Characteristic Analysis of Exponential Compact Higher Order Schemes for Convection-Diffusion Equations

Transcription

1 Ameran Journal of Computatonal Matemats 0 9- do:0./ajm.0.00 Publsed Onlne June 0 (ttp:// 9 Caraterst Analss of Eponental Compat Hger Order emes for Conveton-Dffuson Equatons Abstrat anasraju V... Yedda Naeta Msra Department of Matemats Indan Insttute of Tenolog Madras Cenna Inda E-mal: sredda@tm.a.n msra.naeta@gmal.om Reeved Februar 9 0; revsed Mar 0; aepted Aprl 0 Ts paper loos at te development of a lass of Eponental Compat Hger Order (ECHO) semes and attempts to ompreend ter beavour b ntrodung dfferent ombnatons of dsrete soure funton and ts dervatves. Te araterst analss s performed for one-dmensonal semes to understand te effen of te seme and a smlar analss as been ntrodued for ger dmensonal semes. Fnall te developed semes are used to solve several eample problems and ompared te error norms and rates of onvergene. Kewords: Eponental eme Compat Hger Order eme Caratersts Resolvng Effen Fnte Dfferene. Introduton Man nterestng engneerng problems nvolve te psal proesses and transport penomena tat nlude flud flow eat and mass transfer an be modelled b a general Conveton-Dffuson Equaton (CDE). Ts equaton desrbes te onveton and dffuson aratersts of varous psal quanttes su as momentum energ onentraton et. Ts paper deals wt te numeral soluton of onveton-dffuson equaton of te form au bu u du f ( ) () on R wt boundar ondtons u ( ) g ( ) on () were a b >0 are onstant dffuson d are onstant onveton oeffents and f g are suffentl smoot funtons wt respet to and. If 0<a b << are ver small wen ompared wt and d ten () beomes a onveton domnated equaton for w [-] are some of te eponental semes nown from te lterature. For ger dmensonal problems toug te semes [-] are all fourt order aurate seme presented n [] seems to be gvng better results over te oter two. Te purpose of ts wor s to understand te good features of te seme gven n [] and based on tese features nlude some addtonal ondtons n te development of ECHO semes. ne te development of tese semes s alread been dsussed n [] nstead of repeatng te same n ts wor we fous on understandng te merts of te seme. eton two presents a new lass of ECHO semes for D CDE ter lassfaton and numeral verfaton. Eo semes for D CDE are formulated and ompared n te eton tree and onlusons are drawn n te last seton.. D Conveton-Dffuson Equatons Te one dmensonal equvalent of () b fng b d 0 s gven b au u f ( ) 0< < () wt boundar ondtons u(0) g u() g were g g are some onstants... ECHO emes A general strateg to develop ECHO semes s b startng wt te dfferene equaton Du Du F () ( )/ and Du ( u u were Du u u u )/ over a unforml dstrbuted nodal ponts wt step lengt and F s a lnear ombnaton of te soure term f and ts dervatves at a osen number Coprgt 0 Res.

