Test 2 - UNIVERSITY OF NOTRE DAME Department of Civil...

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UNIVERSITY OF NOTRE DAME Department of Civil Engineering and Geological Sciences CE 30125 Computational Methods December 4, 2008 J.J. Westerink Test 2 NAME ___________________________________________________________ This exam is being conducted under the HONOR CODE. Please sign your exam to indicate that you agree and have adhered to all the stipulations of the honor code. Signed ____________________________________________________________ Notes: 1.) Please highlight your answers with a box. 2.) Clearly show your steps! 3.) Taylor series: 4.) The last two sheets of the exam contain differentiation formulas that may help you to solve certain problems. fx  fa xa df dx ----- = 2 2! ------------------ d 2 f 2 ------- = ++ = 3 3! d 3 f 3 = 1 4! ---- 4 d 4 f ------ x = a x 
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2 Problem 1 (30 pts) Derive the first order accurate backward finite difference approximation to the third derivative at node j: Use interpolating polynomials to derive your result.
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4 Problem 2 (30 pts) Derive a 2 nd order accurate central approximation to the 2 nd derivative and its associated error.
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6 Problem 3 (20 pts) Write the second order accurate finite difference approximation at node (i,j) to where f is a func- tion of f(x,y) 2 f x y ----------- ij 
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7 Problem 4 (20pts) A second order central approximation to the second derivative is estimated using 3 points and a node to node spacing equal to h = 5. If we decrease the grid spacing to h=1, estimate the error associated with the new approximation. f h 5 = 2  23.62 E h 5 = 5.42 =
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8 TABLE OF DIFFERENCE APPROXIMATIONS • First Derivative Approximations • Forward difference approximations: , , ,
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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Test 2 - UNIVERSITY OF NOTRE DAME Department of Civil...

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