1 Model Equation U·u=rV >0, f, gi Despite its apparent simplicity this equation appears in a wide range of dis m heat 7 to financial er we will make extensive use of this equation, and several of the limiting cases contained therein, to illustrate the numerical techniques that will be presented In some cases U, k, and f will be functions of the solution u, in which case the equation is said to be nonlinear Note 1 Derivation of the Convection-Diffusion Equat ion for Heat Transfer We sketch below the derivation of the Convection-Diffusion equation for the particular problem of Heat Transfer in a moving fuid. Consider a velocity field U=(U(a, y), V(, y)) which is(for simplicity) time independent and incom au av v A streamline is a curve which is obtained by integrating the vector field and corresponds to the trajectory of a fuid parcel✽ ✾✿❁❀❃❂✤❄❆❅✛❇✲❈✹❉☎❊✴❋✦✿✢● ❍✴■❏❍ ❑▼▲✕◆✒❖✕P✟◗✝❘❚❙❏▲✕◆❱❯❳❲❨❙❬❩❪❭✗❫❴❙❵▲✕◆ ❛✝❜❞❝❢❡✓❣✐❤ ❥✝❦❥✝❧♥♠♣♦rq✠s❦✹t✙✉s✢✈ ❦ ♠✐✇ ①③② s⑤④⑦⑥ ❥ ❥✝⑧✗⑨ ❥ ❥❶⑩✴❷❹❸ s✢✈✤④ ❥ ✈ ❥✝⑧✈ ♠ ❥ ✈ ❥✝⑩ ✈ ♦❸ ✉✹❺☞❻ ❸ ✇ ❸❽❼❿❾➁➀❿➂➄➃➆➅➈➇➉➃➋➊✦➌➍❾➁➎❿➃➋➏✫➎➐➅ ⑥⑧ ⑨ ⑩ ❷ ➑✓➒❿➓➉➔→➓❚➣❴↔➙↕✒➛➝➜➟➞❴➓❚➣❴↔➙➠✗➓❚➣❳➓❚➡✕➢✝➔➝➛→➒❃➞❞➤➦➥➟➓❚➧✠➛➝➢✝➜ ①☎➨ ➩✢➫❹➭➲➯✝➳❢➵❬➫♥➳❢➵→➭✢➸➺➯➻➯➉➸➐➼➍➫➄➽➋➵✫➭❹➳❢➾✤➯✝➚➪➳➝➶➄➳❢➵➲➹➴➘✤➵❢➷❚➳➬➭♥➫✎➮➄➱➋➸❳➵➲➳➝✃❳➽❆➸➺➯➻➯➉➫✎➸➐➼✎➭✢➳❢➽❐➸✩❒✴➳➝❮❿➫❁➼➍➸❳➽❴❰➻➫✲✃❵ÏÐ❮❳➳➬➭➄➶➄➳❢Ñ ➯✝➚Ò➳❢➽✝➫✦➭✤➼➍➸❳➽❴❰➐➳❢➽❴❰✱Ï✎➼➍✃❳➾⑦Ó✖➫➺➸➐➵ÕÔ➉➼➍➸➐➽➉➭➈Ï✦➫✦➼✗➵Ö✃Ø×✕➳❢➽✝➸➐➽✝➶➄➳➝➸❳➚➋Ù✱➽❞❰❿➳❢➽✝➫✎➫➄➼❹➳❢➽❞❰❞Ú✒➩❪➱❚➼❹➳❢➽❞❰❪➵➈➷❴➳➬➭☎➶✎✃❳➱❚➼✎➭➄➫ ❒✕➫✩❒✴➳❢➚❢➚❱➾♥➸✠Û❿➫✑➫➍Ü➻➵❬➫✦➽➋➭✎➳❢Ý❳➫➆➱❴➭➄➫❃✃❵Ï❁➵➈➷❴➳➬➭❃➫✎➮✦➱➋➸➐➵→➳➝✃➐➽➉➘❪➸❳➽Þ❮✣➭➄➫➄Ý➴➫➄➼➍➸➐➚Õ✃❏Ï✲➵➈➷➉➫✩➚Ò➳❢➾✖➳❢➵➲➳❢➽❞❰❐➶✎➸❳➭✦➫✦➭ ➶✎✃❳➽❶➵Ö➸➐➳❢➽✝➫✎❮Ð➵➈➷➉➫➄➼➍➫➄➳❢➽➉➘✸➵❬✃Ð➳❢➚❢➚➪➱❴➭❹➵➲➼➍➸❳➵❬➫③➵➈➷➉➫❪➽➋➱❚➾Ð➫➄➼❹➳➝➶➺➸➐➚✺➵❬➫➺➶✎➷❴➽❶➳➝➮✦➱➋➫✦➭✧➵❢➷➋➸❳➵✱❒✴➳❢➚❢➚➟ß✎➫Õ➯✝➼➍➫✦➭➄➫✦➽❶➵Ö➫✎❮❿Ú à❹➽✩➭➄✃❳➾Ð➫③➶✎➸➴➭➄➫✦➭ ♦ ➘✓á✝➘✸➸❳➽Þ❮ ✇ ❒✴➳❢➚❢➚✓ß✎➫✴Ï✎➱❚➽Þ➶✦➵➲➳➝✃❳➽➋➭☎✃❵Ï☎➵➈➷➉➫☎➭➄✃❳➚Ò➱❚➵→➳➝✃➐➽ ❦ ➘➙➳❢➽✹❒✟➷❴➳➝➶✎➷✣➶✎➸➴➭➄➫Ø➵➈➷➉➫ ➫✎➮✦➱➋➸➐➵→➳➝✃➐➽✑➳➬➭Ø➭➄➸❳➳➝❮✲➵Ö✃➆ß✎➫✧➽✝✃➐➽➋➚➪➳❢➽Þ➫✎➸❳➼➄Ú âäã➦å❹æ✑ç è✑æ➐é➴ê❏ë➐ì➦å➍ê❬ã❚í✙ã➉î✧å➍ï✸æäð❪ã❚í✴ë❚æ❴ñ❞å➍ê❬ã❚í➟ò❶è✩ê➈ó✗ô❶õöê❵ã❴í♣÷❪ø✟ô❶ì➦å➍ê❬ã❚í❨î❏ã❴é❁ùäæ❿ì➦å úé➴ì➉í✟õ➴î❏æ❳é û ➂✩➏➍ü❿➂➄➌➺➊✎ý❆þ✝➂➄ÿ➁➎✁￾ ➌➍ý➋➂ ✂ ➂☎✄➺❾➀✁✆❳➌➺❾➎➻➃❐➎➐➅✤➌➍ý➉➂✞✝✒➎❿➃❴➀➻➂ö➊❹➌➺❾➎➻➃✠✟✠✡✗❾☞☛✝➇➋➏➍❾➁➎❿➃❐➂✍✌➻➇✎✆❳➌➍❾➁➎❿➃❐➅➈➎✏✄✢➌➍ý➉➂ ✑ ✆✏✄❏➌➺❾➁➊➄➇➉ÿ✒✆✏✄ ✑ ✄➺➎❿þ➋ÿ➂✔✓ ➎➐➅✖✕✤➂✔✆❳➌✘✗✖✄✙✆❿➃➋➏❵➅➈➂✔✄✒❾➁➃✚✆✛✓♥➎❳➀❞❾➁➃➉❼✛✜➋➇➉❾✂✣✢ ✝✒➎❿➃➋➏➍❾✂ ➂☎✄✤✆❪➀➻➂➄ÿ➁➎❚➊✦❾➌✦✥★✧❶➂➄ÿ✂ ♦ t ⑥✪✩♥⑥⑧ ⑨ ⑩ ❷✦⑨✙✫ ⑥⑧ ⑨ ⑩ ❷❏❷✬￾Õý➋❾➁➊✎ý✶❾➬➏ ⑥ ➅➈➎✏✄✖➏➍❾✒✓✑ ÿ ❾➬➊✦❾➌✦✥➉❷☎➌➍❾✒✓♥➂✲❾➁➃✂ ➂✑ ➂➄➃✂ ➂➄➃❞➌✭✆❿➃✂ ❾➁➃➋➊✦➎✮✓★✯ ✑ ✄➺➂ö➏➺➏❏❾➁þ➉ÿ➁➂❿❸❚➂ ✢ ❼ ✢ ❥✩ ❥✝⑧ ♠ ❥✫ ❥❶⑩ t❨❻ ✰ ➏❏➌✱✄➺➂✔✆✏✓♥ÿ ❾➁➃➉➂❁❾➬➏✛✆▼➊✦➇✲✄➺➀❿➂✳￾Õý➉❾➬➊✎ý❐❾➬➏❪➎➻þ❚➌✙✆❿❾➃➋➂✂ þ✠✥▼❾➁➃❞➌➍➂ö❼✏✄✴✆❳➌➍❾➁➃➉❼❃➌➺ý➉➂✲➀➻➂ö➊✦➌➍➎✏✄✵✧➋➂➄ÿ✂ ✆➐➃✂ ➊✦➎✮✄✱✄➺➂ö➏✑ ➎❿➃✂ ➏✕➌➍➎♥➌➺ý➉➂Ø➌✱✄✴✆✁✶❵➂✠➊❹➌➺➎✏✄✙✥Ð➎➐➅✷✆✭✜❶➇➉❾✂ ✑ ✆✏✄➺➊➄➂➄ÿ ✸⑩ ✸⑧ ✹ ✹ ✹ ✹ ✺✼✻✾✽✾✿❁❀✱❂❄❃ ❅ ❆☎✿ t ✫ ✩❈❇ ②