Friday, January 21, 2011

runge kutta method

Runge-Kutta Method of Order $ 4$

We are now ready to state the algorithm for using the Runge-Kutta method of order $ 4$ . This is a generalization of the R-K method of order $ 2$ to higher order methods. Without getting into analytical details, we state the R-K method of order $ 4$ . It is widely used algorithm. For the IVPs ([*]), set $ x_i = a + i h, $ for $ i=0,1,2, \ldots, n$ where $ h = \displaystyle \frac{b-a}{n}$ . Also, we set $ y(a) = y_0$ . The for $ k = 0,1,2,\ldots, n-1$ , we define

$\displaystyle y_{k+1} = y_k + \frac{k_1 + 2 k_2 + 2 k_3 + k_4}{6},$

where

$\displaystyle k_1 = h f(x_k, y_k), \; k_2 = h f(x_k + \frac{h}{2}, y_k + \frac{... ...{h}{2}, y_k + \frac{k_2}{2}) \; {\mbox{ and }} k_4 = h f(x_k + h, y_k + k_3).$
Posted by at

1 comment:

  1. GuruKPOJanuary 21, 2011 at 10:17 PM

    will u tell me why it is called Runge Kutta method,,and also give an example to explain it more n more..

    ReplyDelete

Subscribe to: Post Comments (Atom)