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Final Exam - 2008 Solutions

# Final Exam - 2008 Solutions - 2008 FINAL EXAM SOLUTIONS...

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Unformatted text preview: 2008 FINAL EXAM SOLUTIONS December 15, 2008 Problem 1: Solve the system of equations using LU dccampasitian. M'ake sure that you solve these equations such that accuracy is ensured. .711 —~ mg + 39:3 = 6 211:1 m \$2 + \$3 =- 4 —3.‘1‘1 + 432 —' \$3 = 5 Rearranging for diagonal dominance: L.U a A: IJID EEEE lab-i Lam \___/ II E Hml'J a»; mil—a I—I \..___,/ 1 U 0 U11 1L1 €21 l U U 'uz £31 132 0 0 D “as Solving for U and L: 1 O U 121 1 U 531 1332 0 (LU).X = B L(UX) = B UX = Y LY = B Solving for Y: 111 1112 1113 And ﬁnally, UK 2 Y. so we can solve for X: 4 11 a 5 l0 preamsm #2 5 0.9a SYSTEM ...l.-0?K5. . .HKE ,5; max)" . E!) .[U :w' a.le ms.» ~th CL -_ E'+ :11: 1C: + CM /~../‘4m;x Foam: . l 0 O. o 0 l — 2% __; o o .0. I. 47%. 7‘ 0 _o . 0. l '7-‘35 4* ' o . 9.. 0 I “23% o . ..0. 0 0 o I 24m. 9801!; 5 .. . jﬂ 7+ CHI,ij '0 9 etc». '..;@i:>o' 0”” 4‘45 '-"— f. CST) 7+. ' JCéNEw' :0 " ' < 2W)" ' j ? .." CIT—DJ + [2" Atpjgg/ ’.C;'_~I-JJJ ﬂax) ass—‘Jwejcﬁ? 0.. ‘wa .+(ZT‘E)CSLP. - 0L1] Cs gems? #3 3/». swans mm Ex murmurs” iffy" \ :(a g I l! 5 :q a .. . I .3 :1 .9 . '2 \:0 6 l 5'? Nan£ IS: BI?— "is I “u of; “m °H I on! 01} ~71, xzq _ i=5 4. N ’1? N N i. ..sj' "Io—1mm ; 1W _.= ' 2‘ '. __'5 11.; . (an; 'Zula4‘u‘). "zu'°'* - Z7. . 5(Uu day-W; _+ 2(%- . z2 2‘ 5 Us - 10w“: x 2 (an ‘Zuu 22 ._ . 22- --U7"' qua-Bus 0 ' I (2)(?—) .. .. . ‘uu *‘fum 3“? _—_ O (211) I 3”!“ VUT+ML :: . (13(1) 3.. _3um “Hum! 4U“) =._ O h ' (73(2) Ur =1” (1;, =(0 .. . agave qt: 7".(9 . Urﬁ‘i . . 149:7 .. LUM "7:7 1115.5", Problem 4: You compute a pollutant concentration at a point using a numerical solution to the transient diﬂnsion equation, using Spatial grid size 111 = 20m. Your solution equals CM 2 23.752 ppt. You now compute another solution using a ﬁner grid ha a 5m and. you get CM = 22.437 ppt, Assume that spatial discretization errors dominate temporal discretization error. Also assume that your spatial discretization is fourth orcler accurate. Derive the appropriate formula that estimates the error to both coarse and ﬁne grid solutions. Compute error estimates to both the coarse and ﬁne grid solutions. 9 || CAI + 52(Aa')" + CL. CAI, + a(Aa:1)4 + Cg CAM + 0:028:32)“ + ll Solving for a: CAM + Q(Am1)4 = 0332 + 05(Amalll 0M1 Con-n = all-43532): - (Alllll a _. CAxl CAmn m. _ - 4 EA“ (Am-2)" — (131.1)“ (Ami) = 4.320 _ CA3; _ CA3}: t} EAmn — (Am-2y! _ (Anny; (AT2) = —5.157 * 10-3 Problem 5: Derive a Hemite interpolating polynomial and its associated error estimate for the following data: At mu m 22 f0 — A é” m B r) — c 9(5'3) = ﬂu + Ill-1' + £12912 9" — a1 + ﬂags; H 2 2&2 9"(1') = 2&2 2&2 —- C _ C 0.3 ‘2— al = B — 20 on = A—(23—40)—2C m-m~._WWWWW_Wmqu mum WW "‘ A *— Bwlzszl) - C_Cx:_2 )L 3 WW “WWI m. m m. M w! 1_ e (ﬂag {5:43 __ M I G _. W h __m __ WWW - EQQLZMWWW M W m WWW W} 8C 19mm:— D w) Wm mm .m M I [3 L m MMMMMMMMMMMMM Mg? ......... . 2 ‘ H . N ‘ w . H ‘ ‘ u ‘ u _ u m u _ ﬂ n m ‘ H m mm 3; ‘ m M m m W mmmmmmmmmmmmm wwwww “We 0 “““““““““““““ ‘”(-I}’”“‘ """" "MG “WW” "m" ' ' " ‘ ‘ " ‘ “ ' ” ” ‘ W "m Wm mawgmmr Wm) M m m M m m ........ CM?) 7-“ O WW“ W _ MW m w [ .............................. "a?§.wmw_*‘,wwmmm mm mm w _ .W WWW ﬂ WW _ __ WwJL w m V A# M H , M, Wm ...
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