One-step Methodseng.sut.ac.th/me/2014/document/AutomaticControl/Num ME-2.pdfThe Heun method is a predictor-corrector approach. All the multistep methods to be discussed subsequently

One-step Methods

Numerical Solutions for ODEs

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One-step Methods( ),dy f x y

dx=

New value = old value+ slope×step size

Solving ODEs of the form:

Numerical to solve

1i iy y hφ+ = +

Mathematical term

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One-step Methods

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Taylor Methods:Euler’s method

( ),i if x yφ =At xi

( )1 ,i i i iy y f x y h+ = +

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Example25.1

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Example25.1

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Example25.1

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Effect of step size on the global truncation error

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Taylor Methods:Heun’s method

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Heun’s method

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Runge-Kutta Methods:Midpoint Method

( )1 1 2 1 2,i i i iy y f x y h+ + += +

( )1 1 2 1 2,i i iy f x y+ + +′ =

Slope at the midpoint

The midpoint method

A value of y at the midpoint of the interval

( )1 2 ,2i i i ihy y f x y+ = +

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Examplett

dtdy 2sin423 +−=Problem statement

Initial value y(t=0)=1

Exact solution

32cos23 2 +−−= ttty

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