Runge-Kutta 4th Order Method for Ordinary …

1 Chapter Runge-Kutta 4th Order Method for Ordinary differential equations After reading this chapter, you should be able to 1. develop Runge-Kutta 4th Order Method for solving Ordinary differential equations , 2. find the effect size of step size has on the solution, 3. know the formulas for other versions of the Runge-Kutta 4th Order Method What is the Runge-Kutta 4th Order Method ? Runge-Kutta 4th Order Method is a numerical technique used to solve Ordinary differential equation of the form dy = f (x, y ), y (0) = y 0. dx So only first Order Ordinary differential equations can be solved by using the Runge-Kutta 4th Order Method . In other sections, we have discussed how Euler and Runge-Kutta methods are used to solve higher Order Ordinary differential equations or coupled (simultaneous).

2 differential equations . How does one write a first Order differential equation in the above form? Example 1. Rewrite dy + 2 y = x , y (0 ) = 5. dx in dy = f ( x, y ), y (0) = y 0 form. dx Chapter Solution dy + 2 y = x , y (0 ) = 5. dx dy = x 2 y, y (0 ) = 5. dx In this case f ( x, y ) = x 2 y Example 2. Rewrite dy ey + x 2 y 2 = 2 sin(3 x), y (0 ) = 5. dx in dy = f ( x, y ), y (0) = y 0 form. dx Solution dy ey + x 2 y 2 = 2 sin(3 x), y (0 ) = 5. dx dy 2 sin(3 x) x 2 y 2. = , y (0 ) = 5. dx ey In this case 2 sin(3 x) x 2 y 2. f ( x, y ) =. ey The Runge-Kutta 4th Order Method is based on the following yi +1 = yi + (a1k1 + a2 k 2 + a3 k 3 + a4 k 4 )h (1). where knowing the value of y = y i at xi , we can find the value of y = yi +1 at xi +1 , and h = xi +1 xi Equation (1) is equated to the first five terms of Taylor series dy 1 d2y 1 d3y ( ) ( ) (xi +1 xi ).

3 2 3. yi +1 = yi + xi , yi xi +1 x i + 2 xi , yi xi +1 xi + 3 xi , yi dx 2! dx 3! dx 4. (2). 1d y + 4 xi , yi (xi +1 xi )4. 4! dx dy Knowing that = f ( x, y ) and xi +1 xi = h dx 1 1 1. y i +1 = y i + f ( xi , y i )h + f ' ( xi , y i )h 2 + f '' ( xi , y i )h 3 + f ''' ( xi , y i )h 4 (3). 2! 3! 4! Based on equating Equation (2) and Equation (3), one of the popular solutions used is 1. y i +1 = y i + (k1 + 2k 2 + 2k 3 + k 4 )h (4). 6. Runge-Kutta 4th Order Method k1 = f ( x i , y i ) (5a). 1 1 . k2 = f xi + h, yi + k1h . 2 2 (5b). 1 1 . k 3 = f xi + h, y i + k 2 h (5c). 2 2 . k 4 = f ( x i + h, y i + k 3 h ) (5d). Example 3. A ball at 1200 K is allowed to cool down in air at an ambient temperature of 300 K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by d.

4 = 10 12 ( 4 81 108 ), (0 ) = 1200 K. dt where is in K and t in seconds. Find the temperature at t = 480 seconds using Runge- Kutta 4th Order Method . Assume a step size of h = 240 seconds. Solution d . dt (. = 10 12 4 81 10 8 ). (. f (t , ) = 10 12 4 81 10 8 ). 1. i +1 = i + (k1 + 2k 2 + 2k 3 + k 4 )h 6. For i = 0 , t 0 = 0 , 0 = 1200K. k1 = f (t0 , 0 ). = f (0,1200 ). = 10 12 (1200 4 81 10 8 ). = 1 1 . k 2 = f t0 + h, 0 + k1h . 2 2 . 1 1 . = f 0 + (240 ),1200 + ( ) 240 . 2 2 . = f (120, ). = 10 12 ( 4 81 10 8 ). = 1 1 . k 3 = f t 0 + h, 0 + k 2 h . 2 2 . 1 1 . = f 0 + (240 ),1200 + ( ) 240 . 2 2 . = f (120, ). Chapter (. = 10 12 4 81 108 ). = k 4 = f (t0 + h, 0 + k3 h ). = f (0 + 240,1200 + ( ) 240 ). = f (240, ). (. = 10 12 4 81 108 ). = 1. 1 = 0 + ( k1 + 2 k 2 + 2 k 3 + k 4 ) h 6.

