CSC_349_HANDOUT#35

CSC_349_HANDOUT#35 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 35 NOTES ON THE ANALYSIS OF EULER'S METHOD (Section 25.1.1 : pages 684-689 of 5 th ed. or pages 710-715 of the 6 th ed.) Initial-value problem: b x a x y x f x y = , )) ( , ( ) ( 0 0 ) ( y x y = . Exact solution at i x is denoted by ) ( i x y . Computed approximation at i x is denoted by i y . Definition i i y x y ) ( is called the global truncation error at i x . Definition If the global truncation error is O ( h k ) , the numerical method used to compute the values i y is said to be of order k (or a th k order method). The order of a method is a measure of the accuracy of the computed approximations, or of the rate of convergence of the computed approximations i y to the exact solutions ) ( i x y as 0 h . For any fixed value of the step size h , the larger the order k , the more accurate are the computed approximations. Definition A numerical method is said to be convergent (with respect to the differential equation it approximates) if 0 ) ( max lim 1 0 = i i N i h y x y . (That is, the global truncation error 0 at all grid points i x in [ a , b ] as 0 h .) The total amount of truncation error in each computed approximation 1 + i y (using any numerical method) is composed of two parts: the local truncation error is the amount of truncation error that results from a single application of a numerical method (that is, from the computation of 1 + i y from i y ), whereas the global truncation error
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2 contains the accumulated local truncation errors from all of the steps leading up to the computation of 1 + i y . Definition The local truncation error at any point 1 + i x is the amount of truncation error that would result from using a numerical method with the exact value ) ( i x y rather than the computed approximation
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CSC_349_HANDOUT#35 - COMPUTER SCIENCE 349A Handout Number...

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