N27 - ME – 510 NUMERICAL METHODS FOR ME II Fall 2007...

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Unformatted text preview: ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Runge-Kutta Methods n n n n 2 n+1 n n n n n t ,y t ,y h f f y = y + h f(t ,y ) + + f(t ,y ) + R 2! t y g170 g186 g119 g119 g171 g187 g119 g119 g171 g187 g172 g188 n+1 n n n y = y + a h f(t ,y ) + b h f(t*,y*) + R Taylor Series Order 2: Runge-Kutta Order 2: 2 n+1 n n n n h y = y + h f(t ,y ) + y''(t ) + R 2! t* = t n + g302 h y* = y n + g533 h Find the fractions, a, b, g302 , and g533 such that R’s are the same ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Runge-Kutta Methods g62 g64 n n n n n n n n f f f t h , y hf(t ,y ) = f(t ,y ) + h + h f(t ,y ) + ... t y g68 g69 g68 g69 g119 g119 g14 g14 g119 g119 g62 g64 n+1 n n n n n n n y = y + a h f(t ,y ) + b h f t + h , y + h f(t ,y ) + R g68 g69 Expand f (t*,y*) in Taylor Series around t n , y n : Substitute and equate the coefficients of the same powers of h: For h...
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