5 1. = 1200 + ( + 2( ) + 2( ) + ( ))240. 6. = 1200 + ) 240. (. = K. 1 is the approximate temperature at t = t1. = t0 + h = 0 + 240. = 240. 1 = (240). K. For i = 1, t1 = 240, 1 = K. k1 = f (t1 , 1 ). = f (240, ). (. = 10 12 4 81 10 8 ). = 1 1 . k 2 = f t1 + h, 1 + k1h . 2 2 . 1 1 . = f 240 + (240), + ( )240 . 2 2 . = f (360, ). (. = 10 12 81 108 ). = 1 1 . k3 = f t1 + h, 1 + k 2 h . 2 2 . 1 1 . = f 240 + (240 ), + ( ) 240 . 2 2 . = f (360, ). (. = 10 12 4 81 10 8 ). Runge-Kutta 4th Order Method = k 4 = f (t1 + h, 1 + k3 h ). = f (240 + 240, + ( ) 240). = f (480, ). = 10 12 ( 4 81 10 8 ). = 1. 2 = 1 + ( k1 + 2 k 2 + 2 k 3 + k 4 ) h 6. 1. = + ( + 2( ) + 2( ) + ( )) 240. 6. 1. = + ( ) 240. 6. = K. 2 is the approximate temperature at t = t2. = t1 + h = 240 + 240. = 480. 2 = (480). K. Figure 1 compares the exact solution with the numerical solution using the Runge-Kutta 4th Order Method with different step sizes.

6 1600. Temperature, (K). 1200. h=120. 800 Exact h=240. 400. h=480. 0. 0 200 400 600. -400. Time,t(sec). Figure 1 Comparison of Runge-Kutta 4th Order Method with exact solution for different step sizes. Chapter Table 1 and Figure 2 show the effect of step size on the value of the calculated temperature at t = 480 seconds. Table 1 Value of temperature at time, t = 480 s for different step sizes Step size, h (480) Et | t | %. 480 240 120 60 30 800. Temperature, (480). 600. 400. 200. 0. 0 100 200 300 400 500. -200. Step size, h Figure 2 Effect of step size in Runge-Kutta 4th Order Method . In Figure 3, we are comparing the exact results with Euler's Method ( Runge-Kutta 1st Order Method ), Heun's Method ( Runge-Kutta 2nd Order Method ), and Runge-Kutta 4th Order Method . The formula described in this chapter was developed by Runge.

7 This formula is same as Simpson's 1/3 rule, if f ( x, y ) were only a function of x . There are other versions of the 4th Order Method just like there are several versions of the second Order methods . The formula developed by Kutta is 1. y i +1 = y i + (k1 + 3k 2 + 3k 3 + k 4 )h (6). 8. where k1 = f (xi , yi ) (7a). 1 1 . k 2 = f xi + h, yi + hk1 (7b). 3 3 . 2 1 . k 3 = f xi + h, yi hk1 + hk 2 (7c). 3 3 . k 4 = f (xi + h, y i + hk1 hk 2 + hk 3 ) (7d). Runge-Kutta 4th Order Method This formula is the same as the Simpson's 3/8 rule, if f ( x, y ) is only a function of x . 1400. Temperature, (K) 1200. 4th Order 1000. 800. Exact 600. Heun 400. 200 Euler 0. 0 100 200 300 400 500. Time, t(sec). Figure 3 Comparison of Runge-Kutta methods of 1st (Euler), 2nd, and 4th Order . Ordinary differential equations .

8 Topic Runge-Kutta 4th Order Method Summary Textbook notes on the Runge-Kutta 4th Order Method for solving Ordinary differential equations . Major General Engineering Authors Autar Kaw Last Revised October 13, 2010. Web Sit