2 0.V... YEDIDA ET AL. of stenl (mes) ponts wt equal number of arbtrar onstants (refer [-] for tree su dfferent oes). s taen to be ot wen te onveton a oeffent 0 so tat te dfferene Equaton () s eat for a e oterwse t s equal to a. If F s f ten () s a seond order ompat taen as eponental seme w was dsussed n []. In te development of te ECHO seme te arbtrar onstants n F are obtaned b mang te dfferene Equaton () s eat for In ts wor four dfferent stenls are used for F and te orrespondng onstants ave been omputed b forng te dfferene seme () to be at least fourt order aurate. Te four osen stenls and ter onstants are gven b (refer [] for te omplete dervaton of te omputaton of te oeffents).... tenl- Consder te dsrete soure funton F f f f ( f ) () were f ( f ) are te soure funton and t s dervatve respetvel at te nodal pont. Te a Equaton () s alread eat for e and enforng te eatness also for gves four smultaneous equatons n terms of ts oeffents. olvng tem for & gves a for 0 and wen. 0 0 mlarl for te oter stenls sstem of equatons are obtaned and solved to get te orrespondng oeffents.... tenl- wen F f( f) ( f) ( f) () 0 te oeffents are and 0 for tenl- F f ( f ) ( f ) ( f ) (7) wen 0 te oeffents are ( ) and 0... tenl- for. 0 0 F f f f ( f) ( f) (8) wen 0 te oeffents are ( ) ( ) (0. ) (0.0. ) 0.8 ( ) ( ) (0. ) (0.0. ) ) ( 0. and for 0. emes wt stenls and ontan four parameters and are fourt order aurate wereas te seme wt stenl ontans fve parameters and s st order aurate. Here after we refer te dfferene seme () wt stenls to as semes [ D ] to [ D] respetvel for all te future referenes. Eponental semes of [] are also fourt order aurate wt tree arbtrar parameters and te seme gven n [] uses s parameters to generate a st order seme for te osen one-dmensonal onveton- Coprgt 0 Res.

3 .V... YEDIDA ET AL. dffuson equaton. Te dsrete soure terms of tese semes [-] are gven b. tenl used n []: F f f f (9) wen 0 wen 0. tenl used n []: F f ( f) ( f) (0) ( ) wen 0 0 wen 0. tenl used n []: wen 0 F f f f ( f ) ( f) ( f) te oeffents are 90 ( 90 ) (7. ) 7 (0. ) ( 90 ) (7. ) 7 (0. ) ( ) (. ) (0. ) 0 (0 0 8 ) 0 ( ) (. ) (0. ) 0 and for 0. () Name te seme () wt stenls used n [-] as [ D] [ D] semes and [ D ] respetvel. Tat s 7 a total of seven ECHO se mes ave been ntrodued untl now and out of w fve of tem are fourt order aurate and te oter two are st order aurate. Among te fourt order semes tree of tem developed n ts wor ave four free parameters and te oter two taen from te lterature [] and [] ave tree parameters. Among te st order semes one seme developed n ts wor as fve free parameters and te oter taen from te lterature [ D ] 7 as s parameters. Tat s te seven semes an be lassfed t nto n order semes wt n number of parameters and te oter ontan less tan n number of parameters. Te am of te rest of te wor s to demonstrate usng wave number analss and numeral epermentaton t te n order ECHO semes wt n parameters are more aurate tan te oter lass of semes... Comp arson of te Caraterst Curves Usng te wave number analss resoluton of an numeral seme an be measured wt w one an understand te loseness of te araterst of a dfferene equaton to tat of te dfferental equaton []. ne te stablt of an numeral seme depends on te magntude of te pelet number defned b p n ts wor te aratersts are ompared wt a respet to pelet numbers. Te araterst of te Iw governng Equaton () obtaned b substtutng e n te plae of te dependent varable u s gven b: [ ] D I () p were w w s te wave number I. mlarl te ara tersts of te dfferene semes I are obtaned b substtutng e at u to get (refer [] for more detals). Caraterst Curve for : were w ' sn [D] '' ' [ D] w Iw () z Iz '' w os p ot z ( )sn z ( )os p 0. 0.) ) p p ( p( p p p 0. ( ) ( 0.) p p p p Coprgt 0 Res.

