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2nd order Runge (Dec 3) - December 3 2009*FINAL WILL BE...

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December 3, 2009 **FINAL WILL BE THURSDAY DECEMBER 17, 2009 at 5:30pm **REVIEW ON FRIDAY DECEMBER 11, 2009 at 2:00pm** Tentative** Parameters (the higher the order, the more parameters) a 1 , a 2 , … a n p 1 , p 2 , … p n 2 nd order Runge-Kutta 𝑥𝑥 ( 𝑡𝑡 + 𝛥𝛥𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) + 𝛥𝛥𝑡𝑡𝛷𝛷 𝛷𝛷 = 𝑎𝑎 1 𝑘𝑘 1 + 𝑎𝑎 2 𝑘𝑘 2 𝑘𝑘 1 = 𝑓𝑓 ( 𝑥𝑥 ( 𝑡𝑡 ), 𝑡𝑡 ) 𝑘𝑘 2 = 𝑓𝑓 ( 𝑥𝑥 ( 𝑡𝑡 ) + 𝑞𝑞 11 𝑘𝑘 1 𝛥𝛥𝑡𝑡 , 𝑡𝑡 + 𝑝𝑝 1 𝛥𝛥𝑡𝑡 ) q 11 Exact x(t+ Δ t) 𝑥𝑥 ( 𝜏𝜏 ) = 𝑥𝑥 ( 𝑡𝑡 ) + 𝑑𝑑𝑥𝑥 𝑑𝑑𝜏𝜏 𝑡𝑡 ( 𝜏𝜏 − 𝑡𝑡 ) + 1 2! 𝑑𝑑 2 𝑥𝑥 𝑑𝑑𝜏𝜏 2 𝜏𝜏 = 𝑡𝑡 ( 𝜏𝜏 − 𝑡𝑡 ) 2 + 1 3! 𝑑𝑑 3 𝑥𝑥 𝑑𝑑𝜏𝜏 3 𝜏𝜏 = 𝑡𝑡 ( 𝜏𝜏 − 𝑡𝑡 ) 3 For 𝜏𝜏 = 𝑡𝑡 + 𝛥𝛥𝑡𝑡 𝑥𝑥 ( 𝑡𝑡 + 𝛥𝛥𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) + 𝑑𝑑𝑥𝑥 𝑑𝑑𝜏𝜏 𝑡𝑡 𝛥𝛥𝑡𝑡 + 1 2! 𝑑𝑑 2 𝑥𝑥 𝑑𝑑𝜏𝜏 2 𝜏𝜏 = 𝑡𝑡 𝛥𝛥𝑡𝑡 2 + 1 3! 𝑑𝑑 3 𝑥𝑥 𝑑𝑑𝜏𝜏 3 𝜏𝜏 = 𝑡𝑡 𝛥𝛥𝑡𝑡 3 … ; 𝑑𝑑𝑥𝑥 𝑑𝑑𝜏𝜏 = 𝑓𝑓 ( 𝑥𝑥 ( 𝜏𝜏 ), 𝜏𝜏 ) 𝑑𝑑 2 𝑥𝑥 𝑑𝑑𝜏𝜏 2 𝜏𝜏 = 𝑡𝑡 = 𝑑𝑑 𝑑𝑑𝜏𝜏 𝑑𝑑𝑥𝑥 𝑑𝑑𝑡𝑡 �� 𝜏𝜏 = 𝑡𝑡 = 𝑑𝑑 𝑑𝑑𝜏𝜏 𝑓𝑓 �𝑥𝑥 ( 𝜏𝜏 ), 𝜏𝜏� =
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