Analyse numerique et optimisation : Une introduction a la by Gregoire Allaire

By Gregoire Allaire

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Extra resources for Analyse numerique et optimisation : Une introduction a la modelisation mathematique et a la simulation numerique French

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K∈Z E U IQ Ê Ñ ÖÕÙÓÒ× ÕÙ Ñ Ñ × un ×Ø ÙÒ ÓÒ Ø ÓÒ Ö ÐÐ ¸ Ð × Ó ÒØ× u ˆn (k) Ð × Ö ÓÙÖ Ö ×ÓÒØ ÓÑÔÐ Ü ×º ÍÒ ÔÖÓÔÖ Ø ÑÔÓÖØ ÒØ ÔÓÙÖ Ð ×Ù Ø Ð ØÖ Ò× ÓÖÑ ÓÙÖ Ö × ÓÒ Ø ÓÒ× Ô Ö Ó ÕÙ × ×Ø Ð ×Ù Ú ÒØ × ÓÒ ÒÓØ v n (x) = un (x + ∆x)¸ ÐÓÖ× vˆn (k) = uˆn (k) exp(2iπk∆x)º ÜÔÐ ÕÙÓÒ× Ñ ÒØ Ò ÒØ Ð Ñ Ø Ó ×ÙÖ Ð³ Ü ÑÔÐ Ù × Ñ ÜÔÐ Ø ´¾º¾µº Ú ÒÓ× ÒÓØ Ø ÓÒ׸ ÓÒ Ô ÙØ Ö Ö Ö × Ñ ¸ ÔÓÙÖ 0 ≤ x ≤ 1¸ Y L O P H C E T N −un (x − ∆x) + 2un (x) − un (x + ∆x) un+1 (x) − un (x) +ν = 0. ∆t (∆x)2 LE O ÉC À ÈÁÌÊ ¾º Å ÌÀÇ ¼ È Ö ÔÔÐ uˆ Ø ÓÒ n+1 (k) = ÙØÖ Ñ ÒØ ÉC Ð ØÖ Ò× ÓÖÑ Ø OLE N H EC ÓÙÖ Ö¸ Ð Ú ÒØ Ó Ê Æ Ë ÁÆÁ Ë E U IQ ν∆t 1− (− exp(−2iπk∆x) + 2 − exp(2iπk∆x)) u ˆn (k).

K∈Z E U IQ Ê Ñ ÖÕÙÓÒ× ÕÙ Ñ Ñ × un ×Ø ÙÒ ÓÒ Ø ÓÒ Ö ÐÐ ¸ Ð × Ó ÒØ× u ˆn (k) Ð × Ö ÓÙÖ Ö ×ÓÒØ ÓÑÔÐ Ü ×º ÍÒ ÔÖÓÔÖ Ø ÑÔÓÖØ ÒØ ÔÓÙÖ Ð ×Ù Ø Ð ØÖ Ò× ÓÖÑ ÓÙÖ Ö × ÓÒ Ø ÓÒ× Ô Ö Ó ÕÙ × ×Ø Ð ×Ù Ú ÒØ × ÓÒ ÒÓØ v n (x) = un (x + ∆x)¸ ÐÓÖ× vˆn (k) = uˆn (k) exp(2iπk∆x)º ÜÔÐ ÕÙÓÒ× Ñ ÒØ Ò ÒØ Ð Ñ Ø Ó ×ÙÖ Ð³ Ü ÑÔÐ Ù × Ñ ÜÔÐ Ø ´¾º¾µº Ú ÒÓ× ÒÓØ Ø ÓÒ׸ ÓÒ Ô ÙØ Ö Ö Ö × Ñ ¸ ÔÓÙÖ 0 ≤ x ≤ 1¸ Y L O P H C E T N −un (x − ∆x) + 2un (x) − un (x + ∆x) un+1 (x) − un (x) +ν = 0. ∆t (∆x)2 LE O ÉC À ÈÁÌÊ ¾º Å ÌÀÇ ¼ È Ö ÔÔÐ uˆ Ø ÓÒ n+1 (k) = ÙØÖ Ñ ÒØ ÉC Ð ØÖ Ò× ÓÖÑ Ø OLE N H EC ÓÙÖ Ö¸ Ð Ú ÒØ Ó Ê Æ Ë ÁÆÁ Ë E U IQ ν∆t 1− (− exp(−2iπk∆x) + 2 − exp(2iπk∆x)) u ˆn (k).

E L O Ä × ÓÒ Ø ÓÒ× ÙÜ Ð Ñ Ø × ´¾º½µ Ô ÙÚ ÒØ ØÖ ÔÐÙ× ÙÖ× ØÝÔ ×¸ Ñ × Ð ÙÖ Ó Ü Ò³ ÒØ ÖÚ ÒØ Ô × Ò× Ð Ò Ø ÓÒ × × Ñ ×º Á ¸ ÒÓÙ× ÙØ Ð ×ÓÒ× × ÓÒ Ø ÓÒ× ÙÜ ÐÑØ × Ö Ð Ø ÉC ÕÙ × ØÖ u(t, 0) = u(t, 1) = 0 ÔÓÙÖ ØÓÙØ t ∈ R+ ∗ Ù × ÒØ Ò un0 = unN +1 = 0 ÔÓÙÖ ØÓÙØ n > 0. , N }¸ C E T Y L PO Ò× ÕÙ Ù× Ñ ÑÔÐ Ø n+1 −un+1 un+1 − unj − un+1 j j−1 + 2uj j+1 +ν = 0. ∆t (∆x)2 LE O ÉC HN E U IQ ´¾º¾µ ´¾º¿µ ¾º¾º Á Ê Æ Ë ÁÆÁ Ë ÈÇÍÊ Ä³ ÉÍ ÌÁÇÆ ÁÐ ×Ø Ð Ú Ö Ö ÕÙ Ð × ¹ Ö ÕÙ³ÓÒ Ô ÙØ Ð ÙÐ Ö Ð × Ú Ð Ñ ØÖ ØÖ ÓÒ Ð ÖÖ ⎛ 1 + 2c −c ⎜ −c 1 + 2c ⎜ ⎜ º º ⎜ º ⎜ ⎝ 0 LE O ÉC À Ä ÍÊ ¿¿ E U IQ Ñ ÑÔÐ Ø ´¾º¿µ ×Ø Ø Ú Ñ ÒØ Ò Ò ¸ ³ ×ع Ò ÓÒ Ø ÓÒ × unj Ò Ø¸ Ð ÙØ ÒÚ Ö× Ö Ð ÙÖ× un+1 j Ø ÐÐ N ⎞ 0 ⎟ −c ⎟ ν∆t ⎟ ºº ºº , ´¾º µ ⎟ Ú c= º º ⎟ (∆x)2 −c 1 + 2c −c ⎠ −c 1 + 2c T Y POL ÓÒØ Ð ×Ø × Ú Ö ÓÑ Ò ×ÓÒ ÓÒÚ Ü N H EC Ä ÖÐ Ö Ø Ö Ò ÔÓ× Ø ¸ ÓÒ ÒÚ Ö× Ð º Ò ´¾º¾µ Ø ´¾º¿µ¸ ÔÓÙÖ 0 ≤ θ ≤ 1¸ ÓÒ Ó Ø ÒØ Ð θ¹× × ÒØ ÙÒ Ñ n+1 −un+1 un+1 − unj − un+1 −unj−1 + 2unj − unj+1 j j−1 + 2uj j+1 + θν + (1 − θ)ν = 0.

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