4 .V... YEDIDA ET AL. p (0. p p ). [D] Caraterst Curve for : were z Iz '' ' [ D ] w Iw ' w sn '' w os p ot z ) os ( z ) sn ( p p ) 0. p( ( ) p p p p( ) p p 0. p( ) ( ) p. () Caraterst Curve for were ' w sn [D] : '' ' [ ] w Iw D z Iz '' p w os ot z ( ) z ( p ) p p p p p p p p. Caraterst Curve for were θ(λ) [] ' w sn [D] : '' ' [ ] w Iw D z Iz '' p w os ot z ( sn ) z os ( ) ( p() p ( ) p p (0. ) p (0. 0.)) () () Fgure. Comparson of real and magnar parts of at 0. p. { p p p 0.8 p } p ( p( ) p ( ) p p (0. ) p (0. 0. )) p ( p 0.p ) p p p p. [D] λ Bot real and magnar parts of te aratersts (-) are ompared wt () n Fgures and for pelet numbers 0. 0 and 00 respetvel. For te sae of omparson te aratersts of t e emes [ D] [ D] and [ D ] 7 are also nluded n tese fgures. It s lear from tese omparsons tat te [ D] eme 7 s te best among te osen semes D followed b. Tese omparsons an be quantfed b ntrodung Resolvng effen... Resolvng Effen Te resolvng effen [7] of an numeral seme Coprgt 0 Res.

5 .V... YEDIDA ET AL. Fgure. Comparson of real and magnar parts of at p 0. [D] λ Fgure. Comparson of real and magnar parts of at p 00. [D] λ ma defned b s a number between 0 and. ma ndependent of te grd sze s te mamum valu e of for w s less tan a tolerane. fd eat Resolvng effenes are omputed for varous s emes wt dfferent tolerane lmts and presented n Tables and for pelet numbers 0. 0 and 00 respetvel. It s lear from tese tables tat eme n [ ] as a ver good resolvng effen followed b [ D]. Also a areful loo at tese tables reveals tat for small pelet numbers sa for p 0. all te fourt order semes ave more or less equal resolvng effen owever for p 0 and 00 te fourt order semes wt four parameters ave a mu better resolvng effen tan te emes gven n [] and []. ne Re( ) of tese semes resolved to a mu less value for p 0 and 00 tese are more prone to dsspaton error w ultmatel results nto loss of aura. To demonstrate te effet of te resoluton of varous semes on te aura of te generated numeral soluto ns tese semes ave been used to solve few model problems and ompared ter error norms n te net subseton... Verfaton wt Numeral Eamples Two dstnt one-dmensonal problems wt sarp boundar laers are osen for te purpose of numeral verfaton.... Eample Consder u u snos 0< << / / 0< < for w u ( )sn ( e )/( e ) s te eat soluton wt a sarp boundar laer for small values of towards.... Eample Consder u u ( ) os ( ) sn 0< << 0< < for w u ( ) ln( ) os s te eat soluton. Coprgt 0 Res.

6 .V... YEDIDA ET AL. Table. Resolvng effen of te real and magnar [D] parts of λ at p 0.. [ D] Re( ) [ D] Im( ) eme [ D] [ D] [ D] [ D] eme [] em e [] em e [] Table. Resolv ng effen of te real and ma gnar [D] parts of λ a t p 0. [ D] R e( ) [ D] Im ( ) eme [ D] [ D] [ D] [ D] eme [] eme [] eme [] Table. Resolvng eff en of te r eal and ma- pa rts of λ a t p 00 [D] gnar. [ D] R e( ) [ D] Im ( ) eme [ D] [ D] [ D] [ D] eme [] em e [] em e [] Model problems (...) and (...) are solv ed usng [D] te seven sem es and [ D] 7. To var t e pelet number te number of no des as been vared from to 8 and te dffuson parameter as been v ared between 0 and 0. Te err ors omputed usng te nfnt norm are ompared n te Tables and for problems (...) and (...) respetvel (read.7( 08) as n all tese tables). Te omparson of te error norms for varous semes reveals tat for te pelet number p less tan one te aura of all te fourt order semes are more or less equal owever te aura of te solutons D of te to D beomes better over semes n [] and [] f p s nreased to. Te mprovement n te aura beomes even better better b two demal plaes f p s nreased to 0 or more. Ts beavor supports te araterst analss arred out n te earler seton weren we ave sown tat te resolvng effen of semes n [] and [] s mu smaller tan te oter fourt order semes at large pelet numbers. Ts onludes tat to develop fourt order semes usng four parameters ma mprove te resolvng effen and ene te aura of te numeral semes. Te same s also an be onluded between te st order semes. Te solutons generated usng eme n [] are unforml far superor for te entre range of pelet numbers 0. to 00 over all te semes D were as s omparable onl at low pelet numbers. Furter between te tree developed fourt order D semes as less resolvng effen and te solutons obtaned usng ts seme are slgtl nferor wen ompared wt te oter two owever t s stll as a better performane tan te two estng tree parameter semes.. D Conveton-Dffuson Equatons Effen of ever numeral seme an be establsed omputatonall b solvng a lass of eample problems but analss of te used numeral seme s more mportant to gan onfdene before applng tem for real world problems. Usuall te effen of te ger order ompat semes for one-dmensonal statonar CDE s sown b studng ter monotont or omparng ter araterst urves. For D semes te omparsons ave to be made araterst surfaes. Te development of a D seme for te twodmensonal CDE () s alread presented n [] and usng a smlar proedure D equvalents for te D semes to D an be developed as follows:.. ECHO emes Te development of an ECHO seme for a two dmensonal CDE wll be gven n a general proedure su tat a smlar proedure an be followed for dfferent soure funtons. Wen te onveton oeffents ar e onstant te two-dmensonal equvalent of () s gven b * Du jdu jdu jddu j F j (7) were Du ( u u )/ Du ( u u j j j j j j Coprgt 0 Res.

