lecture31 - Lecture 31 Higher Order Methods The order of a...

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Lecture 31 Higher Order Methods The order of a method For numerical solutions of an initial value problem there are two ways to measure the error. The ±rst is the error of each step. This is called the Local Truncation Error or LTE. The other is the total error for the whole interval [ a, b ]. We call this the Global Truncation Error or GTE. For the Euler method the LTE is of order O ( h 2 ), i.e. the error is comparable to h 2 . We can show this directly using Taylor’s Theorem: y ( t + h ) = y ( t ) + h ˙ y ( t ) + h 2 2 ¨ y ( c ) for some c between t and t + h . In this equation we can replace ˙ y ( t ) by f ( t, y ( t )), which makes the ±rst two terms of the right hand side be exactly the Euler method. The error is then h 2 2 ¨ y ( c ) or O ( h 2 ). It would be slightly more di²cult to show that the LTE of the modi±ed Euler method is O ( h 3 ), an improvement of one power of h . We can roughly get the GTE from the LTE by considering the number of steps times the LTE. For any method, if [ a, b ] is the interval and h is the step size, then n = ( b - a ) /h is the number of steps. Thus for any method, the GTE is one power lower in h than the LTE. Thus the GTE for Euler is O ( h ) and for modi±ed Euler it is O ( h 2 ). By the order of a method, we mean the power of h in the GTE. Thus the Euler method is a 1st order method and modi±ed Euler is a 2nd order method. Fourth Order Runge-Kutta
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This note was uploaded on 02/09/2012 for the course MATH 344 taught by Professor Young,t during the Fall '08 term at Ohio University- Athens.

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lecture31 - Lecture 31 Higher Order Methods The order of a...

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