ ➈➆ ×❛î ➂➞✓➟ ➎ ➞✓➠◆➇ ï ⑤✛➀❡⑦✎❶❅❹⑨⑥✫⑦✏ð❚➝❤⑦❜Ö✠❿☎❿✰❽❚⑦✎Ö❱❹✏➝➘➀❘❾ñ❶◆Õ✂Ö✫❿✰➝✍Ö☛❾➯ðP❹✏➀❡❷❸➝ ➢ ò ó➶ôöõñ÷❧ø❜ù✳ú♥û❤ü✓÷þýÿôôü➡ô ✃✖❐✩❒❰❮♣Ï✶ä✸Ð ✁❾✂❶✦➝❤⑦✦❹❜➝✄✂❚Ö✠❿✰❹❜❶✦⑥✠Ý❘❽æ❹✏➀❡⑥✫❾✆☎ ➀❘❾✩❹✦⑥Pðæ➀❡Ó✖➝☎⑦✦➝☎❾✂❿✰➝✍❶✏❿✎❺❚➝❤❷❸➝ ➂➅➃ ☎➇ ➈➆ ➊ ➂➅ ☎➇ ➈➆ ✝ ✞✄✟ ✠ ✡☞☛ ×✍✌❚➆➈⑨➎ ❻❙⑥✠⑦➐➏ ➑✍➓→➔❪➓➵➣ ➑✍➓↕↔✶➓➐➙ ☎➆☛Ù➄Ú × Û➃ ☎➆ â ➞✓➠ ✌➆ ✎⑥✫❾✂❶◆➀❲❶◆❹✦➝❤❾➯❿✄✏ ï✒✑ ✌➆ ✑ ×❛î ➂➞✓➟ ➎ ➞✓➠◆➇ ➎ ➑✐➓➶↔✶➓➶➙ ✓✕✔✗✖✙✘✛✚✢✜✤✣✥✚✢✦★✧✩✧✫✪✗✬✮✭✯✜✱✰✲✖✳✜✴✚✶✵✷✜✴✚✱✸✺✹✻✖✳✼★✔✗✚✾✽✿✧✗✖❀✜❁✸✺✸❂✧✗✸❃✔✗✬❄✚✶✵✷✜✴✸❂✜✺✬✺✹✻❅❆✚❇✧❉❈✮✽❊✖❋✬✄✜❁✸✺✚✾✽❊✖❍●✛✚✶✵✷✜✆✜❂✣■✔✥✼✄✚ ✬❁✧✗❅❆✹✻✚✾✽✿✧✗✖✆✽❊✖❋✚❏✧❑✚✶✵✷✜❇✭✗✽▲✬❁✼❁✸❂✜❁✚✢✜▼✬❁✼✤✵❋✜✄◆❖✜✫P❘◗✲✧✗✖❋✬★✽▲✬✺✚❏✜✄✖✳✼✄✘❘✽▲✬▼✚❊✵❋✜✄✖❙✧✯✦❁✚✢✔❚✽❊✖❯✜★✭❱✦❁✘❲✸❂✜★❳❁✹✻✽❊✸✺✽❊✖❍●❘✚❊✵❋✔✗✚ ✚✶✵✷✜❖✚✾✸✺✹✻✖❯✼★✔❚✚✱✽✿✧❚✖❀✜✄✸✺✸❂✧❚✸✮✚✢✜❁✖❯✭✗✬❄✚❏✧❃❨✩✜✄✸✤✧❙❩❬✵✷✜❁✖ ➞✓➟ ➎ ➞➡➠❖✚❏✜❁✖❯✭❙✚❏✧✴❨✩✜✄✸❂✧✥P❪❭✳✵■✽▲✬❃✔❚❅❆✚✢✜❁✸✺✖❯✔❚✚✱✽❊❫✗✜ ❴✸✤✧✫✼★✜★✭✗✹✻✸❂✜▼✽▲✬❵✜✤❳❁✹✻✽❊❫✗✔✗❅❛✜❁✖❋✚❜✚✢✧❲✚❊✵❋✔✗✚ ❴✸❂✜✄✬❁✜✄✖✙✚✢✜★✭❝✵❋✜✄✸✤✜ ❴ ✸❂✧✗❫✩✽✿✭✯✜★✭❱✚❊✵❋✔✗✚❬✚❊✵❋✜❝✭❚✽❞❵✜✄✸✤✜✄✖✳✼✤✜▼✬❁✼✤✵❋✜✄◆❖✜ ✽▲✬❡✖❯✧❚✸✺◆✮✔❚❅❆✽❛❨✩✜✤✭❱✽❊✖❃✬★✹❋✼★✵✮❩❢✔❚✘❲✚✶✵✷✔✗✚❜✚✶✵✷✔✗✚❜✚✶✵✷✜❣✭✗✽❞❵✜❁✸❂✜❁✖❯✼★✜❝✚✢✜❁✸✺◆❘✬▼✭✗✽❊✸✤✜✤✼❁✚✾❅❆✘❱✔❴✥❴✸❂✧✫✣✥✽❊◆✮✔❚✚✢✜❣✚✶✵✷✜ ✭✯✜❁✸✺✽❊❫✗✔✗✚✾✽❊❫❤✜✄✬❖✽❊✖✐✚❊✵❋✜❃✭❚✽❞❵✜✄✸✤✜✄✖✙✚✱✽✿✔❚❅❢✜✤❳❁✹❋✔✗✚✾✽✿✧✗✖✳P❥✓✛✹✻❅❆✚✾✽❴❅❆✘✗✽❊✖■●❦✚✶✵■✸❂✧❚✹✯●✩✵✐✦✄✘P➞✓➟❧✧✗✸t➞✓➠❄◆✮✔❚✘ ✸❂✜✄✬✺✹✻❅❆✚❬✽❊✖✛✔✮✭✗✽❞❵✜❁✸❂✜❁✖❯✼★✜❵✬❁✼✤✵❋✜✄◆❖✜♠❈✄✧✗✸❝❩❬✵■✽✿✼★✵✆✚❊✵✻✽▲✬❝✔✗❅❆✚❏✜✄✸✺✖✳✔✗✚✾✽❊❫❤✜ ❴ ✸❂✧✩✼✤✜★✭❚✹✻✸❂✜❇✭✯✧✩✜✺✬❵✖✳✧✗✚✲✔❴✥❴❅❆✘✯P ♥
t lAb erlp/ rolph T tell/ ce△ C pr/a/r A/ n/rbe△rtb6 a Ach grbc/b2bM△ batch/msr/qe/hh△h/ e/r- h en Ab/ 6a /r/ny/ hmn//Tpprbce 1l/. Abh/ ta 4/ 6d/r/nAll q 1ebbb 1lTr1 pyc yi tf Spur ch nb△ abner/m△nb6 ene Ab b r6/ melon C△ l py c. pyi tf spy ug or T h, khfinfi i t3 1 Ia3 Fus 3 uj-1Notetu'1yitc Sy e xacnd-su "N"feier ertu ju h ido t o S/Tr/abmhe6/rene n/r/n realll man//h cement b/t / bnls t 2. 7/t/LG n 1n6 x Ro 5nch 6/q neAn n//6h AbC/e/n/rllen/6 abr lal/t/L hn//hb m abr Tns r/Il,ofE S/nTt/ py h c hs 2y"th m x;P所面m,Exc9 ric r CAEx c g y m, aty s/mhMA/4/rc. gb Ilbo abr hE/th△h6 erb(nla Ab/鄱“/r hb