7 .V... YEDIDA ET AL. Table. Comparson of te err or norms for te eample... N p eme [] eme [] eme [] 9.090( 08).9708( 0).78( 0).8( 07).77( 0).0808( 0).0898( 0) 0 / 0).7( 0).8( 0) /.90( ) 9.70( 07).099( 0).770( 0).087( 07).70( 07).800( 07) 8 /8.( ).09( 08) 9.09( 08).87( ).087( 09).( 08) 9.908( 09) [ D] [ D].8( 09).77( 0).8( 0) 9.099( 09).70( [ D] [ D] ( 08).77( 0).980( 0).( 0).8887( 0).7( 0).7( 0) 0.0( 09).8( 0).90( 0) ( 08).7( 0) 7.90( 0).7( 0)..90( ).0( 0).7990( 0).707( 09).( 07).898( 07).78( 07) 8..( ) 8.80( 07).7( 0).89799( ).897( 09).880( 08).000( 08) ( 08).7( 0).( 0).90( 0).98( 0).7( 0).70( 0) 0 0.8( 09).977( 0).08( 0).( 07).79( 0).0( 0).7( 0).007( ).00( 0).97( 0).080( 09).( 07).8( 07).0890( 07) ( ).7( 0).7988( 0).770( 0).77( 08).87( 08).077( 08) ( 08).9( 0).9( 0).008( 0) 7.( 0).( 0).7090( 0) ( 09).990( 0).770( 0).8( 07).789( 0).078( 0).787( 0) 0.009( ).9708( 0).07( 0).0( 09).07( 07) 7.07( 07).0097( 07) 8.899( ).8880( 0).0( 0).99( 0).8800( 08).79( 08).008( 08) T able. Comparson of te error norm for te eample... N p eme [] eme [] eme [] [ D] [ D] [ D] [ D].890( 07).9( 0).8970( 0).88( 07).87( 0).70( 0).( 0) /.9( 09).7( 0).890( 0).7779( 09).8( 0 0 ).099( 0).90( 0) /.90( ) 7.888( 07).7( 0).09( 0).0( 07).7( 07).980( 07) 8 /8.0( ).90( 08) 7.0( 08).7( ).0( 08).09( 08).0( 08) 0.98( 07) 8.0( 0).09( 0).77( 0).088( 0).0080( 0) 9.89( 0) 0.979( 09) 8.07( 0).( 0).90( 08).90( 0).0( 0).7( 0)..70( ).77( 0) 9.9( 0) 8.799( 0).789( 07) 8.90( 07).7( 07) ( ).( 07).89( 07).770( ).89( 08).07( 08).989( 08) ( 07).98( 0).9( 0).888( 0).89( 0).0( 0).07( 0) ( 09).8( 0).99( 0).980( 08) 9.87( 0).80( 0).709( 0).98( ).997( 0).9( 0).9897( 09).09( 07).9( 0).9( 07) ( ).89( 0).8( 0).887( ).788( 08) 7.7( 08).90( 08) ( 07).( 0).9( 0).89( 0).98( 0).7( 0).08( 0) ( 09).( 0).9( 0).09( 08).009( 0).90( 0).7( 0) 0.9( ).90( 0).997( 0).09( 09).78( 07).7( 0).( 07) ( ).( 0).8( 0).790( ).809( 08) ( 08).9( 08) u j) / and D )/ uj ( u j u j Du j ( u j uju j )/ ove r a unforml dstrbuted nodal ponts wt step lengts and along and dretons respetvel and dsrete soure funton F j s a D e quvalent of te orrespondng D seme. Te development of te D ECHO seme s Coprgt 0 Res.

