COMPACT FINITE DIFFERENCE SCHEMES FOR POISSON EQUATION USING DIRECT SOLVER

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1 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 COMPACT FINITE DIFFERENCE SCHEMES FOR POISSON EQATION SING DIRECT SOLVER Fel M Or Alred E Owl* Department Matematcs Cvenant nverst Ota Ogn State (NIGERIA) *Crrespndng atr: alred_wl@ac ABSTRACT Te Cmpact Fnte Derence Scemes r te sltn ne tw and tree dmensnal Pssn eqatn s cnsdered n ts paper Te dscretatn sng te trncatn errrs te Talr s seres metd are and case te sceme s 9-pnt stencl and te In te ne and tw dmensn cases te stencls are 9-pnt In te tree dmensn sceme s 7-pnt Tw nmercal eperments were cndcted and te reslts cnrm tat Cmpact Fnte Derence Scemes are accrate and ecent metds Ke wrds: Pssn eqatn cmpact derence sceme Talr s seres metd L decmpstn INTRODCTION Te erts t cmpte mre accrate sltn sng lmted grd ses ave drected researcers attentn t develpng g-rder cmpact nte derence scemes Cmpact derence scemes are g-rder mplct metds wc eatre ger-rder accrac and spectral le resltn wt smaller stencls In te past tw decades several strateges ave been devsed r te cnstrctn cmpact derence scemes Te scemes nclde: te Talr seres metd Pade apprmatn metds and Br nterplatn metds Te rgnal Pade metds g bac t te 90s Te Talr s seres metds were pplarsed n te 990s and te nterplatn metds were recentl presented n a paper n te dervatn g-rder cmpact scemes r nn-nrm grds [] Te metds cmpact derence ave been sed wdel n te large area cmptatnal prblems r eample te cnvergence and sltn r te cmpact derence metd n parablc eqatns were dscssed n [ and ] Tere were als sme wrs n applng te cmpact derence sceme r stead cnvectn-dsn prblem [ 7] te Helmlt eqatns [8 9 and 0] and te perblc eqatn [ ] r D and D Pssn In ts paper we present te 9-pnt cmpact scemes and eqatns A g-rder cmpact sceme r dmensnal Pssn eqatn and s als presented Te are 9-pnt and 7-pnt respectvel []Te scemes lead t a large sstem lnear eqatns A b were A s sparse A MATLAB drect slver sng L decmpstn s mplemented r te cmptatn te nmercal eamples FORMLATION OF HIGH-ORDER COMPACT SCHEMES Te arcetpal ellptc eqatn n spatal dmensns s represented b te Pssn eqatn Here we develp scemes r Pssn eqatn r ne tw and tree dmensnal nrm grds n a strctred grd nrm mes se Frst let s ntrdce te llwng ntatns: [see ] and dente te standard rward nte derence and bacward nte derence scemes r rst dervatve Als 0 () s te rst-rder central nte derence wt respect t were Te standard secnd rder central nte derence s dented as X and s dened as 0

2 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 () Derence peratrs and are dened smlarl B sng te Talr s seres epansn a rt and st rder accrate nte derence r te rst and secnd dervatves can be apprmated as llws: d d d d! d d d d d d! d! d as We ma rewrte eqatn () r d d Als d d 0 d d d d 0 d d () 0 () d d d d d d (a) d d d d d 0 d 0 (b) One dmensnal case Fr an llstratn prpse we rst cnsder te ne dmensnal prblem wc can be represented as I I 0 () Frm eqatn (a) te rt rder accrate nte derence estmate r ( v) (7) (v) Te dea bend te ger rder cmpact sceme s t apprmate n eqatn (7) t secnd rder accrac t aceve an verall trncatn accrac rt rder T ts end we smpl dble derentate eqatn () t get ( ) v ( ) ( ) (8) Als applng te central derence sceme t () we ave ( ) (9) Hence rm eqatn (7) we get r (0) sng ts estmate and cnsderng te dscrete sltn eqatn () wc satses te apprmatn we get s

3 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 () were s te dscrete apprmatn t mples satsng te dscrete rmlatn eqatn () wc sng eqatn () eqatn () can be epressed n te rm 0 () Fr te st rder accrate nte derence estmate eqatn () we ave rm eqatn (b): 0 ( v) ( v) () are nclded n eqatn () We apprmate bt tem t cnstrct an sceme Applng v we get Bt and t ( v ) ( v ) ( ) () Sbstttng eqatn () nt eqatn () elds ( v) v 0 T get te cmpact () ( v) ( ) ( ) were apprmatn we agan appl eqatn (9) tat s " " and Insertng ts eqatn nt eqatn () reslts n Agan 0 ( ) 0 ( ) () Or rm eqatn () ( ) 0 (7) were s te dscrete apprmatn t satsng te dscrete rmlatn eqatn () wc mples Te ( ) apprmatn eqatn () can be gven as (8) Tw Dmensnal Case Cnsder te tw dmensnal Pssn eqatn r 0 0 Te central derence sceme r eqatn (9) n tw dmensns can be wrtten as were and and (9) (0)

4 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 0 () We need a cmpact apprmatn te rst sqare bracet n eqatn () Ts can be dne b tang te llwng apprmate dervatve eqatn (9) t get () Sbstttng eqatn () nt eqatn () we btan te alternatve rm r te eact trncatn errr at nde : 0 0 () In r dervatn te sceme we se eqatn () and te epressns te rst sqare bracet n eqatn () tat s: r () Te sceme ma be wrtten eplctl as 8 0 () Ts s te well nwn accrate nne-pnt cmpact sceme Fr te sceme we need a rt-rder apprmatn n eqatn () Ts can be wrtten as () Sbstttng eqatn () nt eqatn () gves 0 r 0 (7) A cmpact epressns te apprmatn s reqred and ts can be dne b rter derentatng eqatn (9) tat s (8) and

