N37 - ME – 510 NUMERICAL METHODS FOR ME II SYSTEM OF...

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Unformatted text preview: ME – 510 NUMERICAL METHODS FOR ME II SYSTEM OF ODE’s Consider: d y(x) = f x, y(x), z(x) dx d z(x) = g x, y(x), z(x) dx , y(0) = y0 z(0) = z0 , All the methods discussed so far are applicable Euler’s Method: y n 1 = y n + h f x n , y n , zn zn 1 = zn + h g x n , y n , zn 3URI 'U )DUXN $UÕQo Fall 2007 ME – 510 NUMERICAL METHODS FOR ME II SYSTEM OF ODE’s Runge-Kutta Order Four: y n 1 = y n + zn 1 = zn + h 6 h 6 K1 + 2 K 2 + 2 K 3 L1 + 2 L2 + 2 L3 + K4 + L4 K1 = f x n , y n , z n h h h· § K 2 = f ¨ x n + , y n + K1, zn + L1 ¸ 2 2 2¹ © h h h· § K 3 = f ¨ x n + , y n + K 2 , zn + L 2 ¸ 2 2 2¹ © L1 = g x n , y n , zn h h h· § L 2 = g ¨ x n + , y n + K1, zn + L1 ¸ 2 2 2¹ © h h h· § L3 = g ¨ x n + , y n + K 2 , zn + L 2 ¸ 2 2 2¹ © K 4 = f x n 1, y n +h K 3 , zn + h L3 3URI 'U )DUXN $UÕQo L 4 = g x n 1, y n +h K 3 , zn + h L3 Fall 2007 ...
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