8 .V... YEDIDA ET AL. gven bellow for dfferent seleton of soure funtons. [D]... eme Consder te soure funton w s an etenson of te seme... gven b * F j f j fj f j ( f ) df df df d( f) j j j j Te trunaton error of te seme (7) wt te soure funton (8) omputed usng Talor seres epanson s gven b TE Eu Gu Hu j j Ku f O j j j ( ) (8) (9) were E dk L G bkl H dk al K bk al K ( ) K ( )/ (0) L ( d d ) d L ( d d )/ Epandng te terms n (9) and (0) sows tat te seme (7) s of seond order aurate. To mae t fourt order te seme and te soure funton s wrtten as D D D dd ED D GD D HD D KD D u F j j F f f f f d f d f d f d f j j j j j j j j j were t e oeffents and d & are gven b d d d d ( 0. ) a a d and d for and d not equal to zero () and 0 d d d d 0 wen d 0. mlarl for te oter seleton of soure funtons remander terms are utlzed to get fourt order aura. For ever se me E G H and K are same as n (0) but K K L and L vares wt te seme. [D]... eme F f ( f) j ( f) j ( f) d ( f) j d ( f ) j d ( f) j Let K K ( ) L d d d L ( d d ) j () Te oeffents n t e dsrete soure funton are gven b d 0. d 0. d 0. 0 for d 0 and d d 0 d wen d eme Let [D] F f ( f ) ( f ) ( f j j j j j d ( f ) d ( f ) d ( f ) () j j j K K L d L d ) Coprgt 0 Res.

9 .V... YEDIDA ET AL. 7 Te oeffents n te dsrete soure funton are gven b ( ) d ( ) d d for and d not equal to zero and 0 0 d 0 d d. 0 wen d 0 [D]... eme F f j fj f j ( f) ( f) df jd fj Let d f d ( f ) d ( f ) f j j () K ( ) K ( ) L d ( d d) L d ( d d) Te oeffents n te dsrete soure funton are gven b ( ) ( ) (0. ) (0.0. ) 0. 8 ( ) ( ) (0. ) (0.0. ) ( 0. ) d ( ) ( ) (0. ) (0.0. ) d 0.8 d ( ) ( ) (0. ) (0. 0. ) d ( 0. ) d for and d not equal to zero and d d 0 0 d 0 d 0 d wen d 0. 0 Tese four dfferent semes are ompared wt te estng ECHO semes gven n [] and []. D eme n [] : Let F f f f j j j df df df f j j j j K ( ) K ( ) () L ( d d) L ( d d) Te oeffents n te dsrete soure funton are gven b ( )( 0.) ( ) ( )( 0.) d ( )( 0.) d ( ) d ( )( 0.) for and d not equal to zero and D eme n [] : Let d d d wen d 0. F f ( f ) ( f ) j j j d ( f ) d ( f ) f () j j K K L d L d T e oeffents n te dsrete soure funton are gven b ( ) d ( ) )) (0. ( d. ) (0 ( ) for and d not equal to zero and 0 Coprgt 0 Res.

