a△t∫a-t a>0 艹1=仍-(奶1-奶-)+(1-2+奶 Note that by introducing the absolute value we are able to write a single eapres sion that takes into account the dependency of the difference stencil on the sign of sLIdE 9 In conservative form 0)-ad(0+1-0) >0 We see that altho ugh the first order upwind method was originally derived using finite difference, and characteristic interpolation arguments, it can also be in terpreted as a conservative finite volame scheme were we solve for cell solution averages rather than pointwise values We note that the upwind scheme written in this manner is precisely a FCTS scheme (forward in time centered in space)with the explicit addition of a second difference term. As we know from the linear analysis this term is required for 2.4.2 Nonlinear case SLIDE 10 In the nonlinear case au af(u) 0 =2(+1)
Ò❙Ó❫Ô✳Õ Ñ Ö × ÒÑ Ö❚ØÚÙ✛Û✆Ü Ó Û✎Ý Þ ÒÑ ÖÓ Ø ÒÑ Ö✙ßÓ Ñ Õ Ù➆à➣á ÒÖÓÔ✳Õ Ø ÒÑ ÖÓ Ù➆â➣á ã❫ä Ò❙Ó✱Ô✗Õ Ñ Ö × ÒÑ Ö✠Ø Ó Ù✁Û✆Ü åÛ✆Ý→æÒÑ ÖÓÔ✳Õ Ø ÒÑ Ö✙ßÓ Õ➼ç❱èêé Ù é Û✆Ü åÛ✎ÝêæÒÑ ÖÓÔ✳Õ Ø åÒÑ ÖÓ è ÒÑ Ö✙ßÓ Õ✙ç ë✎ì✱í➠îïí➏ð➥ñ♠í✦ò➼ó✠ô➏õ➥í➹ö❏ì★÷✱ø➥ù✙ô➏õ✁ú✷í✇ð❤î➡ñ❛òPû➼ì✱ü➮ø✛í➠î➡ý✖ñ✱ü➮ø➥îïþ❱î✎ñ♠ö❏î✆ñ❫ò➼ü✜î➡í➘ì♣þ✞öPô➏í➘î✎ñ✆ûPô➏õ✝ú❫ü✜î✎î➾ÿ✁❙ö❏î✙û✄✂ ûPô❦ì♠õ✍í✇ð❤ñ✱í➵í➘ñ✆☎❛îPû✓ô➏õ➥í➘ì❚ñ❫ù✄ù✄ì♠ø✛õ❘í➵í✇ð❤î➡÷❛î✝➥î✙õ✯÷❫î➼õ❙ù➼ó➆ì✟✞✓í✇ð❤î➡÷✱ô✠✃î➼ö❏î✙õ✯ù✄îïûPí➘î✙õ✯ù✙ô➏ü✫ì♠õ✔í➏ð➥î✓ûPôú✱õ ì✟✞ Ù☛✡ ☞✍✌✏✎✒✑✔✓✖✕ ✗✙✘✛✚✢✜✍✣✥✤✧✦✏★✪✩✬✫✮✭✆✯✰✩✱✦✳✲✝✜✱★✪✴✶✵ ÒÑ Ó✱Ô✗Õ Ö × ÒÑ Ö✠Ø Ó Û✎Ü Û✆Ý ✷✆✸✺✹✼✻Ó Ö Ô✾✽✿ Ø ✸✺✹✼✻Ó Ö✙ß ✿❁❀ ✽ ✸✺✹✼✻ Ö Ô ✿✽ ×❃❂ å Ù✍❄ÒÑ Ö Ô✗Õ è ÒÑ Ö✧❅ Ø å ❂ é Ù é ❄ÒÑ Ö Ô✳Õ Ø ÒÑ Ö✧❅ ✸✹✼✻ Ö Ô❆✽✿ × ÙÒÑ Ö ✸ Ù❚à❆á ✹✼✻ Ö Ô ✿✽ × ÙÒÑ Ö Ô✳Õ Ù❚â❆á ❇î☞û✙î✄î✒í✇ð❤ñ✱í❱ñ♠ü➮í✇ð❤ì✱ø❫ú◆ð➄í✇ð❤î❉❈❱ö✄û✄í➵ì✱ö❏÷❫î➼ö☞ø✧✯þ✞ô➏õ❙÷❋❊✷î✙í✇ð❤ì◆÷✆þ❱ñ✖ûïì♠öPôú❫ô➏õ❙ñ♠ü➏üó➆÷❫î➼öPô➏ý✖î✄÷♣ø✁ûPô➏õ✝ú ❈➵õ❘ô➏í➠î➄÷♠ô✠✃î➼ö❏î➼õ❙ù✄î✁●✓ñ♠õ✯÷✔ù✄ð❤ñ✱ö❏ñ❫ù➼í➠î➼öPô➫ûPí➲ô❦ù➄ô➏õ❘í➠î➼ö❍❤ì✱ü✜ñ♠í➹ô❦ì♠õ❧ñ♠ö➹ú✱ø✮❊✷î✙õ❘í❦û■●❶ô➏íïù✄ñ♠õ❻ñ✱üû➼ì➬ò✄î✷ô➏õ✱✂ í➘î✙ö❍✯ö❏î✙í➘î✄÷➆ñ♠û➡ñ➄ù✄ì♠õ❤û➼î➼öPý✖ñ✱í➲ô➏ý♠î❏❈➵õ❘ô➏í➠î➡ý✖ì✱ü➮ø✮❊✠î✎û➼ù✄ð❤î❑❊✠î✎þ❱î✙ö➾î➡þ❱î✎û➼ì♠ü➮ý♠î▲✞✙ì♠ö✆ù✄î➼ü➏ü✫û➼ì♠ü➮ø✛í➹ô❦ì♠õ ñ♠ý♠î➼ö❏ñ➼ú✝îPû➡ö➾ñ♠í✇ð❤î➼öïí➏ð➥ñ♠õ▼➥ì♠ô➏õ❘í➲þ✞ô➫û➼î✆ý♠ñ♠ü➮ø➥î✙û ✡ ❇î➆õ✯ì♠í➘î➄í➏ð➥ñ♠í✒í➏ð➥î❄ø✧❙þ✞ô➏õ❙÷✍û➼ù➾ð➥î✁❊✷î❄þ✞öPô➏í➹í➠î➼õ→ô➏õ➣í➏ð✛ô➫û◆❊✷ñ♠õ➥õ✯î✙ö✷ô➫û❁❙ö❏î✄ù➼ô➫û✙î➼ü➮ó✏ñP❖❘◗❚❙❱❯ û➼ù➾ð➥î✁❊✷î❳❲❨✞✙ì♠öPþ❱ñ♠ö➾÷✎ô➏õ➄í➲ô✒❊✷î✒ù➾î➼õ➥í➘î➼ö❏î✄÷ïô➏õ❚û❍➥ñ❫ù✄î✟❩➡þ✞ô➏í➏ð❚í➏ð➥î✓î❏ÿ✁❙üô❦ù✙ô➏í➵ñ❫÷❛÷♠ô➏í➹ô❦ì♠õ✍ì✟✞☞ñïû➼î✄ù✄ì♠õ❙÷ ÷♠ô✠☞î✙ö❏î➼õ❙ù✄î❚í➠î➼ö✄❊✡▼❬ û✆þ❱î✺☎✱õ❙ì✱þ❭✞✄ö➾ì❪❊ í✇ð❤î✷ü➮ô➏õ❙î✄ñ♠ö❚ñ✱õ❙ñ♠üó✖û✄ô➫û✷í✇ð✁ô➫û♣í➠î➼ö✄❊ ô➫û✠ö❏î■❫✙ø✛ô➏ö❏î✄÷❆✞✙ì✱ö ûPí➠ñ❛ò✙ô➏üô➏í➲ó ✡ ❴❱❵❜❛✥❵❝❴ ❞✜✍✣✥❡❢✯✰✣❣✦❤✫✮★❥✐❳✫✬✤✆✦ ☞✍✌✏✎✒✑✔✓❧❦♥♠ ✗✙✘P♦q♣✬r❆✘ã✘✬s✉t❜✘✬r✧✈ä❚✇✈✢①qr✢② ③ ③Ò Ü è ③☛④ ❄ ③ Ò ❅ Ý × á ♦q♣⑤r❆⑥⑤⑦✮⑧◆⑨✍r✇➼ã✢⑩r✆① ❶❸❷ ✸✺✹✼✻ Ö Ô ✿✽ ×❹❂ å ✷ Ñ④ Ö Ô✳Õ è Ñ④ Ö ❀ Ø ❂ å é ÑÙ Ö Ô ✿✽ é ❄ÒÑ Ö Ô✳Õ Ø ÒÑ Ö❑❅ ❺
