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COT - December 8 - Error t 2 t 3 t 4 = t 4 December 8 2009...

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December 8, 2009 Error Method Δ t 2 Explicit & Implicit Euler Δ t 3 2 nd order RK Δ t 4 4 th order RK 𝑥𝑥 ̇ = 𝑓𝑓 ( 𝑡𝑡 )[ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑅𝑅 ℎ𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒𝑒𝑒𝑅𝑅𝑒𝑒 Δ t 4 ] “Classical” 4 th order RK 𝑥𝑥 ( 𝑡𝑡 + ∆𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) + ∆𝑡𝑡Φ Φ = 1 6 𝑘𝑘 1 + 1 3 𝑘𝑘 2 + 1 3 𝑘𝑘 3 + 1 6 𝑘𝑘 𝑅𝑅 𝑘𝑘 1 = 𝑓𝑓 ( 𝑥𝑥 ( 𝑡𝑡 ), 𝑡𝑡 ) 𝑘𝑘 2 = 𝑓𝑓 ( 𝑥𝑥 ( 𝑡𝑡 ) + 1 2 ∆𝑡𝑡𝑘𝑘 1 , 𝑡𝑡 + ∆𝑡𝑡 2 ) 𝑘𝑘 3 = 𝑓𝑓 ( 𝑥𝑥 ( 𝑡𝑡 ) + 1 2 ∆𝑡𝑡𝑘𝑘 2 , 𝑡𝑡 + ∆𝑡𝑡 2 ) 𝑘𝑘 4 = 𝑓𝑓 ( 𝑥𝑥 ( 𝑡𝑡 ) + ∆𝑡𝑡𝑘𝑘 3 , 𝑡𝑡 + ∆𝑡𝑡 ) ***Go to programming xcel file for code I.C. 𝑥𝑥 1 (0) = 4 𝑥𝑥 2 (0) = 3 𝑥𝑥̇ 1 (0) = 7 𝑥𝑥 2 ̇ (0) = 5 Example 1 𝑥𝑥̈ 1 + 2 𝑥𝑥 1 ̇ 𝑥𝑥 2 + sin( 𝑥𝑥 2 ) 𝑥𝑥 1 = 0 𝑥𝑥̈ 2 + 2 𝑥𝑥 1 ̇ + 𝑥𝑥 2 𝑥𝑥 2 ̇ + cos( 𝑥𝑥 1 ) 𝑥𝑥 2 = 0 New variables
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