(1) f ( Ω) Keywords: adjoint problem, a posteriori error estimation, global norm of error.

Transcription

1 O a poseriori esimaio of umerical global error orms usig adjoi equaio A.K. Aleseev a ad I. M. Navo b a Deparme of Aerodyamics ad Hea Trasfer, RSC ENERGIA, Korolev, Moscow Regio, 4070, Russia Federaio b Deparme of Maemaics ad C.S.I.T., Florida Sae Uiversiy, Tallaassee, F , USA Absrac A poseriori esimaio of global orms of e umerical error is addressed usig local rucaio error, age liear problem ad adjoi problem loaded by e iformaio from e age problem. Tis approac provides e local sesiiviy of ese orms o rucaio error. Te local rucaio error used i e aalysis is obaied by e acio of a ig order fiie-differece secil o a previously compued field obaied by e mai (low accuracy order) algorim. Keywords: adjoi problem, a poseriori error esimaio, global orm of error.. Iroducio A prese, ere are may wors devoed o a poseriori esimaio of e error of cerai impora fucioals usig e adjoi equaios (for example, [,,3,ad 4]). From viewpoi of oal soluio qualiy esimaio, e meods for error esimaio of global orms usig adjoi equaios are also of ieres. However, e adjoi equaios are raer seldom used for global orms esimaios ad e auors are aware of oly few wors addressig is subjec [5,6,7]. I e prese wor we cosider e esimaio of global orms of e umerical error usig local rucaio error, liear age problem (TP) ad adjoi problem provided wi e iformaio from e age problem. Te use of TP ad correspodig adjoi problem is e mai feaure of e curre approac disiguisig i from above meioed wors [5-7]. Te rucaio error is obaied usig acio of a igly accurae fiie differece secil o e previously compued field.. Algorim oulie e us cosider e brief formal sceme of e adjoi based a poseriori esimaio of e global error. e e problem of our ieres (direc oe) is govered by equaio N f = w i Ω R, () f ( Ω) = f B ( Ω) ( ). Ω Were f deoes pysical parameers ( f H (Ω ) ), N is a oliear differeial operaor ( H ( Ω ) ( Ω ) ), cosidered o be Frece differeiable, e correspodig derivaive is oed as N f. Te umerical soluio is provided by e fiie-differece equaio N f = w, f ( Ω) = f B ( Ω). () As e resul of is soluio, we obai a grid fucio f. We assume e exisece of + a smoo eoug fucio f H (Ω) a coicides wi e grid fucio a e odes. Fiie differeces i N f may be expaded usig Taylor series. Tis provides us wi e differeial approximaio of a fiie-differece sceme

2 Nf + δ f = w, f ( Ω) = f ( Ω). (3) B Here δ f is e rucaio error coaiig leadig erms of Taylor expasio ad i serves as e disurbig erm. Te correspodig disurbaces Δ f = f f are govered by followig problem (TP) N Δ f + δ f = 0, (4) f Ω R, Δf ( Ω) = 0. We are ieresed i compuaio of e orm of error Δ f. (5) = ( Δ f, Δ f ) ( Ω ) ( Ω ) e us iroduce e adjoi parameers ϕ H ( Ω ) ad formulae e agragia = Δ f + ( N Δ f + δ f, ( ) ϕ ). (6) Ω ( Ω ) O e soluio of TP e orm of error is equal o agragia. I may be described usig adjoi parameers as Δ f = δ f ϕ d Ω. (7) ( ) Ω Ω Adjoi equaio for is problem follows: * N f ϕ + Δf = 0, i Ω, ϕ = 0 o Ω. Te adjoi parameer as e sese of sesiiviy of is orm o a local rucaio error ϕ = ( Δf ) / δf. (9) Tis esimaio does o coai ay icompuable cosas. (8) 3. Esimaio of global error orms for fiie differece approximaio of ea coducio Firs, le us cosider a poseriori esimaio of e global orm of emperaure error for e fiie-differece soluio of e useady oe dimesioal ea coducio equaio. T T λ = 0 ; i Q = Ω ( 0, ), f Ω R. x (0) Iiial codiios: T (0, x) = T 0 ( x) ; T 0 ( x) ( Ω); () Boudary codiios T = T x=0 0; x= = 0. x x () Here λ is emperaure coduciviy, T % is emperaure (cosidered ere as exac, operurbed), x-coordiae, -icess, - ime, f - duraio of process, Ω - domai of calculaio, λ = Cos, T (, x) C ( Q). I is space e problem is well-posed [9]. Cosider a fiie-differece approximaio of e firs order i ime ad secod order i space of equaio (): T T T + T + T (3) λ = 0; τ