Abb 5 hph1t计bs 1 q lAbor hb/abruenb 4/r- h br Cb n6Trs ubn6edbmh (" Tk/ Ab/ hu Abn erb( S/ Tbpb△b△M△Cm/t/ r A/ r/uAbrhnep(C/ hpi th Tn6byh6/hnbh△ 6eAbn Cub th m hpy hb en'lsrC nbl/ A△mCg6 /Arb em(√C△mBmC pr/h/6en 4/r-h ba b/2-nbr-b 6/b hayi th2 J P3j/c、ImI s/ Cu nbCnbCAIAa/q rhAbr6/r pCn6 hn//ch hata/b ld Eitc现fr∫l F xr E Fxe E
♦❝♣❆q✶r✷s❚t✯✉✩r✇✈✷①❁②❂r✷③★✈✙④✇③✐♣❆⑤❊q✱q✾♣❛①⑦⑥❁⑤❊q❱⑧❖s✗②❂①✇⑨★s✗⑧❵✈❯♣❆⑤✿⑨★③❚q✢①★⑩❤❶✴❷❢①✴✈✳②❂①✿❸✄①❁②❦q✶r✷①❱❸✄s✗②✺⑧❹✈❯②✤①✺④❁①❁❺❋q❏①✤⑩ r✷①❁②❂①✐⑥★①★⑨★③✗t■④❁①✇⑤❊q❃③✗❻✗s✗⑤✿⑩✗④✇q✶r■⑤▲④❃✈❯②❂s✥⑥✄♣❼①✄⑧❖❽❿❾❱s✗q❏①⑦q❊r❋③✗q❃⑨★s✗❺❋④✺⑤▲④★q❏①❁❺❯⑨❁➀❀②❂①★➁❁t✻⑤❊②❂①✺④✇q❊r❋③✗q❖q✶r✷① q❏①✄②✺⑧❱④❖⑤❊❺➂q✶r✷①❙⑩❚⑤➃❵①✄②✤①✄❺✳⑨✤①✆④❁⑨★r✷①❁⑧✮①❙③★✈✯✈❯②✤s❁➄✥⑤❊⑧✮③❚q✢①✛q❊r❋s❤④❁①✛s❉❸❃q❊r❋①❥⑩❚⑤➃❵①❁②❂①✄❺✙q✱⑤✿③❚♣➅①★➁❁t❋③✗q✾⑤✿s✗❺✳❽ ➆♣❛①★③✗②✺♣❆➀❤❶❧➇➈➊➉ ➋❪➌➍➉ ➋❚➎❬➏❣➐ ➑➒➉➇ ➋ ❶❑⑤▲④❃❺❯s❚q❘⑨✤s❚❺✷④✺⑤▲④✺q✢①❁❺❋q❲③✥⑨✤⑨★s❚②❂⑩✗⑤❊❺■✉⑦q✢s⑦s❚t✻②❥⑩✯①✱➓✲❺✙⑤❊q✱⑤✿s❚❺➔⑥✄t✻q ➈❢➉➇ ➋ ➌➣→✿➉ ➋✯➎❬➏ ➐ ➑➊➉➇ ➋✷↔✤↕❚➙❱➛★❶✲⑤▲④❁❽ ➜ ➝❖➞♠➟❲➠✛➡❁➢❁➡✺➞❯➤ ➥➒➦➨➧ ➩➭➫➲➯❇➳❵➵❉➸✻➵➨➺❢➳ ➻❯➼❍➽❊➾➪➚✇➶✷➶ ➹➅➘✷➴❑➷✻➬❛➮❯➴✫➱❂➴✫✃❋❐✄➴❲❒✤❐★➘✷➴✫❮❄➴ ❰➇ ➋❚➎❬➏ ➌ ➑➇❰➇ ➋ ➬▲❒▼Ï❁Ð✩Ñ✻Ò♠Ó✿Ô❖➬❛Õ❉Ö ×➘✷➴✫➱❂➴❇➴✄Ø✻➬▲❒× ❒❵Ù➊ÚÛ❒➨Ü✙❐★➘ ×➘❋Ý× Þ ➉ ➋ Þ ➌ Þ ➑➇ ➋ ➉ß Þ❣à Ù➊Ú Þ ➉ß Þ Õ✶á✯➱▼Ý❚â❼â ➉ ß✯ã Ý✯✃❋➷✆äæå ➙❱➛ ❒❂Ü❋❐★➘ ×➘❋Ý×❵ç à ä➙❱➛ à❪è é▼ê á✗ë✯➴❑❐❁á✯✃❋➷✻➬× ➬❼á✯✃❙❐❁Ý✯✃ ê ➴❑ì❡➱✤➬×❂× ➴❁✃✛Ý✥❒ Þ ➑í➉ ➇ Þ❣à →❉î➒ï➂ð✴→➙❘➛➨↔❂↔ Þ ➉ Þ ñ①❘③✗②✤①❘⑨★s✗❺❋④★⑤✿⑩✥①✄②✺⑤❊❺■✉❖r✷①❁②❂①❲❺✙t✻⑧✮①❁②✺⑤✿⑨★③✗♣✳④❁⑨★r✷①❁⑧✮①✄④❑❷❬r■⑤✿⑨★r✛⑤❊❺✙❻❤s❚♣❆❻❤①❄s✗❺✙♣❆➀❖q✾❷❢s✮q✱⑤❊⑧❖①❘♣❛①✄❻✗①❁♣ò④❁ó ä ③✗❺✳⑩ ä ï➭î ❽❄ô✳r✻⑤▲④❱⑩✯①✱➓✲❺❋⑤❊q✾⑤✿s✗❺õ❺❯①★①★⑩❤④❑q❏s❃⑥★①❵✉❍①✄❺✳①✄②✤③✗♣❆⑤❛ö✩①★⑩❣❸✄s✗②❑⑧❄t✻♣❆q✱⑤❊♣❼①✄❻✗①✄♣➪④❁⑨✤r❋①✄⑧❖①✺④❁❽ ÷①★⑨★③✗♣❊♣➒q❊r❋③✗q →❉î❝ï✍ø↔ àúù✗û ❸✄s✗②❙③✗❺✙➀❦②❂①★③✗♣ ø✐ü➍➐❑î ❽ ñ①❄r❋③✗❻✗① Þ ➉ ➋ Þ ➌ Þ ➑ý➉ ➇ ➋■þ➲➏ Þ❥à →❉î❑ïÿð✴→➙❱➛➨↔➨↔ Þ ➉ ➋■þ➲➏ Þ✁✂✄ à →❉î❑ïÿð✴→➙❱➛➨↔➨↔ ➋ Þ ➉ ß Þ ❶❖⑥✄t✻q →❉î❇ïÿð❃→➙❱➛➨↔❂↔ ➋ à → ù✆☎✁✝ ↔ ➋➔➌ ù ➋☎✁✝ àíùÚ ➌ Ù➊Ú ❽ ñ①❄❺❯s✗q❏①❱q❊r❋③✗q❡q✶r✷①❄q❏①✄②✺⑧ ð❃→➙❱➛➨↔❖③✗♣❊♣❼s✗❷♠④➊❸✄s❚②❘④❁s❚⑧✮①❖⑨★s✗❺✙q✱②✤s✗♣❊♣❛①★⑩✮✉❚②✤s✗❷æq✶r⑦s❉❸❲q✶r✷①❄❺✙t✻⑧✮①❁②✟✞ ⑤✿⑨★③✗♣❜④❁s✗♣❆t✻q✾⑤✿s✗❺✳❽õô✳r■⑤▲④❄⑤▲④▼✈❋③✗②✺q✾⑤✿⑨✄t✻♣❼③✗②✺♣❆➀❙②❂①❁♣❛①❁❻❤③❚❺❋q❡⑤❸❱❷❢①❘r✷③❚❻❤①✄❶❡⑤❊❺Ûq❊r❋①❖s✗②✺⑤✉✯⑤❊❺❯③✗♣❢①★➁❁t❋③✗q✾⑤✿s✗❺❋❶ ④❁s✗⑧❖①➅❸✄s✗②❂⑨❁⑤❊❺❍✉✛q✢①❁②✺⑧❘④✮s✗②❄⑥★s❚t✻❺❯⑩✯③❚②✺➀❙⑨★s❚❺❯⑩✗⑤❊q✾⑤✿s✗❺❋④❱❷❬r✻⑤✿⑨✤rõ⑧❖③✄✠✥①✮q✶r✷①❱④❁s✗♣❆t✻q✾⑤✿s✗❺❦✉✯②❂s✗❷❢❽ ñ① ③✗♣❆④✄s❵✈❋s✗⑤❊❺✙q➒s✗t✻qæq✶r✷③❚qæ❷❬r✷①❁❺❯①❁❻❤①❁②❇q❊r❋①❣②❂①❁♣❛③✗q✾⑤✿s✗❺❋④➨r✻⑤✈Û⑥★①❁q✱❷❢①★①❁❺ Þ ➉➋❚➎❜➏ Þ ③✗❺✳⑩ Þ ➉ ➋ Þ ⑩✥s✫①✄④❝❺✳s✗q ⑩✯①❏✈❋①✄❺✳⑩✆①✤➄✄✈❯♣❆⑤✿⑨❁⑤❊q✱♣➀✆s✗❺❙➙❘➛★❶✲q✶r✷①❲④✺q✢③✥⑥✄⑤❊♣⑤❊q✱➀✆⑨★s❚❺❯⑩✗⑤❊q✾⑤✿s✗❺Û⑥★①★⑨✤s❚⑧✮①✄④ Þ ➉ ➋❚➎❬➏ Þ❵àÿÞ ➉ ➋ Þ ❽ ✡⑤❊❺❯③❚♣❊♣❆➀❤❶❣❷❢①✆❺✳s✗q❏①✴q✶r✷③❚q❝⑤❸❖❷❢①❙⑩❚⑤❊❻✫⑤✿⑩✥①❥q❊r✻②❂s✗t✯✉❤r❀⑥❁➀☞☛➙✍✌➪❶❝④★q❏③✯⑥❁⑤❊♣❆⑤❊q✾➀õ⑨★s✗t✻♣❛⑩✇③✗♣❆④✄s✇⑥★①❙①❂➄✎✞ ✈❯②✤①✺④★④❁①★⑩❖⑤❊❺✕q❏①❁②✺⑧❘④❲s❉❸❑q✶r✷①✑✏✒✞❏❺✳s✗②✺⑧❖❽▼⑤✿❽❊①✩❽ Þ ➉ ➋❚➎❬➏ Þ✔✓❇àÿÞ ➉ ➋ Þ✔✓ ❽ ➥➒➦✖✕ ✗❖➵✙✘✛✚✯➸✢✜✣✘✛✤▼➫✥✘✧✦✩★✫✪❪➵➨➳✬✤✮✭✰✯✲✱❵➫✴✳ú➫ ñ①❲❷æ⑤❊♣❊♣❬❺❯s✗❷í④❂r❋s✗❷➭q✶r✷③✗q➊q✶r✷①➒➓✲②★④✺q➊s❚②❂⑩✯①❁②❑t✩✈❯❷æ⑤❊❺✳⑩❖④❁⑨★r✷①❁⑧✮①❘⑤▲④❇④★q❏③✯⑥❁♣❛①✩❽ ➻❯➼❍➽❊➾➪➚✇➶✶✵ ❰➇ ➋✯➎❬➏ ✷ ➌ ❰➇ ➋ ✷ ➐ Ù →❰➇ ➋ ✷ ➐ ❰➇ ➋ ✷ þ☞➏ ↔ ➌ →➨î❡➐ Ù↔ ❰➇ ➋ ✷ ï Ù ❰➇ ➋ ✷ þ➲➏ ➌ ✸ ❰➇ ➋ ✷ ï✺✹ ❰➇ ➋ ✷ þ☞➏ ✻