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Test 2 - 2009 Solutions

# Test 2 - 2009 Solutions - UNIVERSITY OF NOTRE DAME...

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Unformatted text preview: UNIVERSITY OF NOTRE DAME Department of Civil Engineering and Geological Sciences CE 30125 Computational Methods December 3, 2009 J .J. Westerink Test 2 NAME 5 0L (/7!ch 5.5 7' . 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. ﬁgmd Notes: 1.) Please highlight your answers with a box. 2.) Clearly show your steps! 3.) Taylor series: 3 7 +ﬁ—ﬂi; 21 ﬂ? ﬁx) = 1(a) + (x- egg ‘t—a x:a l _ 4d4 +m(lﬂa) Etr‘x—i aSﬁSI + (x—‘Ifﬂ 3! df I=ﬂ 4.) The last tWo sheets of the exam contain differentiation formulas that may help you to solve certain problems. Problem 1 130 pts) (a) Derive the fourth order accurate central ﬁnite difference approximation to the second derivative at node i: (b) If you were to use your derived formula to evaluate the second derivatiVe of a 4th degree polynomial ﬁrnction, state the estimated associated error for the given polynomial and very briefly explain your answer. “o0 é,'.EvéL:'/34fé i9" X; “I for-z): All?“f A310" .1912 = - {.r {6;} -307(;+(é-ﬁ_7g “‘1 ”h-L 12‘1"}. (aha f Sue Fora 4.. = Z ._I __ "'2. ‘1) than aye: foiynamr'él. mum HAVE NO 57‘! DEWn-TNE WEKEEFOM 17:95 saga g,UNNECh¢E5\$AIZt{ ~70 DERIVE BECAUSE {‘7’ S 25/25:, \M Ix.) Problem 2130 Hts! Derive a 2nd order accurate backward approximation to the 151 derivative and derive its associated error. 500: —|- 0-”)ATFAJ? (\g—XoYX-ko Ag; + Q-xuxvaIMX-legifg h 'L 2'1"};- IH. h3 5m): A4}. + Ex-xu)+(x-xlﬂA'2-F; + Ex—yojfx-x.) -9 (paw-y!) + (y—yIYx—yzﬂ T "21171 3'. 113 EunLuA'rE Q X =KL 3‘69: Aft”? [aha-hjﬁ df Ezh7‘1A1-o h 21h“- I. I13 5‘09: f—Dru + M4- ‘h 1 k ‘3' 30% “2* ﬂ ‘Vﬁ’ﬂ + w '3) ‘ 2h 3 .4 . W W m h gpcmwnLD '7 V’ EMM’L mmmnmw Problem 3 120 pts} 2 Write the fourth order accurate ﬁnite difference approximation at node (13]) to as E: where f is a func— l‘ J’ (1.1) tion of f(x,y). Draw the associated ﬁnite diﬁ‘erence molecule. a"? _ .8, - . . m '— ay {3+2 ' “6M 2.) I2 AX : - 'F' - +341; —— ' - . . [( WU“ “Jj‘ﬂ lira-UH?” +1D"‘1JJ"’Z)+YC—F"*ZJJ'TI*Y‘FHI,54I‘T'loi-lgj-rl ~0'F' z I "' JJ'H) —BEF' '.~ ' f — ; a :- dﬂ” I + HUJ-l aﬁ—UJF' +{'I'LJJ") +(—£I”JJD-l+%-F'lﬂ,j-Z ’?‘F}—¥JJ'2 “pp. 2 ' 1):] - JJ- Problem 4 lZOnts} Accdﬂ‘ﬂﬁ A fourth order’centrai approximation to the second derivatiVe is estimated using a node to node spacing equal to h "—' 5.00 2) Its. = 5.00 Eh=5.ﬂ0 = 3-42 If We decrease the grid spacing to h=0.01, estimate the error associated with the new approximation. Show how you arrived at your estimate. (t. Fm h: 5 E: g'ghqﬂ Um: pf"): 2.“:(“9 = ass??? 5 o = "' _ F2 11 0,0! E n g; #41 -.: 21%- (0.0;)"(osqsq'tﬂ 2 3372* lo” ...
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Test 2 - 2009 Solutions - UNIVERSITY OF NOTRE DAME...

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