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Lecture 39

Lecture 39 - Truncation Errors There are Local truncation...

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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

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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
Euler’s 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 + + + = Euler’s Method ) ( ! ) , ( 2 2 i i a h O h 2 y t f E = =

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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 Heun’s method Midpoint (or improved polygon) method Third-order Runge-Kutta methods Fourth-order Runge-Kutta methods
Improvements of Euler’s method - Heun’s method Euler’s method - derivative at the beginning of interval is applied to the entire interval Heun’s method uses average derivative for the entire interval A second-order Runge-Kutta Method Heun’s Method Heun’s Method

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