mws_gen_ode_ppt_runge2nd(1)

# mws_gen_ode_ppt_runge2nd(1) -...

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Unformatted text preview: 1/10/2010 http://numericalmethods.eng.usf.edu 1 Runge 2 nd Order Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates Runge-Kutta 2 nd Order Method http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu 3 Runge-Kutta 2 nd Order Method Runge Kutta 2nd order method is given by ( ) h k a k a y y i i 2 2 1 1 1 + + = + where ( ) i i y x f k , 1 = ( ) h k q y h p x f k i i 1 11 1 2 , + + = For ) ( ), , ( y y y x f dx dy = = http://numericalmethods.eng.usf.edu 4 Heun’s Method x y x i x i+1 y i+1 , predicted y i Figure 1 Runge-Kutta 2nd order method (Heun’s method) ( ) h k y h x f Slope i i 1 , + + = ( ) ( ) [ ] i i i i y x f h k y h x f Slope Average , , 2 1 1 + + + = ( ) i i y x f Slope , = Heun’s method 2 1 1 = a 1 1 = p 1 11 = q resulting in h k k y y i i + + = + 2 1 1 2 1 2 1 where ( ) i i y x f k , 1 = ( ) h k y h x f k i i 1 2 , + + = Here a 2 =1/2 is chosen http://numericalmethods.eng.usf.edu 5 Midpoint Method Here 1 2 = a is chosen, giving 1 = a 2 1 1 = p 2 1 11 = q resulting in h k y y i i 2 1 + = + where ( ) i i y x f k , 1 = + + = h k y h x f k i i 1 2 2 1 , 2 1 http://numericalmethods.eng.usf.edu 6 Ralston’s Method Here 3 2 2 = a is chosen, giving 3 1 1 = a 4 3 1 = p 4 3 11 = q resulting in h k k y y i i...
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mws_gen_ode_ppt_runge2nd(1) -...

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