i每1+1≠每 Here f, denotes f(i,). The above choice of a guarantees that one sided approx mation is obtained. i.e >0 F First Order Upwind Scheme The first order upwind scheme is conser vative, and for At sufficiently small, it can be shown to be convergent (later on in this lecture we will dis scuss for convergence). The Lax-Wendroff the hat it will converge to a weak solution We show below that the lar-Wen droff and Beam-Warming algorithms are also ervative schemes and therefore admit a finite volume interpretation 2.5 Lax-Wendroff SLIDE 1 =2(+)- (1+1-a aii is again defined as For the linea ("+1--)+(2+1-2+-) =a△/△t
❻❼✢❽q❾➀❿➁❳➂➄➃ ➅➆✙➇❢➈ ❿■➉ ➅➆✙➇ ➅➊➇❢➈ ❿■➉ ➅➊➇ ➋➍➌ ➎❻ ❽q❾❣➏✺➐➂ ➎❻ ❽ ➑✼➒❍➓➎❻ ❽✧➔ ➋➍➌ ➎❻ ❽q❾❣➏ ➂ ➎❻ ❽ →✺➣❑↔q➣ ➑❻❽➙↕➣❑➛✍➜♥➝✝➣✁➞ ➑❉➓➎❻ ❽ ➔✆➟❋➠✼➡ ➣✺➢✢➤■➜❪➥❪➣✺➦➡ ➜♥➧✰➦➨➣❳➜✟➩ ❻❼◆➫✢➭➢❪↔q➢♥➛⑤➝✙➣■➣✄➞❸➝➡ ➢♥➝▲➜❪➛✼➣➯➞✄➧↕➣↕ ➢➨➲➳➲✍↔q➜✆➵➳➸ ➧✒➺▼➢❪➝❍➧✰➜❪➛➻➧❝➞✺➜➳➤✁➝✙➢❪➧✒➛✼➣↕❪➼ ➧ ➟ ➣ ➟ ➽❽q❾ ❿➁ ➂ ➃ ➑❻❽ ❻❼❽q❾ ❿➁➚➾➶➪ ➑❻❽q❾✥➏ ❻❼❽q❾ ❿➁➚➹➶➪ ➘✖➴☛➷✄➬P➮ ➱❘✃✰❐✪❒➳➷➚❮❁❐➳❰❣➬♥❐ÐÏ❚Ñ▲Ò❆✃❢ÓÔ❰ÖÕØ×✁ÙÔ➬♥Ú➶➬ ÛÝÜ✬ÞPß⑤à■á✟â▼ã➳à➨ä✮Þ✆à➚å✬æ❤ç➋❜èä➶á➨é■Ü✬Þ❑ê➚Þ ➋ á◆é✁ãè áëÞ✆àqì❪í❪â ➋ì➳Þ✢îïíè ä ➌ã✢à◆ð❳ñ❳áqå✮ò➙é➋Þè âqó✉ôõáqê▼í♥ó✉ó❢î ➋â❳é❑íè✖ö Þ◆áqÜ✬ã❪çè