10 8.V... YEDIDA ET AL. d 0 d wen d 0... Comparson of te Caraterst urfaes Te araterst surfae of a dfferental equaton s obtaned n terms of pel et numbers p and a d p n and dretons respetvel. a Te araterst of te governng Equaton () ( ) obtaned b substtutng I w w e n te plae of te dependent varable u s gven b: [ D] d p r I pr a pp r r (7) were w w are pase angles and w w are te wave numbers and are step lengts and I. mlarl te araterst surfae of an dfferene seme s also obtaned b substtutng I( j ) e for u n te dfferene seme. Followng j ts proedure te araterst surfaes of te D [ D] semes to [ D ] are omputed and te same are gven b [D] Caraterst urfae for : d D a pp z Iz z Iz (8) z r(os ) (os ) r p pr ( ) ( ) sn sn ; pr p r r (os )(os ) r rp p r p z prsn sn r r pr ( ) sn ( os ) ; p pr p p r ( ) ( ) ( os ) sn rp r p z ( ) ( )os ( )os z were ( )sn ( )sn p p p p ot ot ( ) p p p ( ) ( ) p p p ( ). p p mlarl ( ) ( ) p p p ( ) p p p ( ). p p Te terms z and z n te denomnator of (8) are te ontrbutons due to te soure funton of te seme and ene var from seme to seme. However te numerator of all te eponental semes are same as n (8). To justf ts one an epand K K L and L wt ter parameters for ever seme and n ea ase te appear le K a K aa ( ) b bb ( ) L L. d d Terefore te aratersts of te semes are dffer b ter denomnator w ontans te ontrbuton of te soure funton of te seme. Te araterst surfaes of te remanng tree semes are [D] Caraterst urfae for : [ D] d z Iz (9) a pp z Iz z ( ) sn ( ) os z ( ) os ( ) os p p p ( ) p p p p p p p p p ( ) p p p p p Coprgt 0 Res.

11 .V... YEDIDA ET AL. 9 ; p p p [D] Caraterst urfae for : z p p d D a pp z Iz z z Iz (0) p p p p p p Caraterst urfae for [D] : D d z Iz a pp z Iz z ( ) ( )os ( )os ( )sn ( )sn z ( ) ( ) 0 p p p p ( ) ( ) p p ( ) ( ) 0 p p p p () ( ) ( ) ; p p p p ( ) ( ) p p p 0 p ( ) ( ) p p ( ) ( ) 0 p p p p ( ) ( ) ; p p p p mlarl te aratersts of te semes n [] and [] are derved and gven b Caraterst urfae for eme []: d z z a pp z z [] z ( ) ( )os ( )os z ( )sn ( )sn ( ) p p p. p Caraterst urfae for eme []: d z z a pp z z [] z z p p p. p () Te araterst surfaes defned n (8-) are smmetr or antsmmetr n te regon [0 ] [0 ] dependng on weter [ D] s an even or odd funton of p and p. Furter te are also perod [ D] wt perod. Tese surfaes wt respet to 0 p and 0 p an be plotted togeter for te sae of omparson owever unle n one dmensonal ase t s dffult to vsualze te loseness of tese surfaes. Alternatvel omparsons are made at dfferent angular ross setons f rom te orgn. Furter f p p te are also smmetr wt respet to urve te r efore n te present ase te values of te aratersts are ompar ed at 0 and ross setons. Te aratersts at te tree osen ross setons are plotted aganst te eat one n Fgures and for p p 0 and p p 00 respetvel. Te omparsons of te real parts of te aratersts are nluded n te frst olumn of tese fgures wle te omparsons of te magnar parts are sown n te seond olumn. Te tree rows n tese fgures stand for te omparsons at 0 and ross setons respetvel. It an be seen learl n ea of tese fgures tat te aratersts of te estng tree parameter D semes are far awa from te eat urve ompared to te four parameter semes w ave been developed n ts wor. Interestngl te devaton s nreased wt angle and also wt pelet number gvng a ver substantal devaton at p p 00. Partularl eme n [] s devated more at te enter and also produed a sgnfant oversoot for all most all te ross setons. Among te present four parameter based fourt order D semes [ D ] produed mnmum and [ D ] produed mamum dsspaton errors. However wen Coprgt 0 Res.

