ME 501A Seminar in Engineering Analysis Page 1

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1 Accurac, Stabilit ad Sstems of Equatios November 0, 07 Numerical Solutios of Ordiar Differetial Equatios Accurac, Stabilit ad Sstems of Equatios Larr Caretto Mecaical Egieerig 0AB Semiar i Egieerig Aalsis November 0, 07 Outlie Review last class Aalsis of umerical algoritms Stabilit of umerical solutios Explicit versus implicit approaces Step size variatio for error cotrol Error cotrol for multistep metods wit costat ad variable step size Review Implicit Metods Metods discussed previousl are called explicit Ca fid + i terms of values at Use predictors to estimate values betwee ad + Implicit metods use f + i algoritm Usuall require approximate solutio Have better stabilit but require more work ta explicit metods Trapezoid metod is a example Review Trapezoid Metod ) Algoritm f f / O( Have to adle f + depedece of + Simple iteratio f f x /, f f x, f f x, Talor series for f + Newto iteratio f f x f 4 Review Multistep Metods Multistep metods use iformatio from previous steps for improved accurac wit less work ta sigle step metods Need startig procedure tat is a sigle step metod Derivatio based o iterpolatio polomials wic are te itegrated Predictor-corrector process Derivatio provides error estimate Review Adams Metods Predictor corrector metod Predictor equatio uses four poits 4 P f 9 f 7 f 9 f Corrector equatio uses four poits icludig poit + wit predicted P 4 C P 9 f x, 9 f f f 6 ME 0A Semiar i Egieerig Aalsis Page

2 Accurac, Stabilit ad Sstems of Equatios November 0, 07 Review Step Size Cotrol Get estimate of trucatio error, E C, from predictor-corrector differece 9 70 If e mi E C e max, do ot cage If E C < e mi double step size, If E C > e max alf step size, P C EC 7 Review Grid Size Cages Keep extra values f i-4 ad f i- i memor to be read for grid doublig f i-,ew = f i- ; f i-,ew = f i- ; f i-,ew = f i- ; f i,ew = f i+ Grid alvig requires iterpolatio for missig values i old grid f i-,ew = f i- ; f i,ew = f i f i, ew f i 4 8 f i 70 f i 40 f i f i 8 f i, ew f i 4 6 f i 4 f i 4 f i f i 64 8 Review Extrapolatio Metods Use ifiite series trucatio error depedece o to get better estimate from results o two values of Aalze trucatio error as ifiite series ad elimiate lowest order term True value, t = () + A m + B m+a m t B m a ma 9 Review Midpoit Metod Take big step from x to x + H i steps Start wit results at x ad defie z 0 = (x) Compute z = z 0 + f(x, z 0 ) Cetral differece itermediate steps z m+ = z m- + (x+m,z m ) m =,,.. - Fial value at x + H, called, is a average of te cetral differece value, z, ad a backward differece value z - + f(x+h,z ) = [ z + z - + f(x+h,z ) ] / 0 Review Bulirsc-Stoer Metod Tree mai ideas Use large step size H ad compute results at x + H for several values of te extrapolate results to = 0 Use midpoit metod wose trucatio error is A + B + + C +4 to improve accurac of iterpolatio process Use ratioal fuctio approximatio istead of simple polomial iterpolatio for extrapolatig to = 0 Some Basic Cocepts A fiite differece equatio is cosistet wit te correspodig differetial equatio if bot equatios give te same result as 0 A umerical metod is coverget wit te solutio of te ODE if te umerical solutio approaces te actual solutio as 0 (wit icrease i umerical precisio at smaller ) Mail teoretical cocepts ME 0A Semiar i Egieerig Aalsis Page

