2.29 Numerical Fluid Mechanics Spring 2015

Size: px

Start display at page:

Download "2.29 Numerical Fluid Mechanics Spring 2015"

Rebecca Gilbert
5 years ago
Views:

1 2.29 Sprg PFJL Lecture 2, 1

2 REVIEW Lecture Sprg 2015 Lecture 2 1. Syllabus, Goals ad Objectves 2. Itroducto to CFD 3. From mathematcal models to umercal smulatos (1D Sphere 1D flow) Cotuum Model Dfferetal Equatos => Dfferece Equatos (ofte uses Taylor expaso ad trucato) => Lear/No-lear System of Equatos => Numercal Soluto (matrx verso, egevalue problem, root fdg, etc) 4. Error Types Roud-off error: due to represetato by computers of umbers wth a fte umber of dgts (sgfcat dgts) ad to arthmetc operatos (choppg or roudg) Trucato error: due to approxmato/trucato by umercal methods of exact mathematcal operatos/quattes Other errors: model errors, data/parameter put errors, huma errors PFJL Lecture 2, 2

3 REVIEW Lecture 1, Cot d Approxmato ad roud-off errors 2.29 Sgfcat dgts: Numbers that ca be used wth cofdece Absolute ad relatve errors Iteratve schemes ad stop crtero: For dgts correct, base 10: E xˆ xˆ, a a a s a xˆ xˆ xˆ xˆ a xˆ xˆ a 1 s Number represetatos Iteger represetato Floatg-Pot represetato: Cosequece of Floatg Pot Reals: Lmted rage (uderflow & overflow ) Lmted precso (Quatzg errors) Relatve error costat, absolute error growths wth umber For t = sgfcat dgts wth roudg: x = m b e x x 2 ε = b 1-t = Mache Epslo 2.29 PFJL Lecture 2, 3

4 TODAY s Outle Approxmato ad roud-off errors Sgfcat dgts, true/absolute ad relatve errors Number represetatos Arthmetc operatos Errors of arthmetc/umercal operatos Examples: recurso algorthms (Hero, Horer s scheme) ad other examples Order of computatos matter Roud-off error growth ad ()-stablty Trucato Errors, Taylor Seres ad Error Aalyss Taylor seres Use of Taylor Seres to derve fte dfferece schemes (frst-order Euler scheme ad forward, backward ad cetered dffereces) Error propagato ad error estmato: Dfferetal Formula ad Stadard Error (statstcal formula) Error cacellato Codto umbers Referece: Chapra ad Caale, Chaps ad PFJL Lecture 2, 4

5 Arthmetc Operatos 1. Addto ad Subtracto Shft matssa of smallest umber, assumg, Result has expoet of largest umber: Absolute Error Relatve Error Ubouded for m = m 1 ± m Multplcato ad Dvso Multplcato: Add exp, multply matssa, ormalze ad chop/roud Dvso: Subtract exp, dvde matssa, ormalze ad chop/roud Relatve Error Bouded 2.29 PFJL Lecture 2, 5

6 fucto c = radd(a,b,) % % fucto c = radd(a,b,) % Dgtal Arthmetcs Fte Matssa Legth % Adds two real umbers a ad b smulatg a arthmetc ut wth % sgfcat dgts, ad roudg-off (ot choppg-off) of umbers. % If the puts a ad b provded do ot have dgts, they are frst % rouded to dgts before beg added. %--- Frst determe sgs sa=sg(a); sb=sg(b); radd.m Lmted precso addto MATLAB %--- Determe the largest umber (expoet) f (sa == 0) la=-200; %ths makes sure that f sa==0, eve f b s very small, t wll have the largest expoet else la=cel(log10(sa*a*(1+10^(-(+1))))); %Ths determes the expoet o the base. Celg s used %sce 0<log10(matssa_base10)<=-1. The 10^etc. term just %properly creases the expoet estmated by 1 the case %of a perfect log:.e. log10(m b^e) s a teger, %matssa s 0.1, hece log10(m)=-1, ad %cel(log10(m b^e(1+10^-(+1))) ~< cel(e +log10(m)+log10(1+10^-(+1)))=e. ed f (sb == 0) lb=-200; else lb=cel(log10(sb*b*(1+10^(-(+1))))); ed lm=max(la,lb); 2.29 PFJL Lecture 2, 6