12 0.V... YEDIDA ET AL. (a) Fgure. Comparson of te (a) real and (b) magnar parts of te araterst at (b) p p 0. [ D] te pelet number s nreased to 00 oversot te eat araterst n ts real part but tere s no su [ D] abnormalt wt respet to. A smlar oversoot n ts real part s also been observed n [ D ] at least [ D] along ross seton. For te magnar parts as te mnmum and [ D ] as te g devaton Coprgt 0 Res.

13 .V... YEDIDA ET AL. (a) Fgure. Comparson of te (a) real and (b) magnar parts of te araterst at (b) p p 00. [ D] gvng lttle more dsperson error. To onlude ma be relatve a better one among te developed four parameter semes. However bot fve and s parameter based semes are ndstngusable and tese [ D] are better resolved for real ase as omparable to [ D] and almost lose to for magnar ase. Coprgt 0 Res.

14 .V... YEDIDA ET AL. Table. Comparson of te error norm and rate of onvergene for te eample... N ( p p ) eme [] Rate Rate [ D] Rate [ D] Rate [ D] Rate [ D] ( ).78( 0).( 0).0( 0).9( 0) 7.79( 0) 0 ( /).08( 0).98.08( 0).98.70( 0).0.8( 0).98.7( 0).0 (/ /).99( 07).99.7( 07).99.0( 07).0 9.8( 07).0.98( 07) (/ /) 8.8( 09).00 8.( 09).99.90( 08).0.8( 08).00.8( 08).00 (0 0).080( 0) 8.09( 0).( 0) 8.( 0).77( 0) 0 (0 ).99( 0).0 9.( 0).8.79( 0)..7( 0)..( 0).07 ( /).787( 0)..8( 0).7.9( 0).0.880( 0)..90( 0) (/ /).98( 0).8.( 0).7.80( 0) ( 0).9.0( 0).88 (0000).99(0).( 00).089( 0).09(0).0( 00) 0 (000).79(00)..7( 0).00.8( 0).87.9(0).0.08( 00).00 (0).99(00).7.070( 0).0.0( 0)..0(00).0.798( 0) ( /).0( 0).7.( 0).9.( 0).0.8(00)..9( 0). (//).( 0).7.098( 0).8.9( 0) 0..99( 0). 7.00( 0).7 (//8). 7798( 0).80.09( 0)..9( 0) ( 0).08.07( 0).79.. Numeral Verfaton Consder te followng two-dmensonal problems wt sarp boundar laers.... Eample u u u u ( ) e ( ) 0< << n te regon 0 wt eat soluton u ( ) e ( ).... Eample u u u u ( )ep 0< << n te regon 0 wt eat soluton u ( ) ( ) ep. Te eample problems (...) and (...) are solved [ D] usng to [ D ] and also wt te seme gven n []. Te results are ompared n te form of error norm and te rate of onvergene n Tables and 7 for problems (...) and (...) respetvel. As epeted [ D] te seme gven n [] and te seme pro dued ger rate of onvergene for Eample... and better aura for Eam ple... T ese omparsons are one agan onfrm te aura of te araterst an alss made n t e prevous subseton. For onveton domnated problems all most all te semes produed same aura owever t as been sown n [ ] tat te seme gven n [] as performed better tan te semes gven [ ] and []. Loon g at te araterst analss an d numeral verfaton t an be onluded tat t s better to use n parameter based D semes to d evelop n t order semes over semes wt less parameters. Coprgt 0 Res.

