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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
labi
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
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.. 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 Conn = all43532):  (Alllll
a _. CAxl CAmn m. _  4
EA“ (Am2)" — (131.1)“ (Ami) = 4.320 _ CA3; _ CA3}: t}
EAmn — (Am2y! _ (Anny; (AT2) = —5.157 * 103 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 + Ill1' + £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 mm~._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
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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
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........ CM?) 7“ O WW“ W _ MW m w
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WWW ﬂ WW _ __
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 Fall '08
 Westerink,J

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