7 radd.m, cotued %--- Shft the two umbers magtude to obta two tegers wth dgts f=10^(); %ths s used cojucto wth the roud fucto below at=sa*roud(f*sa*a/10^lm); %sa*a/10^lm shfts the decmal pot such that the umber starts wth 0.somethg %the f*(*) the rases the umber to a power 10^, to get the desred accuracy %of dgts above the decmal. After roudg to a teger, ay fgures that %rema below are wped out. bt=sb*roud(f*sb*b/10^lm); % Check to see f aother dgt was added by the roud. If yes, crease % la (lb) ad reset lm, at ad bt. reset=0; f ((at~=0) & (log10(at)>=)) la=la+1; reset=1; ed f ((bt~=0) & (log10(bt)>=)) lb=lb+1; reset=1; ed f (reset) lm=max(la,lb); at=sa*roud(f*sa*a/10^lm); bt=sb*roud(f*sb*b/10^lm); ed ct=at+bt; %adds the two umbers sc=sg(ct); %The followg accouts for the case whe aother dgt s added whe %summg two umbers... e. f the umber of dgts desred s oly 3, %the = 1002, but to keep oly 3 dgts, the 2 eeds to be wped out. f (sc ~= 0) f (log10(sc*ct) >= ) ct=roud(ct/10)*10; % 'ct' ed ed %-----Ths bascally reverses the operato o le 34,38 % (t brgs back the fal umber to ts true magtude) c=ct*10^lm/f; 2.29 PFJL Lecture 2, 7

8 Matlab addtos ad quatzg effect EXAMPLES radd (100,4.9,1) = 100 radd (100,4.9,2) = 100 radd (100,4.9,3) = 105 >> radd (99.9,4.9,1)= 100 >> radd (99.9,4.9,2)= 100 >> radd (99.9,4.9,3) = 105 NOTE: Quatzg effect peculartes >> radd (0.095,-0.03,1) =0.06 >> radd (0.95,-0.3,1)= 1 Dfferece come from MATLAB roud: >> roud(10^1*0.095/10^(-1)) 9 >> roud(10^1*0.95/10^(0)) 10 But ote: >> roud(10^1*(0.095/10^(-1))) PFJL Lecture 2, 8

9 Issues due to Dgtal Arthmetc Large umber of addtos/subtractos (recurso), e.g. add 1 100,000 tmes vs. add ,000 tmes. Addg large ad small umbers (start from small to large) Subtractve cacellato Roud-off errors duced whe subtractg early equal umbers, e.g. roots of polyomals Smearg: occurs whe terms sum are larger tha the sum e.g. seres of mxed/alteratg sgs Ier products: very commo computato, but proe to roud-off errors Some examples of the above provded followg sldes 2.29 PFJL Lecture 2, 9

10 Recurso: Hero s Devce Numercally evaluate square-root Ital guess x 0 Test a=26; %Number for whch the sqrt s to be computed =10; %Number of terato recurso g=2; %Ital guess % Number of Dgts dg=5; MATLAB scrpt sq(1)=g; hero.m for =2: sq()= 0.5*radd(sq(-1),a/sq(-1),dg); ed ' value ' [ [1:]' sq'] hold off plot([0 ],[sqrt(a) sqrt(a)],'b') hold o plot(sq,'r') plot(a./sq,'r-.') plot((sq-sqrt(a))/sqrt(a),'g') leged('sqrt','x','s/x','relatve Err') grd o Mea of guess ad ts recprocal Recurso Algorthm 2.29 PFJL Lecture 2, 10

11 Recurso: Horer s scheme to evaluate polyomals by recursve addtos ( b 0 =a 0 ) Goal: Evaluate polyomal + Horer s Scheme horer.m % Horer s scheme % for evaluatg polyomals a=[ ]; =legth(a) -1 ; z=1; b=a(1); % Note dex shft for a for =1: b=a(+1)+ z*b; ed p=b >> horer Geeral order p = 55 Recurrece relato For home suggesto: utlze radd.m for all addtos above ad compare the error of Horer s scheme to that of a brute force summato, for both z egatve/postve 2.29 PFJL Lecture 2, 11

12 Recurso: Order of Operatos Matter Teds to: 0 1 If N=20; sum=0; sumr=0; b=1; c=1; x=0.5; x=1; % Number of sgfcat dgts computatos dg=2; dv=10; for =1:N a1=s(p/2-p/(dv*)); recur.m a2=-cos(p/(dv*(+1))); % Full matlab precso x=x*x; addr=x+b*a1; addr=addr+c*a2; ar()=addr; sumr=sumr+addr; z()=sumr; % addtos wth dg sgfcat dgts add=radd(x,b*a1,dg); add=radd(add,c*a2,dg); % add=radd(b*a1,c*a2,dg); % add=radd(add,x,dg); a()=add; sum=radd(sum,add,dg); y()=sum; ed sumr Result of small, but sgfcat term destroyed by subsequet addto ad subtracto of almost equal, large umbers. Remedy: Chage order of addtos ' delta Sum delta(approx) Sum(approx)' res=[[1:1:n]' ar' z' a' y'] hold off a=plot(y,'b'); set(a,'lewdth',2); hold o a=plot(z,'r'); set(a,'lewdth',2); a=plot(abs(z-y)./z,'g'); set(a,'lewdth',2); leged([ um2str(dg) ' dgts'],'exact','error'); recur.m Cotd PFJL Lecture 2, 12

