Ch E 310 - Fall 10 - Lecture 22

Ch E 310 - Fall 10 - Lecture 22 - Lecture 22 November 11,...

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Lecture 22 – November 11, 2010 Agenda: Initial Value Problems (solving one type of ODE) Midpoint Method Runge-Kutta Methods o Second Order o Fourth Order Systems of Equations o Euler’s Method o Fourth Order Runge-Kutta Work on In-class 9 (from last time) and/or HW 5
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Ordinary Differential Equations We are looking at solving “initial value problems” (ODEs) that have the general form: From the perspective of a “forward” method with an initial condition specified, this requires solving for subsequent time points y ( t ) As we saw last time, we can develop an iterative structure to accomplish this: new value = old value + ( slope ) × ( time increment )   y t f dt dy , h y y i i 1
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Midpoint Method Another modification of Euler’s Method uses the Euler procedure to predict y at the midpoint of the interval: This is then used to predict the slope at the midpoint: Finally, we calculate an estimate of y i+1 a full step forward: This method is related to the simplest Newton-Cotes integration formula (see Chapra Table 17.4)   2 , 2 / 1 h y t f y y i i i i   2 / 1 2 / 1 2 / 1 , i i i y t f y   h y t f y y i i i i 2 / 1 2 / 1 1 ,
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Midpoint Method Graphically, the Midpoint Method looks like this: The Midpoint Method is more accurate than the Euler Method, but has similar order error
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(slope) is an increment function ; generally: where a ’s are constants and k ’s are: and p ’s and q ’s are constants Note that the k ’s are recurrence relationships (lower k ’s appear in higher k equations) Runge-Kutta Methods n n k a k a k a ... 2 2 1 1         1 2 1 11 1 3 2 21 1 22 2 1 1,1 1 1,2 2 1, 1 1 , , , , ii n i n i n n n n n k f t y k f t p h y q k h k f t p h y q k h q k h k f t p h y q k h q k h q k h
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Second order Runge-Kutta formula : Second-order Taylor Series: The above two should match in order to properly apply the approximation (find values for a and k ) Second Order Runge-Kutta   h k a k a y y i i 2 2 1 1 1     2 1 2 , , h y t f h y t f y y i i i i i i
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Second order Runge-Kutta formula (result): where We find a 1 , a 2 , p 1 , and q 11 by setting the first equation equal to a second-order Taylor Series This gives (full derivation in the text): Second Order Runge-Kutta   h k a k a y y i i 2 2 1 1 1     h k q y h p t f k y t f k i i i i 1 11 1 2 1 , , 2 1 2 1 1 11 2 1 2 2 1 q a p a a a
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Ch E 310 - Fall 10 - Lecture 22 - Lecture 22 November 11,...

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