3 Accurac, Stabilit ad Sstems of Equatios November 0, 07 More Basic Cocepts We caot kow te accurac of umerical solutios, but we ca use error approximatios to cotrol step size We kow te order of te global trucatio error Stabilit refers to te abilit of a umerical algoritm to damp a errors itroduced durig te solutio Ustable solutios grow witout boud More o Stabilit Fiite differece equatios i umerical algoritms, we iterated, ma umericall icrease witout boud Stabilit usuall is obtaied b keepig step size small, sometimes smaller ta te required for accurac For most ODEs stabilit is ot a problem, but it is for stiff sstems of ODEs ad for partial differetial equatios 4 Stabilit of Exact Solutios Exact solutios to differetial equatios ma be ustable Solutios of te form Ce at wit a > 0 are ustable because te grow witout boud as t Judge stabilit of a umerical metod b test o a exact solutio tat is stable Test = -a wose solutio is = e -at, were a is a positive costat Euler Solutio of Decaig Expoetial Euler algoritm errors move umerical solutio of d/dx = - to solutio for aoter iitial coditio. Aaltical solutio, decaig expoetial, is stable (0) = (0) = (0) = (0) = 4 (0) = Euler x Euler Solutio for Icreasig Expoetial Euler algoritm errors move umerical solutio of d/dx = - to solutio for aoter iitial coditio. Aaltical solutio, icreasig expoetial is ustable x (0) = (0) = (0) = (0) = 4 (0) = Euler 7 Examie Euler Stabilit Look at test equatio wit = f = -a Exact solutio is = 0 e -ax so tat / 0 is a fuctio of ax Euler metod: + = + f Wit f = -a te Euler metod equatio for +, + = + f, becomes + = + (-a ) = ( a) Compare various umerical solutios to exact solutio for differet values of a i followig plot of / 0 versus ax 8 ME 0A Semiar i Egieerig Aalsis Page

4 Accurac, Stabilit ad Sstems of Equatios November 0, 07 /0 Stabilit of Euler Metod ax Exact a =. a = a =. a = a =. 9 Cart Observatios Used Euler metod: + = + f to solve = -a For a, metod looks psicall realistic if ot accurate For < a, metod is ot psicall realistic but is bouded (stable) Metod is ustable for a > Not sow o cart is tat we usuall eed a << for accurac i Euler s metod 0 Geeral Stabilit Look at trial ODE = f = a Defie growt or amplificatio factor, G = + / Euler metod as + = ( + a) so G = + / = + a For a (G ), metod was psicall realistic if ot accurate ad metod was ustable for a > (G > ) Geeral approac is to seek relatio for (or a) tat keeps G stable Geeral Stabilit II Use same test equatio wit f = -a wit positive a (egative a is ustable ODE) Fid amplificatio factor G for metod If growt is bouded for a combiatio of psical parameters ad step size,, metod is ucoditioall stable Coditioall stable metod is stable ol for some combiatio of step size ad psical parameters givig step size limit Implicit Metods Cotrast betwee implicit ad explicit metods discussed previousl Explicit metods fid + i terms of values at x (ma use estimated values betwee x ad x + ) Implicit metods use f + i algoritm Require iterative solutio or series expasio of derivative expressio for f Examie stabilit of trapezoid metod for usual test problem = a Implicit Stabilit Trapezoid metod equatio from previous class basic equatio ad computatio wit series expasio for f + f f O( ) f f x f 4 ME 0A Semiar i Egieerig Aalsis Page 4

5 Accurac, Stabilit ad Sstems of Equatios November 0, 07 Implicit Example For d/dx = f = a, f/x = 0 ad f/ = a f f x f a a 0 ( a) a a a a Here + = G wit G = ( a)/( + a) G < if a > 0; stable for a if a > 0 /0 Trapezoid Metod Stabilit ' = -a for a > ax Exact a =. a= a =. a = a =. a= a=4 6 Cart Observatios Trapezoid metod results muc closer to exact solutio ta Euler results Expected because of O( ) local error For values of a >, solutios udersoot te fial value of = 0 Solutios remai stable, but urealistic, givig oscillatios aroud = 0 Stabilit is ot te same as accurac Must ave bot Error Cotrol How do we coose to maitai desired accurac? Wat to obtai a result wit some desired small global error Ca just repeat calculatios wit smaller util two results are sufficietl close Ca use algoritms tat estimate error ad adjust step size durig te calculatio based o te error 7 8 Ruge-Kutta Error Cotrol Cotrol error b doig itegratio wit ad alog all te itegratio Itegratio wit step requires additioal fuctio evaluatios per steps Aalze local trucatio error, wic is O( ) for bot steps, at eve step locatios 6 ( x ) A B 6 ( x ) A 0 B 6 A O( ) 9 Ruge-Kutta Error Cotrol II = is measure of trucatio error User specifies D, te desired error Ma was to specif tis, sigle value, relative values, relative to icremets for i oe step Sice error scales as, we ca adjust step size suc tat ew = old D / / Tpicall use safet factor to avoid makig ew too large 0 ME 0A Semiar i Egieerig Aalsis Page