13 recur.m >> recur b = 1; c = 1; x = 0.5; dg=2 delta Sum delta(approx) Sum(approx)..\codes_2\recur.pg res = PFJL Lecture 2, 13

14 Order of Recurrece - Error Propagato Numercal Istablty Example Evaluate Itegral Backward Recurrece Recurrece Relato: Proof : 3-dgt Recurrece: > y2!! < 0!! Exercse: Make MATLAB scrpt Correct 2.29 PFJL Lecture 2, 14

15 Order of Recurrece - Error Propagato Sphercal Bessel Fuctos ps: Bessel fuctos are oly used as example, o eed to kow everythg about them for ths class PFJL Lecture 2, 15

16 Order of Recurrece - Error Propagato Sphercal Bessel Fuctos Forward Recurrece Forward Recurrece Ustable Backward Recurrece Mller s algorthm wth N ~ x+20 Stable 2.29 PFJL Lecture 2, 16

17 Error Propagato: Roud-off ad Trucato Errors Dfferetal Equato Euler s Method Example Dscretzato Fte Dfferece (forward) Recurrece Cetral Fte Dfferece euler.m 2.29 PFJL Lecture 2, 17

18 Trucato Errors, Taylor Seres ad Error Aalyss Taylor Seres: Provdes a mea to predct a fucto at oe pot terms of ts values ad dervatves at aother pot ( the form of a polyomal) Hece, ay smooth fuctos ca be approxmated by a polyomal Taylor Seres (Mea for tegrals theorems): 2 3 x x x 1 f ( x ) f ( x ) x f '( x ) f ''( x ) f '''( x )... f ( x ) R 2! 3!! x ( 1) ( 1 ) x x t R f ( ) f ( t) dt 1!! = costat + le + parabola + etc x 2.29 PFJL Lecture 2, 18

19 Taylor seres, Δx costat: Taylor Seres to Derve Fte Dfferece Schemes 2 3 x x x 1 f ( x ) f ( x ) x f '( x ) f ''( x ) f '''( x )... f ( x ) R 2! 3!! 1 x ( 1) R f ( ) 1! Forward fte-dfferece estmate of f (x ) wth 1 st order accuracy 2 x 3 f ( x 1) f ( x) x 2 f ( x 1) f ( x ) x f '( x ) f ''( x) O( x ) f '( x) f ''( x) O( x ) 2! x 2! Cetered fte-dfferece estmate of f (x ) wth 2 d order accuracy Forward Backward 2 3 x x 4 5 f ( x 1) f ( x ) x f '( x ) f ''( x ) f '''( x ) O( x ) O( x ) 2! 3! 2 3 x x 4 5 f ( x 1) f ( x ) x f '( x ) f ''( x ) f '''( x ) O( x ) O( x ) 2! 3! f ( x ) f ( x ) x f x f x O x 2x 3! '( ) '''( ) ( ) Order p of accuracy dcates how fast the error s reduced whe the grd s refed (ot the magtude of the error) 2.29 PFJL Lecture 2, 19

20 Uvarate Case Recall: Hece, Dervato of Geeral Dfferetal Error Propagato Formula 2 3 x x x 1 f( x ) f( x ) x f '( x ) f ''( x ) f '''( x )... f ( x ) R 2! 3!! 1 x ( 1) 1 R f ( ) O( x ) 1! y f f( x ) f( x ) x x x xf'( x) f''( x) f'''( x)... f ( x) R 2! 3!! 1, f xf '( x ) O ( x ) xf '( x) 2 For x 1, f xf'( x ) O( x) xf'( x ) Thus, for a error o x equal to Δx such that x 1, we have a error o y equal to : Multvarate case y f( x) y f x x f '(x x ) x f '( x ) f '( x ) y y f( x1, x2, x3,..., x ) Dervato doe class o the board For x 1, f( x,..., x ) 1 1, y 1 x 2.29 PFJL Lecture 2, 20

21 Geeral Error Propagato Formula (The Dfferetal Formula) Note: ot to scale. For ths large plotted Δx, secod dervatve ot eglgble Absolute Errors Fucto of oe varable? > y~f (x)x x = x - x y~f (x)x Fct. of var., Geeral Error Propagato Formula x x 2.29 PFJL Lecture 2, 21

22 Error Propagato Example wth Dfferetal Approach: Multplcatos Multplcato Error Propagato Formula => => => ε r y ε r => Relatve Errors Add for Multplcato Aother example, more geeral case: ε r y ε r 2.29 PFJL Lecture 2, 22

23 MIT OpeCourseWare Sprg 2015 For formato about ctg these materals or our Terms of Use, vst:

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 2

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 2 REVIEW Lecture 1 1. Syllabus, Goals and Objectives 2. Introduction to CFD 3. From mathematical models to numerical simulations (1D Sphere in 1D flow)