3 ad 3 Here T is e approximae soluio of fiie differece equaio, τ is emporal sep - spaial sep size. e us expad e mes fucio T i a Taylor series ad subsiue o (3). Herei we imply a ere exiss e smoo eoug fucio T (, x) a coicides wi T a all grid pois [0]. Te e umerical solvig of equaio (3) is equivale o solvig of disurbed equaio (4). T T (4) + λ + δt = 0. x Herei e source erm δ T is e local rucaio error a ca be calculaed usig a differeial approximaio of fiie-differece sceme [0] or by especial posprocessor [,,ad ]. Te rasfer of error ( T = T + ) is deermied via iear Tage Problem ΔT ΔT (5) + λ + δt = 0. x Te global orm of error as a appearace: ( Q ) ΔT = ( ΔT ) dωd. (6) ( Q) e us iroduce a agragia comprised of e esimaed value ad e wea saeme of problem (5) a is equal o e orm o e soluio of TP ΔT ΔT (7) = ΔT ΔTdΩd + + λ + δt dωd x ϕ. Usig sadard eciques icludig iegraio by pars [8] we obai e followig adjoi problem ϕ ϕ (8) + λ + ΔT = 0, ϕ = 0 o Ω. x O is problem soluio e orm of error may be expressed as: ΔT = δt ϕdωd (9) Ceraily, if e rucaio error (residual) δ T is ow, oe may direcly compue e orm of error from equaio (5). However, e adjoi esimaio eables deermiaio of sesiiviy of is orm o a local rucaio error ( ΔT ) / δ T = ϕ. (0) a may be useful for e grid adapaio i lie wi [4]. Tus, for sesiiviy esimaio we eed o coduc ree calculaios: forward problem (3), iear Tage Problem (5), ad adjoi problem (8). For e error esimaio i orm e correspodig expressios ave followig appearace: Δ = ΔT dωd = sig( ΔT ) ΔTdΩd = FΔTdΩd, F = sig( ΔT ) = ±. () T = sig( ΔT ) ΔTdΩd + ΔT ΔT + λ + δt dωd x ϕ () Adjoi equaios ϕ ϕ + λ + F = 0, ϕ = 0 o Ω. x (3) Te orm of error:

4 ΔT = δt ϕdωd. (4) 4 Tis expressio formally coicides wi (9) aloug e differe adjoi emperaure is implied. 4. Numerical ess. Te umerical ess are based o e emperaure field evoluio egedered by a poiwise ea source ( 0, ξ -are e iiial ime ad e coordiae of e poi source, respecively). q ( x ξ ) T a (, x) = exp ( ) (5) πλ 0 4λ ( 0 ) We use e daa f = T0 ( x ) calculaed by (5) as e iiial daa we solvig (3). Te leg of spaial ierval is cose so as o provide a egligible effec of e boudary codiio compared wi e effec of approximaio. Te roud-off errors were esimaed by comparig calculaio wi sigle ad double precisio, ad e differece was foud o be egligible. A implici meod (implemeed by e Tomas algorim) was used for soluio bo of e ea rasfer equaio ad e age ad adjoi equaios. Te spaial grid cosised of odes, e emporal iegraio coaied seps. Te illusraios, preseed erei, ave bee carried ou wi odes ad ime seps. Temperaure coduciviy was ae as λ = 0 7 m /s. A four order accuracy (over ime ad spaial variables) secil was used for e esimaio of residual: T + 8T 8 f + f T + + 6T + 30T + 6T T ( δ T ) = Cρ λ (6) τ.5.0 dt Fig.. Temperaure error. -iear Tage Problem, - differece bewee umerical ad aalyical soluios.