15 .V... YEDIDA ET AL. Table 7. Comparson of te error norm and rate of onvergene for te eample... [ D] [ D] [ D] N ( p p ) eme [] Rate Rate Rate Rate Rate [ D] ().8( 0).7( 0).87( 0).987( 0).900( 0) 0 (/ /).0( 0).00.08( 0).00.9( 0).0.0( 0).0.797( 0).0 (/ /) 8.79( 07) ( 07) ( 07).0.( 0).00.( 0) (/8 /8).87( 08).99.87( 08) ( 08).99 9.( 08) ( 08).99 (00) 9.0( 0).89( 0).0( 0).0( 0).9( 0) 0 ().7( 0).0.988( 0).7.00( 0).8.( 0) ( 0).8 (/ /) 9.89( 0) ( 0) ( 0)..008( 0).0. 8( 0). 8 8 (/ /) 9.0( 0) ( 0).0.99( 0)..8( 0)..( 0). ( 0000).7( 0).8( 0).97( 0).788( 0).09( 0) 0 (00).0( 0)..0879( 0) ( 0) ( 0)..9( 0) 0.7 ().70( 0).99.8( 0) ( 0) ( 0).9 7.( 0) (/ /). 70( 0)..7( 0)..9( 0). 8.( 0).7.( 0).08 (/ /) 8.( 0) ( 0).8.0( 0)..9( 0).8.8( 0). (/8 /8).08( 0).0.8( 0) ( 0)..7( 0)..9( 0).. Conlusons In ts wor we ave developed and made aratersts based omparsons for eponental ompat semes. Te araterst omparsons are also been etended for two dmensonal problems. It an be onluded from ts sort analss tat wen eponental ompat ger order semes a re generated b evaluatng te soure term as a lnear ombnaton of ts values at te surroundng nodal ponts and ts dervatves t s bet ter t to use n parameters to generate an n order seme so tat te resultant seme wll ave a better resoluton and e ne an produe more aurate solutons. For te same reason te fourt order ECHO semes developed n t s wor ar e more aurate tan te estng semes []. Te same s also true wen te ECHO semes are etended for D CDE te orrespond ng tree pa rameter semes are omparatvel less effent tan te four or s parameter based semes.. Anowledgements Autor Naeta Msra s greatl ndebted to te Counl of entf and Industral Resear for te fnanal support 09/08(089)/00-EMR-I.. Referenes [] E. C. Gartland Jr. Unform Hg-Order Dfferene emes fo r a ngularl Perturb ed Two-Pont Boundar Value Problem Matemats of Computaton Vol. 8 No pp. -. do:0.090/ [] A. C. R. Plla Fourt-Order Eponental Fnte Df- ferene Metods for Boundar Value Problems of Convetv e Dffuson Tpe Internatonal Journal for Numeral Metods n Fluds Vol. 7 No. 00 pp do:0.00 /fld.7 [] Z. F. Tan and. Q. Da Hg-Order Compat Eponental Fnte Dfferene Metods for Conveton-D- Coprgt 0 Res.

16 .V... YEDIDA ET AL. ffuson Tpe Problems Journal of Computa tonal P- Vol. 0 No. 007 pp do:0.0/j.jp ss [] Y. V... anasraju and N. Msra petral Resolu- toned Eponental Compat Hger Order eme (RE- CHO ) for Conveton-Dffuson Equaton Computer Metods n Appled Means and Engneerng Vol. 97 No pp do:0.0/j.ma [] A. M. Il ' n A Dfferene eme for a Dfferental Equaton wt a mall Parameter Multp lng te Hgest Dervatve (Russan) Matematese Zamet Vol. No. 99 pp [] D. You A Hg-Order Padé ADI Metod for Unstead Conveton-Dffuson Equat ons Journal of Computa- tonal Pss Vol. No. 00 pp. -. do:0. 0/j.jp [7]. K. L ele Compat Fnte Dfferene emes wt petral-le Resoluton Journal of Computatonal Pss Vol. 0 No. 99 pp. -. do:0.0/00-999(9)90-r Coprgt 0 Res.