Lecture 39 - There are Local truncation errors - error from...

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Unformatted text preview: There are Local truncation errors - error from application at a single step Propagated truncation errors - previous errors carried forward The sum is global truncation error Truncation Errors x y o x i x i+1 y i y i+1 Local error x i x i+1 x i+2 y i y i+1 Global & Local Errors Global error x y o Error accumulates n*h 2 => h Eulers method uses Taylor series with only first order terms The true local truncation error is Approximate local truncation error - neglect higher order terms (for sufficiently small h ) ) ( ... ! ) , ( 1 n 2 i i t h O h 2 y t f E + + + = Eulers Method ) ( ! ) , ( 2 2 i i a h O h 2 y t f E = = Runge-Kutta Methods Higher-order Taylor series methods-- need to compute the derivatives of f(t,y) Runge-Kutta Methods-- estimate the slope without evaluating the exact derivatives Heuns method Midpoint (or improved polygon) method Third-order Runge-Kutta methods Fourth-order Runge-Kutta methods Improvements of Eulers method - Heuns method Eulers method- derivative at the beginning of interval is applied to the entire interval Heuns method uses average derivative for the entire interval A second-order Runge-Kutta Method Heuns Method Heuns Method Heuns method is a predictor-corrector method Predictor Corrector (may be applied iteratively) h y t f y y i i i 1 i ) , ( + = + h 2 y t f y t f y y 1 i 1 i i i i 1 i ) , ( ) , ( + + + + + = Heuns Method Heuns Method Predictor Corrector Iterate the corrector of Heuns method to obtain an improved estimate Heuns Method with Iterative Correctors Heuns Method with Iterative Correctors Heuns Method with Iterative Correctors...
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Lecture 39 - There are Local truncation errors - error from...

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