5 5 Fig.. preses a compariso of emperaure error obaied from iear Tage Problem ad differece bewee umerical ad aalyical soluios as a fucio of grid umber. Oe ca see a e soluio of iear Tage Problem (5) wi e source erm i form (6) provides a admissible approximaio of e emperaure error. (D) 3 Apr 008 Tage Fig.. Temperaure field (D) 3 Apr 008 Tage

6 6 Fig. 3. Trucaio error calculaed by acio of secil (6) Fig.. displays rucaio error calculaed by acio of secil (6) o e emperaure field of Fig.. (D) 3 Apr 008 Tage Fig. 4. Temperaure error field. (D) 3 Apr 008 Tage Fig. 5. Adjoi emperaure field from Eq. (8) for

7 7 Fig. 4 preses e field of emperaure error calculaed by problem (5) wi e rucaio error from Fig. 3. Te correspodig adjoi field is provided i Fig. 5. Te resul of calculaio usig Eq. (5) is.56 wile e resul obaied usig e adjoi equaio (8) is.5647, a cofirms e accuracy of bo problems soluio. Te adjoi field preseed i Fig. 5 may be cosidered as a sesiiviy of is orm o e local rucaio error ϕ = ( ΔT ) / δt a provides a sigifica addiioal iformaio if compared wi TP solvig (Eq. 5). (D) 3 Apr 008 Tage Fig. 6. Adjoi emperaure field from Eq. (3) for Similar calculaios are coduced for adjoi equaio (3) aimed a esimaio of. Correspodig field of e adjoi emperaure is preseed i Fig. 6. Calculaio of usig TP (5) yields a value of.9896 wile usig adjoi problem by Eq. (3) provides a value of.008, wic verifies e soluio of (3). 5. Coclusio Global orms of e umerical error of a fiie differece soluio may be esimaed usig local rucaio error, age liear problem ad adjoi problem loaded by e

8 8 iformaio from e age problem. Tis approac provides e local sesiiviy of e orm o rucaio error. Te umerical ess for ea rasfer equaio demosraed e validiy of discussed approac. REFERENCES. J. T. Ode ad S. Prudomme, Esimaio of modelig error i compuaioal mecaics, Joural of Compuaioal Pysics, 8, pp (00).. M. B. Giles ad E. Suli, Adjoi meods for PDEs: a poseriori error aalysis ad posprocessig by dualiy, Aca Numerica, Cambridge Uiv. Press, pp (00). 3. R. Becer, R. Raacer A opimal corol approac o a poseriori error esimaio i fiie eleme meods. I A. Iserles, edior, Aca Numerica, Cambridge Uiv. Press, pp. -0 (00). 4. Vedii D. ad Darmofal D. Grid Adapaio for Fucioal Oupus: Applicaio o Two-Dimesioal Iviscid Flow, J. Compu. Pys., v.76, pp , (00) 5. S. Repi ad M. Frolov, A poseriori error esimaes for approximae soluios of ellipic boudary value problems. Compuaioal Maemaics ad Maemaical Pysics, v. 4, N., pp , (00). 6. M. Ruer, E. Sei, O e Dualiy of Global Fiie Eleme Discreizaio Small Srai Newoia ad Eselbia Mecaics, TECHNISCHE MECHANIK, v. 3, NN -4, pp (003). 7. S. Prudomme, J. T. Ode, T. Weserma, J. Bass ad M. E. Boi, Pracical meods for a poseriori error esimaio i egieerig applicaios, I. J. Numer. Me. Egg: v. 56, pp (003). 8. G. I. Marcu. Adjoi Equaios ad Aalysis of Complex Sysems. Kluwer Academic Publisers, Dordrec, Hardboud (995). 9. adyzesaja O. A., Soloiov V.A. ad Ural ceva N. N., iear ad Quasiliear Equaios of Parabolic Type, Tras. Ma. Moograp 3, America Maemaical Sociey, Providece, RI, (968) 0. Yu.I. Soi, Meod of differeial approximaio. Spriger-Verlag (983).. Aleseev A.K., ad Navo I.M., O a-poseriori poiwise error esimaio usig adjoi emperaure ad agrage remaider, Compuer Meods i Appl. Mec. ad Eg., Vol 94, NN 8-0, pp -8, (005).. Aleseev A.K., ad Navo I.M., A Poseriori Error Esimaio by Posprocessor Idepede of Flowfield Calculaio Meod, Compuers & Maemaics wi Applicaios, v. 5, pp , (006). Figure capios Fig.. Temperaure error. -iear Tage Problem, - differece bewee umerical ad aalyical soluios. Fig.. Temperaure field Fig. 3. Trucaio error calculaed by acio of secil (6) Fig. 4. Temperaure error field. Fig. 5. Adjoi emperaure field from Eq. (8) for Fig. 6. Adjoi emperaure field from Eq. (3) for