Runge-Kutta 4th Order Method For Ordinary Differential .
Chapter 08.04Runge-Kutta 4th Order Method forOrdinary Differential EquationsAfter reading this chapter, you should be able to1. 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 methodWhat is the Runge-Kutta 4th order method?Runge-Kutta 4th order method is a numerical technique used to solve ordinary differentialequation of the formdy f (x, y ), y (0) y 0dxSo only first order ordinary differential equations can be solved by using the Runge-Kutta 4thorder method. In other sections, we have discussed how Euler and Runge-Kutta methods areused to solve higher order ordinary differential equations or coupled (simultaneous)differential equations.How does one write a first order differential equation in the above form?Example 1Rewritein08.04.1dy 2 y 1.3e x , y (0 ) 5dxdy f ( x, y ), y (0) y 0 form.dx
08.04.2Chapter 08.04Solutiondy 2 y 1.3e x , y (0 ) 5dxdy 1.3e x 2 y, y (0 ) 5dxIn this casef ( x, y ) 1.3e x 2 yExample 2Rewriteeyindy x 2 y 2 2 sin(3 x), y (0 ) 5dxdy f ( x, y ), y (0) y 0 form.dxSolutiondy x 2 y 2 2 sin(3 x), y (0 ) 5dxdy 2 sin(3 x) x 2 y 2 , y (0 ) 5dxeyIn this case2 sin(3 x) x 2 y 2f ( x, y ) eyThe Runge-Kutta 4th order method is based on the followingyi 1 yi (a1k1 a2 k 2 a3 k 3 a4 k 4 )hwhere knowing the value of y y i at xi , we can find the value of y yi 1 at xi 1 , andeyh xi 1 xiEquation (1) is equated to the first five terms of Taylor series3dy1 d2y1 d3y2()()(xi 1 xi )yi 1 yi xxxx xi , yii 1ii 1i2 xi , yi3 xi , yidx2! dx3! dx41d y(xi 1 xi )4 4 xi , yi4! dxdyKnowing that f ( x, y ) and xi 1 xi hdx111y 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 42!3!4!Based on equating Equation (2) and Equation (3), one of the popular solutions used is1y i 1 y i (k1 2k 2 2k 3 k 4 )h6(1)(2)(3)(4)
Runge-Kutta 4th Order Method08.04.3k1 f ( x i , y i )(5a)11 k2 f xi h, yi k1h 22 (5b)11 k 3 f xi h, y i k 2 h 22 k 4 f ( x i h, y i k 3 h )(5c)(5d)Example 3A 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 ofthe ball is given bydθ 2.2067 10 12 (θ 4 81 108 ), θ (0 ) 1200 Kdtwhere θ is in K and t in seconds. Find the temperature at t 480 seconds using RungeKutta 4th order method. Assume a step size of h 240 seconds.Solutiondθ 2.2067 10 12 θ 4 81 10 8dtf (t , θ ) 2.2067 10 12 θ 4 81 10 81θ i 1 θ i (k1 2k 2 2k 3 k 4 )h6For i 0 , t 0 0 , θ 0 1200K()(k1 f (t0 , θ 0 )) f (0,1200 ) 2.2067 10 12 (1200 4 81 10 8 ) 4.557911 k 2 f t0 h, θ 0 k1h 22 11 f 0 (240 ),1200 ( 4.5579 ) 240 22 f (120,653.05) 2.2067 10 12 (653.05 4 81 10 8 ) 0.3834711 k 3 f t 0 h, θ 0 k 2 h 22 11 f 0 (240 ),1200 ( 0.38347 ) 240 22 f (120,1154.0 )
08.04.4Chapter 08.04()()()() 2.2067 10 12 1154.0 4 81 108 3.8954k 4 f (t0 h, θ 0 k3 h ) f (0 240,1200 ( 3.894 ) 240 ) f (240,265.10) 2.2067 10 12 265.10 4 81 108 0.00697501θ 1 θ 0 ( k1 2 k 2 2 k 3 k 4 ) h61 1200 ( 4.5579 2( 0.38347 ) 2( 3.8954 ) (0.069750 ))2406( 1200 2.1848) 240 675.65 Kθ1 is the approximate temperature att t1 t0 h 0 240 240θ1 θ (240) 675.65 KFor i 1, t1 240, θ1 675.65 Kk1 f (t1 , θ1 ) f (240,675.65) 2.2067 10 12 675.65 4 81 10 8 0.4419911 k 2 f t1 h, θ1 k1h 22 11 f 240 (240),675.65 ( 0.44199)240 22 f (360,622.61) 2.2067 10 12 622.614 81 108 0.3137211 k3 f t1 h, θ1 k 2 h 22 11 f 240 (240 ),675.65 ( 0.31372 ) 240 22 f (360,638.00 )( 2.2067 10 12 638.00 4 81 10 8)
Runge-Kutta 4th Order Method08.04.5 0.34775k 4 f (t1 h, θ1 k3 h ) f (240 240,675.65 ( 0.34775) 240) f (480,592.19) 2.2067 10 12 (592.19 4 81 10 8 ) 0.253511θ 2 θ 1 ( k1 2 k 2 2 k 3 k 4 ) h61 675.65 ( 0.44199 2( 0.31372 ) 2( 0.34775) ( 0.25351)) 24061 675.65 ( 2.0184 ) 2406 594.91 Kθ 2 is the approximate temperature att t2 t1 h 240 240 480θ 2 θ (480) 594.91 KFigure 1 compares the exact solution with the numerical solution using the Runge-Kutta 4thorder method with different step sizes.Temperature, θ(K)16001200h 120800Exacth 240400h 48000200400600-400Time,t(sec)Figure 1 Comparison of Runge-Kutta 4th order methodwith exact solution for different step sizes.
08.04.6Chapter 08.04Table 1 and Figure 2 show the effect of step size on the value of the calculated temperature att 480 seconds.Table 1 Value of temperature at time, t 480 s for different step sizesStep size, h4802401206030θ (480)Et-90.278 737.85594.91 52.660646.16 1.4122647.54 0.033626647.57 0.00086900 εt , θ(480)80060040020000100200300400500-200Step size, hFigure 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 ordermethod), Heun’s method (Runge-Kutta 2nd order method), and Runge-Kutta 4th ordermethod.The formula described in this chapter was developed by Runge. This formula is same asSimpson’s 1/3 rule, if f ( x, y ) were only a function of x . There are other versions of the 4thorder method just like there are several versions of the second order methods. The formuladeveloped by Kutta is1(6)y i 1 y i (k1 3k 2 3k 3 k 4 )h8where(7a)k1 f (xi , yi )11 k 2 f xi h, yi hk1 33 21 k 3 f xi h, yi hk1 hk 2 33 k 4 f (xi h, y i hk1 hk 2 hk 3 )(7b)(7c)(7d)
Runge-Kutta 4th Order Method08.04.7This formula is the same as the Simpson’s 3/8 rule, if f ( x, y ) is only a function of x .Temperature, θ(K)140012004th 00Time, t(sec)Figure 3 Comparison of Runge-Kutta methods of 1st (Euler), 2nd, and 4th order.ORDINARY DIFFERENTIAL EQUATIONSTopicRunge-Kutta 4th order methodSummaryTextbook notes on the Runge-Kutta 4th order method forsolving ordinary differential equations.MajorGeneral EngineeringAuthorsAutar KawLast Revised October 13, 2010Web Sitehttp://numericalmethods.eng.usf.edu
Oct 13, 2010 · 08.04.1 Chapter 08.04 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 vers
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