1.3 3D Finite differences +1,k 7×n× n points bandwidth b=n This means that we we halve the grid spacing, we will have 8 times(23)more unknowns and the cost of soluing the problem will increase by a factor of 128 (2). It is apparent that, at least for practical three dimensional problems, faster methods are needed 2 Basic iterative methods 2.1J aconI 2.1.1 Intuiti d of sol at starting from an arbitrary u(a, 0) That is, we ecpect the solution af the time dependent problem to converge to the lution of Solution by relaxation By adding the time dependent term at, we have now a parabolic equation. If kept fixed, we expect the solution to"ce to the steady state solution (i.e 0). Recall that the time dependent heat transfer problem is modeled by this equation. In this case, we expect the temperature to settle to a steady state distribution provided that the heat source,f, and the boundary conditions, do not depend on t➤✷➥➞➦ ➦✷➧✍➨✕➩❑➫❸➩❺➭❖➯❼➧❥➩❾➲❿➯✰➳❯➯❆➫❸➵➂➯❆➸ ➺❀➻❂➼❄➽❆➾➇➚ ➪r➶✗➪r➶✗➪r➹❀➘●➴❛➷♣➬❲➮ ➱ 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 725 ➪✰✃☎➶✓➪✰✃➑❐✕❒➒➬❳❮✥➴❴❰ Ï ❒❱➷■Ð❖Ñ❘➴❛Ð❖➬❳Ò④Ó❘ÔÕ➪✰Ö ×➘❂➮❾➬❘➘●Ø✷Ù➑❒●Ú❚➮❳➮❳➴▼❒❱➷✟Û❩Ü❴➴❛❐✈➴❴➷■❒➒➬❳➴❛➘●➷ÞÝ✈ß➞ÓÖ ➪✰✃❩àáÔ⑥â➇ã❾ä❘å➡æáç è➂é➡ê▼ë✦ì✈í❲î➒ï❚ë✽ðñé❯î❱ð❘òáí✈òáí❪é❯î❱óõô➒í✕ðñé❯í❿ö●÷✝êùø✗ë➣ú❯î♣û☛ê❄ï➡ö➒ü✙òáí✈ò✷ê❄ó❄ó❆é❚î➒ô➒í✂ý✮ð➣ê❄ì✈í☛ë✗þñÿ ✃✁￾ ì✄✂➒÷❳í ☎ï✝✆❱ï✞✂❱ò✷ï❯ë✂î➒ï➂ø ðñé❯í✓û✟✂➒ë✝ð✠✂☛✡✕ë☞✂❱óõôê❄ï❂ö➇ð❄é❚í➑ú➂÷✌✂✎✍✭ó❴í✭ì➔ò✷ê❄ó❄ó✫ê❄ï❀û✭÷❳í❲îë✭í✏✍☞✑➇î✒✡☛î●û✭ð✓✂❱÷✔✂☛✡✖✕✘✗♣ý þñÿ✘✙ ￾✎✚✜✛ð✛ê▼ë✙î❲ú●ú❚î➒÷❳í✭ï❚ð✻ðñé❯î❱ðùü✷î➒ð✛ó❴í❲îë✝ð✢✡☞✂❱÷✻ú❀÷✥î●û✭ð➞êùû❲î➒ó■ð❄é❖÷❳í❲í❸ø❱ê❄ì✈í✭ï❯ë✝ê✣✂❱ï❀î➒ó●ú❀÷✟✂✎✍✭ó❴í☛ì✽ë❲ü✢✡☛î➒ë❲ð❺í✭÷ ì✕í☛ðñé✤✂øë❪î➒÷✥í❿ï➂í✥í❲ø♣í✥ø✚ ✥ ✦★✧✪✩✬✫☞✭✯✮✞✰✬✱✳✲✴✧✠✰✬✫☞✵✶✱✸✷✱✹✰✬✺✼✻✾✽✿✩ ❀♦➥❑➤ ❁❃❂✻➵❅❄❇❆❸➩ ❈❊❉✣❋✞❉✣❋ ●☞❍❅■❑❏▼▲✣■◆▲✣❖◗P❘●❙❍❅■◆P✢❚❱❯▼❚❲P✢■◆❳❨■❑▲✣❩✞❍ ➺❀➻❂➼❄➽❆➾❭❬ ❪➷❚➮❑➬❳Û❒●Ð✗➘❱Ø✷➮❑➘♣Ü❴❫➡➴❛➷✤❵ ❛❝❜❅❞◆❞ Ô❢❡❤❣ Ñ✢Û➑➮❳➘●Ü✐❫●Û ❥ ❥✞❦❜ Ô ❜❞❑❞✳❧ ❡❤❣ ♠✒♥ ➮❑➬✥❒❱❮❳➬❳➴❛➷✤❵✽Øñ❮❳➘♣❐ ❒●➷✓❒●❮Ï ➴❴➬❳❮❲❒❱❮✟♦ ❜ ßq♣sr✟t♣à☞✉ ✈Û☎Û✭❰➡➹❀Û◆✇☛➬ ❜ ß✣♣sr ❦ ➱②①à ➱ ❜ ßq♣➂à✁✉ è➂é❯î❱ð✑ê▼ë❲ü✛òáí☎í✟③☛ú❯í❲û✭ð✷ð❄é❚í❸ë☞✂❱ó☎ð➣ê✣✂➒ï④✂☛✡❸ð❄é❚í✺ð➞ê❄ì✕í☎ø●í❺ú❚í☛ï➂ø●í✭ï❚ð❀ú❀÷✟✂✎✍✭ó❴í☛ì ð✓✂✈û⑤✂➒ï■ô í✭÷➞ö❂í➑ð✓✂✽ðñé❯í ë❙✂➒ó☎ð➣ê✣✂➒ï⑥✂☛✡➑ðñé❯í❿ë✝ðí❲î●ø✘✑✕ë✝ðî❱ðí♦ú❀÷✌✂⑦✍☛ó❛í☛ì✚ ⑧❭⑨❶⑩✁❷❘❸ ❹✬⑨❨❺✌❻▼⑩❽❼☛⑨✝❾➀❿⑦➁➃➂⑦❷✘❺➅➄❨➆s➄❶⑩❽❼☛⑨✝❾ ➇♦✓❒♣Ð❯Ð❖➴❛➷✤❵✂➬✥Ò❯Û❿➬✥➴❴❐✈Û✈Ð❖Û✭➹❀Û✭➷■Ð❖Û✭➷❂➬❡➬✥Û✭❮✥❐➉➈◆➊ ➈◆➋ ❣❀Ñ✢Û❪Ò❚❒❱❫♣Û❿➷❯➘➒Ñ ❒✂➹❚❒●❮✥❒Ï ➘♣Ü❴➴➌✇➑Û◆➍❂Ú❚❒❱➬❳➴❛➘●➷❊✉ ❪Ø ➬❳Ò❚Û Ï ➘♣Ú❯➷❚Ð❯❒●❮✌♦➎✇☛➘●➷■Ð❖➴❭➬✥➴❴➘♣➷❚➮❿❒●❮❳Û➐➏●Û❩➹❖➬➒➑❯❰❖ÛÐ❶❣✡Ñ✢Û✕Û☛❰❖➹❀Û◆✇☛➬☎➬❳Ò❯Û✟➮❳➘●Ü❛Ú❖➬✥➴❴➘♣➷➠➬✥➘➃➓✌✇✭➘●➷✝❫●Û❩❮✌❵♣Û◆➔ ➬❳➘➊➬✥Ò❯Û④➮❑➬❳Û❩❒♣Ð❨♦❈➮❑➬✥❒❱➬❳Û④➮❳➘●Ü❛Ú❖➬❳➴❛➘●➷ ßñ➴→✉ Û✎✉✐❣ ➈❑➊ ➈◆➋ Ô➣t❂à✁✉↕↔❘Û❑✇✭❒●Ü❴Ü❘➬✥Ò❚❒➒➬✕➬✥Ò❯Û ➬✥➴❴❐✈Û④Ð❖Û❩➹■Û❩➷❚Ð❖Û✭➷❂➬ Ò❯Û❒➒➬✂➬✥❮✥❒●➷❚➮❾ØñÛ❩❮✂➹❯❮✥➘Ï Ü❛Û✭❐ ➴▼➮✂❐✈➘❖Ð❖Û✭Ü❛Û❩Ð Ï ♦❈➬❳Ò❯➴▼➮✟Û❑➍❂Ú❚❒➒➬✥➴❴➘♣➷❊✉ ❪➷Õ➬❳Ò❯➴▼➮✔✇❩❒●➮❳Û✎❣♦Ñ✢Û④Û☛❰❖➹❀Û◆✇✝➬ ➬❳Ò❚Û☎➬❳Û❩❐✦➹❀Û✭❮❲❒➒➬✥Ú❯❮✥Û☎➬❳➘✂➮❳Û☛➬❳➬❳Ü❛Û☎➬❳➘✟❒✟➮❾➬✥Û❩❒●Ð✤♦✗➮❑➬✥❒➒➬✥Û❪Ð❖➴▼➮❾➬✥❮❳➴ÏÚ❖➬✥➴❴➘♣➷ ➹❯❮✥➘❲❫❂➴▼Ð❖ÛÐ✟➬✥Ò❚❒➒➬❡➬❳Ò❯Û❿Ò❚Û❩❒➒➬ ➮❳➘●Ú❯❮⑤✇☛Û⑦❣❨❡❤❣❯❒●➷❚Ð✗➬❳Ò❯Û Ï ➘●Ú❚➷❚Ð❯❒❱❮✟♦✖✇✭➘●➷❚Ð❖➴❴➬❳➴❛➘●➷■➮❙❣❯Ð❖➘✈➷❯➘●➬❡Ð❖Û✭➹❀Û✭➷■Ð✗➘♣➷ ❦ ✉ ÿ