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Runge-Kutta 2nd Order Method Notes

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

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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 0 0 , , 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 0 , 3 . 1 2 = = + - y e y dx dy x in 0 ) 0 ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 0 , 3 . 1 2 = = + - y e y dx dy x ( 29 5 0 , 2 3 . 1 = - = - y y e dx dy x In this case 08.03.1
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08.03.2 Chapter 08.03 ( 29 y e y x f x 2 3 . 1 , - = - Example 2 Rewrite ( 29 5 0 ), 3 sin( 2 2 2 = = + y x y x dx dy e y in 0 ) 0 ( ), , ( y y y x f dx dy = = form. Solution ( 29 5 0 ), 3 sin( 2 2 2 = = + y x y x dx dy e y ( 29 5 0 , ) 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 Euler’s method is given by ( 29 h y x f y y i i i i , 1 + = + (1) where 0 0 = x ) ( 0 0 x y y = i i x x h - = + 1 To understand the Runge-Kutta 2nd order method, we need to derive Euler’s 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 Euler’s 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.
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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 0 , 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 ( 29 dx dy y y x f x y x f y x f + = , , , (5) ( 29 ( 29 [ ] ( 29 y e y e y y e x x x x 3 3 3 2 2 2 - - + - = - - - ( 29 y e e x x 3 ) 3 ( 2 2 2 - - + - = - - y e x 9 5 2 + - = - The 2nd order formula for the above example would be ( 29 ( 29 2 1 , ! 2 1 , h y x f h y x f y y i i i i i i + + = + ( 29 ( 29 2 2 2 9 5 ! 2 1 3 h y e h y e y i x i x i i i + - + - + = - -
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