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lecture 11

# lecture 11 - CE 30125 Lecture 11 LECTURE 11 NUMERICAL...

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CE 30125 - Lecture 11 p. 11.1 LECTURE 11 NUMERICAL DIFFERENTIATION • To find discrete approximations to differentiation (since computers can only deal with functional values at discrete points) • Uses of numerical differentiation To represent the terms in o.d.e.’s and p.d.e.’s in a discrete manner Many error estimates include derivatives of a function. This function is typically not available, but values of the function at discrete points are. • Notation • Nodes are data points at which functional values are available or at which you wish to compute functional values • At the nodes fx i  f i f i-2 f i-1 f i f i+1 f i+2 x i-2 x i-1 x i x i+1 x i+2 f(x) x node i-2 i-1 i i+1 i+2

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CE 30125 - Lecture 11 p. 11.2 • Node index i indicates which node or point in space-time we are considering (here only one spatial or temporal direction) • For equi-spaced nodal points, Taylor Series Expansion for f(x) About a Typical Node i _______________________________________________ i 0 1 2 i 2 0 21 N=22 i=1, N i=0,N hx i 1 + x i = fx  i xx i f 1 x i i 2 2! ------------------- f 2 x i i 3 3! f 3 x i ++ + = i 4 4! + f 4 x i i 5 5! f 5 x i i 6 6! f 6 x i   + +++
CE 30125 - Lecture 11 p. 11.3 •Fo r th e present analysis we will consider only the first four terms of the T.S. expansion (may have to consider more) where • If the Taylor series is convergent, each subsequent term in the error series should be becoming smaller. fx  i xx i f 1 x i i 2 2! ------------------- f 2 x i i 3 3! f 3 x i E ++ + + = E i 4 4! f 4 x i i 5 5! f 5 x i i 6 6! f 6 x i   +++ = E i 4 4! f 4 = x i x  E i 4 4! f 4 x i EO i 4

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CE 30125 - Lecture 11 p. 11.4 • The terms in the error series may be expressed • Exactly as • We note that the value of is not known • This single term exactly represents all the truncated terms in the Taylor series • Approximately as • This is the leading order truncated term in the series • This approximation for the error can also be thought of as being derived from the exact single term representation of the error with the approximation • In terms of an order of magnitude only as • This term is often carried simply to ensure that all terms of the correct order have been carried in the derivations.
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lecture 11 - CE 30125 Lecture 11 LECTURE 11 NUMERICAL...

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