6 Accurac, Stabilit ad Sstems of Equatios November 0, 07 Ruge-Kutta-Felberg Uses two equatios to compute +, oe as O( ), te oter O( 6 ) error Requires six derivative evaluatios per step (same evaluatios used for bot equatios) Te error estimate ca be used for step size cotrol based o a overall t order error Cask-Karp versio ad Ruge-Kutta- Verer use same idea Ruge-Kutta-Felberg II Oe algoritm o followig slides Tpical formula compoets below + = + (6k / + 666k /8 k = f(x + /8, + k / + 9k /) Error = k /60-8k /47 ew = old E Desired /Error /4 E Desired is set b user RKF4 code b Watts ad Sampie, RKF4 k Equatios 4, 4 8, 9, , , RKF4 Equatios for + / ew Differece betwee + ad * + used for error estimate to adjust step size R max is user-specified maximum error per step Solvig Simultaeous ODEs Appl same algoritms used for sigle ODEs Must appl eac part of eac algoritm step to all equatios i sstem before goig o to ext step Ke is avig cosistet x ad values i determiatio of f i (x,) All i values i must be available at te same x poit we computig te f i E.g., i Ruge-Kutta we must evaluate k Ruge-Kutta for ODE Sstem () is vector of depedet variables at x = x k (), k (), k (), ad k (4), are vectors cotaiig itermediate Ruge-Kutta results f is a vector cotaiig te derivatives k () = f = f(x, () ) k () = f(x + /, () + k () /) k () = f(x + /, () + k () /) k (4) = f(x +, () + k () ) (+) = (k () + k () + k () + k (4) )/6 for all equatios before fidig k 6 ME 0A Semiar i Egieerig Aalsis Page 6

7 Accurac, Stabilit ad Sstems of Equatios November 0, 07 ODE Sstem b RK4 d/dx = + z ad dz/dx = z wit (0) = ad z(0) = - wit =. Details of first step from 0 to k () = [- + z] = 0.[- + (-)] = -. k ()z = [ - z] = 0.[ - (-)] =. k () = [-(+ k () /) + z + k ()z /] = 0.[ -( + -0./) + (- +./)] = -.8 k ()z = [(+ k () /) (z + k ()z /)] = 0.[( + -0.)/ - (- +./)] =.8 7 ODE Sstem b RK4 II k () = [-(+ k () /) + z + k ()z /] = 0.[ -( /) + (- +.8/)] = -.8 k ()z = [(+ k () /) (z + k ()z /)] = 0.[( )/ - (- +.8/)] =.8 k (4) = [-(+ k () ) + z + k ()z ] = 0.[ -( ) + (- +.8)] = -.66 k (4)z = [(+ k () ) (z + k ()z )] = 0.[ ( ) - (- +.8)] =.66 8 ODE Sstem b RK4 III i+ = i + (k () + k () + k () + k (4) )/6 = + [ (.) + (.8) + (.8) + (.66)]/6 =.887 (ere i = 0) z i+ = z i + (k ()z + k ()z + k ()z + k (4)z )/6 = + [(.) + (.8) + (.8) + (.66)]/6 =.887 Cotiue i tis fasio util desired fial x value is reaced Note all k m computed before a k m+ No x depedece for f i tis example 9 ME 0A Semiar i Egieerig Aalsis Page 7