âqã ö Þ◆é✁ãè ì✢Þ❑à➨÷✢Þè â ➓ ó✉í♥âqÞ❑à✺ãèÐ➋✉è âqÜ➋ á✾ó✉Þ✆é✄â➨å✬à➨Þ➙ç▲Þ▼ç➋ ó✉ó▲ä➋ áqé❑å⑤á➨á❆âqÜ✬Þ à➨Þ✆ø✏å➋ àqÞ✆ê❋Þè â➨á ➌ã✢à➙é❑ãè ì✢Þ✆àq÷➳Þè é❑Þ ➔✄ù ÛÝÜ✬Þ✳ú✥í❪û❤ü❍ý✖Þè ä✮àqã✢þ➶âqÜ✬Þ✆ã✢à➨Þ❑êÿâ➨Ü✬Þ❑à➨Þ➌ã➳àqÞ Þè áëå✬à➨Þ✆á âqÜ✱í❪â ➋â❚ç➋ ó✉ó❱é✁ãè ì➳Þ❑à➨÷✢Þ➀âqã▼í❋ç▲Þ✧í✁ áqã✢ó✉å✮â ➋ãè ù ✂✛➣❁➞➡ ➜☎✄ ➤■➣✝✆❜➜☎✄ ➝➡ ➢❪➝❏➝➡ ➣✟✞✥➢❑➵➳➸☎✂✛➣✁➛ ↕↔q➜✡✠ ➢♥➛↕☞☛➣■➢❪➺❋➸☎✂✛➢❪↔✄➺❋➧✒➛➫ ➢☎✆➫➜♥↔✄➧✒➝➡➺✺➞❋➢♥↔q➣➚➢☎✆❨➞❑➜ ➦■➜❪➛⑤➞✁➣❑↔✄➥❪➢❪➝❍➧✒➥✪➣✺➞❑➦➡ ➣✁➺▼➣✁➞❁➢♥➛↕ ➝➡ ➣✁↔q➣❢➩✁➜❪↔➨➣❋➢ ↕➺❋➧✒➝❚➢✍✌Ô➛✱➧✒➝✝➣❁➥✪➜✎✆➭➺▼➣❁➧✒➛✱➝✝➣❑↔❍➲✍↔q➣❑➝✙➢❪➝❍➧✰➜❪➛➟ ✏✒✑✔✓ ✕✗✖✙✘✛✚✢✜✤✣✦✥★✧★✩✫✪✭✬ ✮✰✯✲✱✴✳✦✵✷✶✫✶ ➽✹✸✰✺ ❽q❾ ❿➁ ➂ ✻ ✼ ✽ ❻➑❽q❾✥➏✿✾ ❻➑❽✝❀❂❁ ✻ ✼ ❻❼❄❃ ❽q❾ ❿➁ ð❳ñ ð❆❅ ➓➎❻ ❽q❾❣➏✛❁ ➎❻ ❽✆➔ ❻❼❽q❾ ❿➁ ➧❝➞❳➢➫➢♥➧✒➛ ↕➣❇✌ï➛✍➣↕ ➢❪➞ ❻❼❽q❾ ❿➁ ➂ ➃ ➅➆✟➇❢➈ ❿■➉ ➅➆✙➇ ➅➊➇❢➈ ❿■➉ ➅➊➇ ➧➩ ➎❻ ❽q❾✥➏ ➐➂ ➎❻ ❽ ➑✼➒❍➓➎❻ ❽ ➔ ➧➩ ➎❻ ❽q❾✥➏ ➂ ➎❻ ❽ ❈✬ã➳àÝâqÜ✬Þ✾ó ➋✉èÞ✆í✢à❘Þ✆ø✏å⑤í❪â ➋ãè ➎✰❉❻ ❾❣➏ ❽ ➂ ➎❻ ❽ ❁❋❊✼❍●➎❻ ❽q❾✥➏ ❉ ❁ ➎❻ ❽❉➉ ➏❏■ ✾❑❊❃ ✼▲●➎❻ ❽q❾✥➏ ❉ ❁ ✼➎❻ ❽❉ ✾ ➎❻ ❽❉➉ ➏❏■ ❊ ➂ ❼ð▼❅❖◆♥ð❳ñ P
2.6 Beam-Warming (-++3+1-3+-)+号计+计(+-+1一+一1 For the linear equation -立(-3+41+1-12)+(+2-213+1+)a<0 2.7 Entropy Solutions SLIDE 13 Do these schemes converge to the entropy satisfying solution? EXAMPLE: Consider a non-physical solution to Burgers'equation 1x<0 either 1 2.7.1 Example SLIDE 14 First order upwind =2(1+1+1) +t(u+1-a a,+1 or ij+1-ij