Lecture 6: =1=Truncation Error, Consistency, and Convergence
Lecture 6: =1=Truncation Error, Consistency, and Convergence
Lecture 6: =1=Truncation Error, Consistency, and Convergence
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ExampleTruncation errorThe numerical method applied to the exact solution, y n , isgiven byz ⋆ = y n + βhf (t n , y n )z n+1 = z ⋆ + (1 − β)hf (t n + βh, z ⋆ )We first need to exp<strong>and</strong> f (t n + βh, z ⋆ ) in a Taylor series around(t n , y n ) :f (t n + βh, z ⋆ ) =f (t n , y n ) + (f t ) n (t n + βh − t n ) + (f y ) n (z ⋆ − y n )+ 1 2 β2 h 2 (f tt ) n (t n + βh − t n ) 2 + 1 2 (f yy) n (z ⋆ − y n ) 2+ βh(f ty ) n (t n + βh − t n )(z ⋆ − y n ) + O(h 3 )() Truncation <strong>Error</strong>, <strong>Consistency</strong>, <strong>and</strong> <strong>Convergence</strong> 8 October 2012 42 / 54
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