ࡱ> #` m bjbj\.\. .>D>DeTTTTXXXl    0 l&P p    L&N&N&N&N&N&N&$i(h*r&X    r&TT K&  T X L&  L& "JX$ D pʧ$ً + Xz#L&&0&#6Q+ .Q+,$$Q+XX%       r&r&    &    lll$llllllTTTTTT Homework 2: AE601, Computational Fluid Mechanics February 11, 2009 Due: February 18, 2009, beginning of class Problem 1: Consider the function f(x) on the interval [0,3]  EMBED Equation.DSMT4  EMBED Equation.DSMT4  Determine analytically  EMBED Equation.DSMT4 and plot it in a graph on the interval x=[0,3] Derive the following third order approximation to  EMBED Equation.DSMT4   EMBED Equation.3 , by fitting the function f to a polynomial and differentiating the resulting interpolant. Approximate  EMBED Equation.DSMT4 with the central difference scheme and the third order approximation. Determine the approximate first order derivative using the central difference AND the third order approximation and f(x) evaluated on grid points on the interval [0,3]. Use several grids with uniform grid spacing of 1, 0.3, 0.03, 0.003, and 0.0003. You can use grid points that lie outside the interval [0,3] to approximate the derivative at x=0, and x=3. Determine the average absolute error over [0,3] of the approximations as compared to the analytical derivative on all grids. Plot the average error versus the grid spacing in a log-log plot. Explain the trend of the curve. This homework must be in the Computer-homework format as described in the handout (keep it brief and to the point). Use Matlab for your programming and plotting.   012CDIZpy{|}~   ~q_#jJ hohCDCJUVaJjhohCDEHU#jHJ hohCDCJUVaJjhohCDEHU#jJ hohCDCJUVaJjhohCDU hohCD hohh?hCDCJOJQJ^JaJhV h<~5>*h<~h&+h2Q=hJ&h[ohEJ h2Q=h2Q=#12Dop}~J  | d e f g h^hgdRV & Fgdogdo 7$8$H$gdh?gdVgd2Q=gd2Q=gd2Q=e l    D E J | }    3 4 5 6 X { ô՟Ր{i\UU hohvjhohCDEHU#jJ hohCDCJUVaJhRVhRV6j hRVhEEHUjK* hoh`8 hohv hohCDhRV&g h i j k l m gdV,1h/ =!"#$% Dd 6  3 A?2ʷ>>]vvo:~lD`!dʷ>>]vvo:~  =2xMAN@RHR*1FcdPwa!RtTi bSp7^+•+̾)h|3o/$?6C"?ujF1eY__ *Ӣ#dʌ>*DHR?PSW=~.@*GGO :o {n^vaWaDǏƢ:0rzO֖ɤ'$0N8 i< A~sE{1ŖU?K=ke/<{֎3lȶxڕSkAlVl$?MAM h{I &ECLFId#&K*A4F$áӅM_YFWO0tl3C׀N{-5;sxe>!9iOS^ պaY/m%]}>W9[e=7|2hG4V6h"#}St ʈguħ]>:A9{+3}r5u0NiA8s!-ڢGO/=NA̕o׷`'гcqG_8G e|Nb䞓KTw۩ZmdUi+=4,^q5!5m>nT'[3f0f"H/| .rl ;c_{EF@vc[ l /׊[X ~8UxD%jItHQM1w8PNY:r/Y0y ]\Crq48%;pρkzWDd lb  c $A? ?3"`?2]26Kr;;^TѺ\9`!126Kr;;^TѺ\@xڥSkAMm؍UAJ5 z6 zI &Ee6 $CAl B<?' /xYfg7z8|ެ, HKBHd2q9Q◥gt㌁L3aZb,Ymg̰6q[h*DhЬ&Ehozu l}ki=궽`cNRc_dxArE@jIvq*vR_*-L"bRUðhe2Tŝ=vvc#@Z Mkr>qZ^]#>QI[^MRUza \{lN%XT ^ǫ.#(,Z l6x^ l⮹P=\,A<8! JD(4*IحObƌq@_ELox2鎘8Y?E|kQU}>lo+qv]FQ4Dd lb  c $A? ?3"`?2~4N(/ vXԳZ`!R4N(/ vXԳ>(+ xڥSkAf&& 즭Z̮6Zw"xIͮ.GI"&XK5U{Twdh}FЧ](;]A7IA_B*y8_L@u`Dd @b  c $A? ?3"`?2X@Q-;õid `!~X@Q-;õid`M(PLxڥkA{iMڄ4i m Aj{1 )إ""/"x"QSEy3]vyfU?~&?"D/Br]WFXA^NP΋hKWr27ED|.j ̝UڐHU#avL$>'m) w㭬IIvrv\6^py@{콴qhqP?f\t7k. 6Ot.BwjV#Y\S钧Xb3}c7;6R߲0q<e2>|P>>p]Ѡ\yEöXOL 0/kGԼث؊3k}[xK|l?fC)szog} 햿>Lp$TAGu=ߣW?