is zero Vj Be ther f;=fi F+i=2 F一F The entropy-violating solution is preserved 5 Entropy Satisfying Solutions It turns out that the first order upwind, the Lax Wendroff and the beam Warming schemes allow for entropy violating solutions. These sche distinguish between shocks and expansions To determine in advance if a general numerical scheme will only produce en tropy satisfying solutions is very difficult. One possible approach is to derive an
◗✒❘✔❙ ❚❱❯❳❲✭❨❑❩✢❬✤❲✙❭❪❨❴❫❛❵✍❜ ❝✰❞✲❡✴❢✦❣✷❤❥✐ ❦✭❧♥♠♦q♣✭rs✉t ✈✇②①❇③⑤④⑥ ♦✔♣✲⑦⑨⑧☞⑩ ④⑥ ♦✔♣ ✈ ⑧☞⑩ ④⑥ ♦ ③❶④⑥ ♦❸❷ ✈✡❹ ③ ④❺ ⑦ ♦q♣✙rs☞❻♥❼ ✇❻✫❽ ① ④❾♦q♣✲⑦ ③ ④❾♦q♣ ✈ ⑧ ④❾♦ ③ ④❾♦❸❷ ✈❸❹ ③ ❿➁➀➁➂ rs ✇➃①❇③❂④⑥ ♦q♣✲⑦⑨⑧➄⑩ ④⑥ ♦q♣ ✈ ③ ⑩ ④⑥ ♦✰⑧ ④⑥ ♦➅❷ ✈✡❹ ⑧➇➆♦✔♣ rs ❺ ⑦ ♦q♣ rs❱❻✫❽ ✇❻♥❼ ① ④❾♦q♣✲⑦ ③ ④❾♦✔♣ ✈ ③ ④❾♦⑨⑧ ④❾♦➅❷ ✈❸❹ ➈❏➉❛➊➌➋➍▼➎❍➏➉❛➊➌➋➍✲➐♥➑ ➏ ➉❛➊➒➋➍⑨➑ ➓✫➔➣→✒↔❛↕✫➙✹➛➝➜➝➞✫➙✢➟❥→➠➙✢➡✲➢♥➟☎↔➤➜➥➔➣➞ ➧✝➨❸➩☎➫ ➦ ➭ ➯ ➧➦ ➭➳➲▼➵ ➨ ➸✹➺➼➻➧➦ ➭✦➲✛➽ ➨ ➧➦ ➭❇➾➨ ➫✎➚ ➧➦ ➭❇➾✝➪➹➶ ➨ ➚ ➵ ➪ ➸➘➺➧➦ ➭✙➲ ➨ ➸➧➦ ➭❇➾➨ ➫❥➚ ➧➦ ➭❇➾✝➪➹➶ ➨ ➴✙➷➒➬ ➧➦ ➨❸➩☎➫ ➭ ➯ ➧✝➨➦ ➭➳➲ ➵ ➸ ➺➝➲➮➻➧✝➨➦ ➭ ➚ ➽➧✝➨➦ ➭➩☎➫ ➲ ➧✝➨➦ ➭➩ ➪q➶ ➚ ➵ ➪➸ ➺➧✢➨➦ ➭➩ ➪➮➲ ➸➧✢➨➦ ➭➩☎➫❥➚ ➧✝➨➦ ➭❏➶➱➴✙✃➒➬ ❐✛❒➹❮ ❰ÐÏ②Ñ❄Ò♥Ó✿Ô✛Õ×Ö✛ÓÙØ❛Ú➠Ñ✫Û❸Ó②Ï★Ü Ý✰Þ✲ß✴à✦á✷â➣ã äæå❱ç✢è✙é✲êëéÐêëìëè✙é❄íîé❄ê❆ì➣å❪ï⑨ð✰é✲ñóò✰éÐçëå❱ç✢è✙éôé❄ï⑨ç✝ñ☎å♥õ❖ö÷êëø➮ç✢ù✔ê✢úqö❳ù❇ï✙ò❱ê✢å✰û❇ü➳ç✢ù✔å❪ï✿ý þ✿ÿ✁✄✂✆☎✞✝✦þ✄✟ ✠☛✡✌☞✎✍✑✏✓✒✕✔✗✖✙✘✚☞✛✡✜☞✕✢✤✣✛✥✧✦✧✍★✏✓✩✗✘✫✪✬✍★✡✌✪✮✭✕✯★✏✮✡✌☞✰✯★✡✲✱✳✭✛✖✵✴✌✔✶✖✵✍✶✷✧✔✶✸✹✭✺✘✻✯✵✏✼✡✜☞ ✟ ✽✿✾❁❀❃❂✑❄✑❅☛❆❈❇❊❉ ❀●❋■❍ ❏ ❉ ❀●❑■❍ ✏▼▲ ✔✌▲❖◆✽◗P❘ ✏✓✍✙✔✗✏✼✯★✥✛✔✶✖ ❉ ✡✌✖ ❏ ❉❚❙ ❯ ❘ ❆❊❱❲❨❳✕❩ ❬✬❭❫❪✬❭❵❴ ❛❝❜ø✫íõû❇é Ý✰Þ✲ß✴à✦á✷â✻❞ ❡✏✼✖❢✍✑✯❣✡✜✖✵✒✛✔✗✖❣✭✛✣✧❤✙✏✼☞✎✒ ✟ ✐❦❥♠❧ ❘★♥♣♦q ❆ ❉rts ◆❯ ❘★♥ ❱✈✉ ◆❯ ❘✶✇ ❏ ❉r✈① ◆②❘★♥③♦q ① ✾ ◆✽❘★♥ ❱ ❏ ◆✽❘ ❅ ④✧✏✮☞✺✩⑤✔❣✔✶✏⑥✯✵✥✛✔✗✖✄◆② ❘★♥ ♦q ✡✌✖⑦◆✽❘★♥ ❱ ❏ ◆✽❘ ✏✓✍✿⑧✗✔✗✖✵✡ ❳✛❩⑩⑨❷❶❢❸❢❹✻❺✧❻✗❶✁❶⑤❼❫❽❵❾✕❶✗❿ ◆❯ ❘ ❆ ◆❯ ❘★♥ ❱➁➀➃➂ ❿ ◆✽❘ ❆ ◆✽❘★♥ ❱✌➄ ❙ ✐❥♠❧ ❘★♥ ♦q ❆❊❱❲❨❳✕❩ ❙ ✐❥♠❧ ❘★♥ ♦q ❏ ✐❥◗❧ ❘⑤➅ ♦q ❆➆❍ ❳✕❩ ❙ ◆✽P♥ ❱ ❘ ❆ ◆✽◗P❘ ➇è✙éôé❄ï⑨ç✝ñ☎å❪õ❖ö◗➈❛ð❳ù❇å✰û❇ø✫ç✢ù❇ï✙ò êëå❪û✔ü✦çëù❇å✰ï✷ù✔ê❆õ✙ñ☎é❄ê✢é✲ñóð❪é✧➉ ➊✁➋ ➌t➍✬➎➐➏✆➑ ➒❝➓➔➎★→✜➍✌➣✬↔➙↕➜➛➜➎✑➝❵➞➠➟❁↔❃➝❵➓❃➡■↕➃➍✕➢★➤❃➎✑➝➥➍✧➓➃➞ ➦✯➧✯★✭✛✖✵☞✺✍➧✡✜✭✕✯➧✯★✥✺✘✫✯➨✯★✥✛✔t➩✺✖❢✍➥✯➨✡✜✖✵✒✕✔✶✖➧✭✛✣✧❤✙✏✼☞✺✒➜➫❝✯✵✥✛✔ ✝✘✫➭✹✢✤➯t✔✗☞✺✒✕✖✵✡✫➲➳✘✌☞✺✒➵✯✵✥✛✔➸✱✳✔✶✘✌➺❦✢ ➯➻✘✫✖✵➺➼✏✮☞✛✴➽✍✵✩❢✥✛✔✗➺✚✔➁✍✄✘✫✪✮✪✮✡✻❤➚➾❁✡✜✖✄✔✗☞✹✯✵✖★✡✜✣✹✦✰➪✧✏✼✡✜✪✮✘✫✯★✏✮☞✛✴➶✍★✡✌✪✮✭✕✯★✏✮✡✌☞✺✍✶▲✙➹❣✥✛✔✶✍★✔⑩✍✵✩❢✥✛✔✗➺✚✔➁✍✄✩✗✘✌☞✛☞✛✡✌✯ ✒✕✏✓✍➥✯✵✏✼☞✺✴✌✭✛✏✓✍✑✥➴➘♠✔⑤✯➥❤✳✔✗✔✗☞➨✍★✥✛✡✕✩❢➷✕✍✳✘✌☞✺✒➴✔⑤➭✕✣✺✘✌☞✺✍✑✏✮✡✌☞✎✍✗▲ ➹❃✡●✒✕✔⑤✯✵✔✗✖✵➺✚✏✼☞✛✔➶✏✼☞➸✘✜✒✕➪✻✘✫☞✺✩✗✔➽✏⑥➾✄✘➧✴✜✔✗☞✛✔✶✖✵✘✌✪✈☞✧✭✛➺✚✔✗✖✵✏✮✩✶✘✫✪✳✍★✩❢✥✛✔✶➺✚✔✲❤✙✏✼✪✮✪✞✡✜☞✛✪✮✦➬✣✛✖✵✡✕✒✕✭✺✩✗✔✲✔✗☞✕✢ ✯★✖✵✡✌✣✧✦➼✍★✘✫✯★✏✓✍➥➾❁✦✧✏✮☞✛✴⑦✍✑✡✜✪✼✭✛✯★✏✮✡✌☞✺✍☛✏✮✍☛➪✌✔✗✖✵✦❦✒✛✏⑥➮➽✩⑤✭✺✪⑥✯➁▲☛➱✄☞✺✔❷✣✎✡✹✍★✍★✏✮➘✛✪✼✔❝✘✫✣✛✣✛✖✵✡✜✘✜✩❢✥❦✏✓✍➔✯✵✡⑩✒✕✔✶✖★✏✮➪✌✔❷✘✌☞ ✃