lecture 12

lecture 12 - CE 30125 - Lecture 12 LECTURE 12 DERIVATION OF...

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CE 30125 - Lecture 12 p. 12.1 LECTURE 12 DERIVATION OF DIFFERENCE APPROXIMATIONS USING UNDETERMINED COEFFICIENTS • All discrete approximations to derivatives are linear combinations of functional values at the nodes • The total number of nodes used must be at least one greater than the order of differentia- tion to achieve minimum accuracy . • To obtain better accuracy, you must increase the number of nodes considered. • For central difference approximations to even derivatives, a cancelation of truncation error terms leads to one order of accuracy improvement f i p () a α f α a β f β a λ f λ ++ + h p ------------------------------------------------------------------ E + = p Oh
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CE 30125 - Lecture 12 p. 12.2 Forward second order accurate approximation to the first derivative • Develop a forward difference formula for which is accurate • First derivative with accuracy the minimum number of nodes is 2 • First derivative with accuracy need 3 nodes • The first forward derivative can therefore be approximated to as: • T.S. expansions about are: f i 1 () EO h 2 = Oh 2 i i+1 i+2 2 df dx ----- xx i = E α 1 f i α 2 f i 1 + α 3 f i 2 + ++ h ------------------------------------------------------------- = x i f i f i = f i 1 + f i hf i 1 h 2 2 f i 2 h 3 6 f i 3 4 + +++ = f i 2 + f i 2 i 1 2 h 2 f i 2 4 3 -- h 3 f i 3 4 + + =
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CE 30125 - Lecture 12 p. 12.3 • Substituting into our assumed form of and re-arranging • Desire and 2 nd order accuracy coefficient of must equal unity and coeffi- cients of and must vanish f i 1 () α 1 f i α 2 f i 1 + α 3 f i 2 + ++ h ------------------------------------------------------------- = α 1 α 2 α 3 h ------------------------------------ f i α 2 2 α 3 + f i 1 + α 2 2 ----- 2 α 3 + ⎝⎠ ⎛⎞ hf i 2 + 1 6 -- α 2 4 3 α 3 + h 2 f i 3 Oh 3 f i 1 f i 1 f i f i 2 α 1 α 2 α 3 h ------------------------------- 0 = α 2 2 α 3 + 1 = α 2 2 2 α 3 + h 0 =
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CE 30125 - Lecture 12 p. 12.4 • Solving these simultaneous equations , , • Thus the equation now becomes • The forward difference approximation of 2 nd order accuracy where α 1 3 2 -- = α 2 2 = α 3 1 2 = 3 2 f i 2 f i 1 + 1 2 -- f i 2 + + h ---------------------------------------------------------- 0 () f i 21 f i 1 0 f i 2 ++ = 1 6 2 4 3 1 2 ⎝⎠ ⎛⎞ h 2 f i 3 Oh 3 f i 1 3 f i –4 f i 1 + f i 2 + + 2 h ---------------------------------------------- 1
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lecture 12 - CE 30125 - Lecture 12 LECTURE 12 DERIVATION OF...

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