@şQ`+^ :y.׻|x^-Fy)ix!oLy AyARSy"I>W8 b8 =d/Dd lb  c $A? ?3"`?2]26Kr;;^TѺ\9W`!126Kr;;^T $ !"#&'%(*+,-./0123456789:;<=>Root Entry F`$ً Data &WordDocument.ObjectPoolpY$ً`$ً_1251190299FpY$ًpY$ًOle ObjInfo_1251190344 FpY$ًpY$ً "#$%'()*+,-/01235 FMathType 5.0 Equation MathType EFEquation.DSMT49q, ,DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_Ole CompObj iObjInfo Equation Native 9APAPAE%B_AC_A %!AHA_D_E_E_A  f(x)==e "-x cos(5x) FMathType 5.0 Equation MathType EFEquation.DSMT49q, ,DSMT5WinAllBasicCodePages_1251190407Fpʧ$ًpʧ$ًOle  CompObj  iObjInfoEquation Native _1295784779 Fpʧ$ًpʧ$ًOle CompObjiTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   dfdx(x) FMathType 5.0 Equation MathType EFEquation.DSMT49qObjInfoEquation Native  _1295785547Fpʧ$ًpʧ$ًOle , ,DSMT5WinAllBasicCodePagesTimes New RomanSymbolCourier NewMT Extra!/'_!!/G_APAPAE%B_AC_A %!AHA_D_E_E_A   dfdx(x i ) FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfo Equation Native !1Table)}+@]1. dfdx() i =2f i+1 +3f i "6f i"1 +f i"2 6x+Ox 3 ()Oh+'0  Ѻ\@xڥSkAMm؍UAJ5 z6 zI &Ee6 $CAl B<?' /xYfg7z8|ެ, HKBHd2q9Q◥gt㌁L3aZb,Ymg̰6q[h*DhЬ&Ehozu l}ki=궽`cNRc_dxArE@jIvq*vR_*-L"bRUðhe2Tŝ=vvc#@Z Mkr>qZ^]#>QI[^MRUza \{lN%XT ^ǫ.#(,Z l6x^ l⮹P=\,A<8! JD(4*IحObƌq@_ELox2鎘8Y?E|kQU}>lo+qv]FQSummaryInformation(&DocumentSummaryInformation8.LCompObj4q( H T ` lx Run 1: M=2, zero degree anglegjacobsNormalgjacobs18Microsoft Office Word@A@B؋@@ً՜.+,0 hp  San Diego State University c Run 1: M=2, zero degree angle Title  FMicrosoft Office Word Document MSWordDocWord.Document.89q@@@ NormalCJ_HaJmH sH tH Z@Z 2Q= Heading 1$<@&5CJ KH OJQJ\^JaJ V@V 2Q= Heading 3$<@&5CJOJQJ\^JaJDA@D Default Paragraph FontRi@R  Table Normal4 l4a (k@(No Listm m 12Dop}~J|defghijkn00(0(00E0E0E0E0E0E0E 0E 000 0 0 00000@00000h00?12Dop}~J|dn 00*0*00E0E0E0E0E0E0E 0E 0E 0E 0E00D m  g m  l |35m::::::UV/X$b$+Xy@/ cqW@U 0(  B S  ?m:@en " en33333enenY\>}"귆/jdV.:$I64-M"B2XKdl}!`Q^`o() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.h hh^h`hH.h 88^8`hH.h L^`LhH.h   ^ `hH.h   ^ `hH.h xLx^x`LhH.h HH^H`hH.h ^`hH.h L^`LhH.h^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hHh^`OJQJo(hHhpp^p`OJQJ^Jo(hHoh@ @ ^@ `OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHhPP^P`OJQJ^Jo(hHoh  ^ `OJQJo(hH^`OJQJ^Jo() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.^`o() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.^`o() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.64}"/2XKYdl}M"BdV.                                                                        E|V n/H'gZ~K>ZY[ 6+g76h?y6QQ( S&+L&>'*<*R,8/T 6`8H92Q=hBEJ RRV1f[o>x<~ @7J&s[?64huV2WPo 3`vCDEND SAQtjAR3/A@0m@UnknownGz Times New Roman5Symbol3& z Arial5& cmmi103& cmr10?5 z Courier New;Wingdings"qhR&SFSF  !24cc 2QHP)?P2Run 1: M=2, zero degree anglegjacobsgjacobs,