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 ✻