Numerical Differentiation

Transcription

1 Part 5 Capter 19 Numercal Derentaton PowerPonts organzed by Dr. Mcael R. Gustason II, Duke Unversty Revsed by Pro. Jang, CAU All mages copyrgt Te McGraw-Hll Companes, Inc. Permsson requred or reproducton or dsplay.

2 Capter Objectves Understandng d te applcaton o g-accuracy numercal derentaton ormulas or equspaced data. Knowng ow to evaluate dervatves or unequally spaced data. Understandng ow Rcardson etrapolaton s appled or numercal derentaton. Recognzng te senstvty o numercal derentaton to data error. Knowng ow to evaluate dervatves n MATLAB wt te d and gradent unctons. Knowng ow to generate contour plots and vector elds wt MATLAB.

3 Introducton to Derentaton Te one dmensonal orms o some consttutve laws commonly used Law Equaton Pyscal Area Gradent Flu Fourer s law dt qk d Heat conducton Temperature Heat Proportonal constant Termal conductvty t dc Fck s law J D d Mass duson Concentraton Mass Dusvty D Arcy s law d qk d Flow troug porous meda Head Flow Hydraulc conductvty Om s dv J Electrcal l Current low Voltage Current law d conductvty Newton s du vscosty t Fluds Vl Velocty Sear stress d law Hooke s law L E L Elastcty Deormaton Stress Dynamc vscosty Young s modulus

4 Derentaton Te matematcal denton o a dervatve begns wt a derence appromaton: y and as : ndependent varable s allowed to approac zero, te derence o or y: dependent varable becomes a dervatve: dy d lm 0

5 Hg-Accuracy Derentaton Formulas Taylor seres epanson can be used to generate g-accuracy ormulas or dervatves by usng lnear algebra to combne te epanson around several ponts. l l d Tree categores or te ormula nclude orward nte-derence, backward ntederence, and centered nte-derence.

6 Derentaton Tere are also backward derence and centered derence appromatons, dependng on te ponts used: Forward: ' 1 Backward: ' 1 1 Centered: O O ' O 2

7 Derentaton cont te rst order dervatve Forward derence appromaton o O O te rst order dervatve Backward derence appromaton o O O te rst order dervatve Centered derence appromaton o O O O Hger-Accuracy

8 Forward Fnte-Derence Hger-Accuracy

9 Backward Fnte-Derence

10 Centered Fnte-Derence

11 Eample /2 Q. Recall tat n E. 4.4 we estmated te dervatve o at =0.5 usng orward derences and a step sze o =0.25. Te results are summarzed n te table below. Te eact value o 0.5= Repeat te computaton wt g accuracy ormulas. Backward O Centered O 2 Forward O Estmate t 21.7% -2.4% -26.5%

12 Sol. 2 2 Eample / Forward derence o O 2 s computed as t =5.82 % Backward derence o O 2 s computed as t =3.77 % Centered derence o O 4 s computed as t =0 % Backward Centered Forward O O 2 O t 21.7% -2.4% -26.5%

13 Rcardson Etrapolaton As wt ntegraton, te Rcardson etrapolaton can be used to combne two lower-accuracy estmates o te dervatve to produce a ger-accuracy estmate. For te cases were tere are two O 2 estmates and te nterval s alved = /2, an mproved O estmate may be ormed usng: D 4 3 D D 1 For te cases were tere are two O 4 estmates and te nterval s alved 2 = 1 /2, an mproved O 6 estmate may be ormed usng: D D D 1 For te cases were tere are two O 6 estmates and te nterval s alved 2 = 1 /2, an mproved O 8 estmate may be ormed usng: D 64 D D 1

14 Eample 19.2 Q. Usng te same uncton as n E.19.1, estmate te rst dervatve at =0.5 or a step sze o 1 =0.5, and 2 =0.25. Use te Rcardson etrapolaton to compute mproved estmate. Te eact soluton s Sol. Te rst dervatve wt centered derence D t = 9.6% D t = 2.4% Usng te Rcardson etrapolaton, te mproved Estmate s 4 1 D D 2 D D

15 Unequally Spaced Data One way to calculated dervatves o unequally spaced data s to determne a polynomal l t and take ts dervatve at a pont. As an eample, usng a second-order Lagrange polynomal to t tree ponts and takng ts dervatve yelds:

16 Eample 19.3 Q. A temperature s measured nsde te sol as sown below. Compute te eat lu nto te ground at te ar-sol nterace. q z 0 dt k dz z 0 were q=eat lu W/m2, k=termal conductvty or sol =0.5 W/m K, T=TemperatureK, z=dstance measured rom te surace nto te sol K / m W W W qz mk m m 2

17 Dervatves and Integrals or Data wt Errors A sortcomng o numercal derentaton s tat t tends to amply errors n data, wereas ntegraton tends to smoot data errors. One approac or takng dervatves o data wt errors s to t a smoot, derentable uncton to te data and take te dervatve o te uncton. a Data wt no error b Resultng numercal derentaton o curve a c Data moded slgtly d Resultng numercal derentaton o curve a > Small data errors are ampled -> Small data errors are ampled by numercal derentaton.

18 Numercal Derentaton wt MATLAB MATLAB as two bult-n unctons to elp take dervatves, d and gradent: d Returns te derence between adjacent elements n dy./d Returns te derence between adjacent values n y dvded by te correspondng derence n adjacent values o

19 Numercal Derentaton wt MATLAB = gradent, Determnes te dervatve o te data n at eac o te ponts. Te program uses orward derence or te rst pont, backward derence or te last pont, and centered derence or te nteror ponts. s te spacng between ponts; omtted =1. Te major advantage o gradent over d s gradent s result s te same sze as te orgnal data. Gradent can also be used to nd partal dervatves or matrces: [, y] = gradent,

20 Vsualzaton MATLAB can generate contour plots o unctons as well as vector elds. Assumng and y represent a mesgrd o and y values and z represents a uncton o and y, contour, y, z can be used to generate a contour plot [, y]=gradentz, can be used to generate partal dervatves and quver, y,, y can be used to generate vector elds