Runge-Kutta 2nd Order Method Notes

Runge-Kutta 2nd Order Method Notes - Chapter 08.03...

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Unformatted text preview: Chapter 08.03 Runge-Kutta 2nd Order Method for Ordinary Differential Equations After reading this chapter, you should be able to : 1. understand the Runge-Kutta 2nd order method for ordinary differential equations and how to use it to solve problems. What is the Runge-Kutta 2nd order method? The Runge-Kutta 2nd order method is a numerical technique used to solve an ordinary differential equation of the form ( 29 ( 29 , , y y y x f dx dy = = Only first order ordinary differential equations can be solved by using the Runge-Kutta 2nd order method. In other sections, we will discuss how the Euler and Runge-Kutta methods are used 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 1 Rewrite ( 29 5 , 3 . 1 2 = = +- y e y dx dy x in ) ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 , 3 . 1 2 = = +- y e y dx dy x ( 29 5 , 2 3 . 1 =- =- y y e dx dy x In this case 08.03.1 08.03.2 Chapter 08.03 ( 29 y e y x f x 2 3 . 1 ,- =- Example 2 Rewrite ( 29 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y in ) ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 ), 3 sin( 2 2 2 = = + y x y x dx dy e y ( 29 5 , ) 3 sin( 2 2 2 =- = y e y x x dx dy y In this case ( 29 y e y x x y x f 2 2 ) 3 sin( 2 ,- = Runge-Kutta 2 nd order method Eulers method is given by ( 29 h y x f y y i i i i , 1 + = + (1) where = x ) ( x y y = i i x x h- = + 1 To understand the Runge-Kutta 2nd order method, we need to derive Eulers method from the Taylor series. ( 29 ( 29 ( 29 ... ! 3 1 ! 2 1 3 1 , 3 3 2 1 , 2 2 1 , 1 +- +- +- + = + + + + i i y x i i y x i i y x i i x x dx y d x x dx y d x x dx dy y y i i i i i i ( 29 ( 29 ( 29 ... ) , ( ' ' ! 3 1 ) , ( ' ! 2 1 ) , ( 3 1 2 1 1 +- +- +- + = + + + i i i i i i i i i i i i i x x y x f x x y x f x x y x f y (2) As you can see the first two terms of the Taylor series ( 29 h y x f y y i i i i , 1 + = + are Eulers method and hence can be considered to be the Runge-Kutta 1st order method. The true error in the approximation is given by ( 29 ( 29 ... ! 3 , ! 2 , 3 2 + + = h y x f h y x f E i i i i t (3) So what would a 2nd order method formula look like. It would include one more term of the Taylor series as follows. Runge-Kutta 2nd Order Method 08.03.3 ( 29 ( 29 2 1 , ! 2 1 , h y x f h y x f y y i i i i i i + + = + (4) Let us take a generic example of a first order ordinary differential equation ( 29 5 , 3 2 =- =- y y e dx dy x ( 29 y e y x f x 3 , 2- =- Now since y is a function of x , ( 29 ( 29...
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This note was uploaded on 05/01/2011 for the course CHBE 2120 taught by Professor Gallivan during the Spring '07 term at Georgia Institute of Technology.

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Runge-Kutta 2nd Order Method Notes - Chapter 08.03...

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