2nd order Runge (Dec 3)

2nd order Runge - q 11 =p 1 December 3 2009*F I NAL WILL BE T H URSDAY DECE at 5:30pm*REV IEW ON FR IDAY DECE at 2:00pm Tentative 2 nd order

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Unformatted text preview: q 11 =p 1 December 3, 2009 **F I NAL WILL BE T H URSDAY DECE MBER 17, 2009 at 5:30pm **REV IEW ON FR IDAY DECE MBER 11, 2009 at 2:00pm** Tentative** 2 nd order Runge-Kutta + = + xt Δt xt ΔtΦ = + Φ a1k1 a2k2 = ( , ) k1 f xt t = ( + , + ) k2 f xt q11k1Δt t p1Δt Parameters (the higher the order, the more parameters) a 1 , a 2 , … a n p 1 , p 2 , … p n q 11 Exact x(t+ Δ t) = + - + ! = ( - ) + ! = ( - ) … xτ xt dxdτtτ t 12 d2xdτ2τ t τ t 2 13 d3xdτ3τ t τ t 3 For = + τ t Δt + = + + ! = + ! = …; = ( , ) xt Δt xt dxdτtΔt 12 d2xdτ2τ tΔt2 13 d3xdτ3τ tΔt3 dxdτ f xτ τ = = = = d2xdτ2τ t ddτdxdtτ t , = ( ) + ddτfxτ τ dfdxτdx τ dτ dfdτx Aside: (Verifying ^ ^ ^) Let , = + fxτ τ 3sinx2τ 2τ = xτ 4τ2 Left hand side , = + ddτfxτ τ ddτ3 sin4τ22 2τ = + 3cos16τ464τ3 2 = + 192cos16τ4τ3 2 Right Hand Side = + dfdxτ 3cosx2τ2xτ 0 = + 3cosx22x8τ 2 = + 3cos4τ2224τ228τ 2 = + 192cos16τ4τ3 2 ( ) = + dx τ dτ 0 2 = dfdτx 8τ *E=Explicit Euler...
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This note was uploaded on 01/08/2010 for the course COT 3502 taught by Professor Hawkins during the Fall '08 term at University of Florida.

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2nd order Runge - q 11 =p 1 December 3 2009*F I NAL WILL BE T H URSDAY DECE at 5:30pm*REV IEW ON FR IDAY DECE at 2:00pm Tentative 2 nd order

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