RK2-4 2010 - 10/6/2010 RungeRunge-Kutta Method Math...

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10/6/2010 Math Modeling 2 (Xin Li) 1 Runge Runge-Kutta Method Kutta Method Math Modeling II 2010 Euler Euler Method Method Euler’s method can be viewed as: First estimate the derivative Then, update value () , n n 1 n n n n y h y y y t f y + = = + Note: 1. We are using only one value of f in estimating the derivative. 2. Runge-Kutta methods are based on the idea that using more values of f (at two, three, or four points) may lead to improvements. Runge Runge-Kutta Methods Kutta Methods Runge-Kutta methods are very popular because they have higher order of convergence. The idea is based on using more values of f to get better estimate of the updated value. They are single-step methods, as the Euler methods.
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10/6/2010 Math Modeling 2 (Xin Li) 2 Runge Runge-Kutta Kutta Methods Methods Let’s look at the 2 nd order more closely. 2 2 1 1 n 1 n K w K w y y + + = + () 1 n n 2 n n 1 , , K y h t hf K y t hf K β α + + = = Exact solution The initial conditions are: y t f dt dy , = 0 , y a y b t a = The Taylor series expansion of the exact solution: () ( ) . . . ! 2 2 n 2 2 n n 1 n T O H dt t y d h dt t dy h t y t y + + + = + Runge Runge-Kutta Kutta Methods Methods Expand the derivatives: [] f f f dt dy f f y t f dt d dt y d y t y t 2 2 , + = + = = So, the Taylor expansion becomes n t n n f f f h t y t hf t y t y + + + = + y t 2 n 1 n 2 1 )) ( , ( ) ( ) (
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10/6/2010 Math Modeling 2 (Xin Li) 3 Runge Runge-Kutta Kutta Methods Methods From the proposed Runge-Kutta Expend the last term at ( t n ,y n ) () hf y h t hf w hf w y y β α + + + + = + n n 2 1 n 1 n , So, . . . , y t n n T O H f hf hf f hf y h t f + + + = + + ( ) [] y 2 2 2 2 2 1 n y t 2 1 n 1 n f f h w f h w hf w w y f hf hf f h w hf w y y t + + + + = + + + + = + Comparing the Coefficients From the Runge-Kutta (evaluated at ( t n ,y n )) Taylor of true solution (evaluated at ( t n ,y ( t n ))): y 2 2 2 2 2 1 n 1 n f f h w f h w hf w w y y t + + + + = + This leads to the equations for 4 unknowns: 2 1 , 2 1 , 1 2 2 2 1 = = = + w w w w ( ) n t n n f f f h t y t hf t y t y
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This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.

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RK2-4 2010 - 10/6/2010 RungeRunge-Kutta Method Math...

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