5 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 (9) Sbstttng eqatns (8) and (9) nt eqatn (7) gves ) (0 0 0 Te cmpact st rder apprmatn te tw dmensnal Pssn eqatn can te be btaned as ) ( 90 0 were s te Laplacan peratr and s te barmnc peratr In eqatn () we assme te dervatve can be determned analtcall In te case were s nt nwn analtcall we need nl a rt rder accrate apprmatn and a secnd-rder accrate apprmatn and [9] A rt-rder accrate apprmatn can be btaned sng t get 0 A secnd-rder accrate apprmatn can als be btaned as () A nne-pnt sceme r D Pssn ma be epressed as () v N M L G E D C v v v v N M G E D C Tree-Dmensn Case In ts sectn we perrm a smlar dervatn te g-rder derence sceme r Pssn eqatn n spatal dmensns wc s gven as: () r Here s taen as a cbc sld Te central derence sceme r eqatn () can be wrtten as: ()

6 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 were and ) ( 0 We tae apprmatn te rst sqare bracet n eqatn () Ts s dne b tang apprprate dervatve eqatn () wc s (7) Eqatn (7) wen sbsttted nt eqatn () gves te alternatve rm r te eact trncatn errr at mde Tat s ) (8 0 ) (9 Te rder r D sceme te Pssn eqatn ma be wrtten eplctl as ) (0 Fr te sceme te D prblem we need a rt rder apprmatn te llwng: () We sbsttte eqatn () nt eqatn (8) t get:

7 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 ) ( 0 0 Gettng a cmpact st rder apprmatn reqres cmpact epressns r te nne dervatves rder s n eqatn () wc can be dne b rter derentatng eqatn () tat s () Als () Sbsttte eqatns () and () nt eqatn () gves ) ( 0 ater smplcatn te epressns n te secnd sqare bracet sng eqatn() and eqatn () te cmpact st-rder apprmatn te tree-dmensnal Pssn eqatn can ts be btaned as ) ( NMERICAL EXAMPLES In ts sectn we perrmed tw nmercal eperments t slve a dmensnal Pssn eqatn (9) n te nt sqare dman 0 0 In bt eamples pre Drclet bndar cndtns are prescrbed n all sdes te nt sqare

8 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 In rder t cmpare te nmercal sltn t te eact sltn we sed vectr e dene as e N N e 0 were e Prblem F and N s te nmber ndes sn sn sn sn 0 Te eact sltn s Prblem F( ) cs( )sn( ) cs Te eact sltn s sn CONCLSIONS 0 L nrm te errr In ts paper we present a cmpact nte derence scemes r ne tw and tree dmensnal Pssn eqatn Te dscretatn are and Te tree dmensnal eqatn are 7-pnt stencl r and 7-pnt r Te ne and tw dmensnal eqatns are 9-pnt Or nmercal eperments cnrm tat cmpact nte derence scemes are accrate and ecent metds Ts wr s etendable t 9-pnt sceme r tree dmensnal Pssn eqatns Table Cmptatn reslt r test prblem Errrs Lcatn( ) Eact Sltn Sceme (00) 0 79E-0 8E-08 (000) E-0 97E-08 (070) 0 8E-0 8E-08 (000) 070 0E E-08 (0000) E-0 0E-07 (0700) 0 09E-0 87E-07 (007) 0 7E-0 78E-07 (0007) 0 8E-0 8E-08 (0707) E-0 90E-08 Table Cmptatnal reslts r test prblem Errrs Sceme Lcatn( ) Eact Sltn Sceme (00 00) 07 78E-07 E-09 (00 00) 08 E-07 87E-09 (00 00) -08 7E-07 9E-09 (080 00) -07 9E-07 E-09 (00 00) 079 9E-07 07E-09 (00 00) 099 8E-07 79E-09 (00 00) E-07 8E-09 (080 00) E-07 78E-09 (00 00) E-07 E-09 (00 00) E-07 8E-09 (00 00) E-07 87E-09 (080 00) E-07 97E-09 (00 080) 07 E-0 9E-09 (00 080) 08 7E-0 78E-08 (00 080) -08 E-07 7E-08 ( ) -07 8E E-09 Sceme 7

9 Jrnal Matematcs and Tecnlg ISSN: N Agst 00 REFERENCES RK Sla X Zng Dervatn g-rder cmpact nte derence scemes r nn-nrm grd sng plnmal nterplatn J Cmpt Ps 0(00) 0-9 JZa W Da S Zang Frt-rder cmpact scemes r slvng mltdmensnal eat prblems wt Nemann bndar cndtns Nmer Metds Partal Derental Eqatns (007) - 78 M Ma W Ma X Wang A cmpact alternatve drect mplct derence metd r slvng parablc eqatn mlt-dmensn J Cmpt Appl Mat 87 (00) -70 K Omran A secnd-rder accrate derence sceme n nn nrm meses r nnlnear eatcndctn eqatn Far East J Appl Mat 0 (00) - H Han Z Sn X W Cnvergence a derence sceme r te eat eqatn n a lng strp b artcal bndar cndtns Nmer Metds Partal Derental Eqatns (008) 7-9 A L Pardanan WF Spt GF Care A stable mltgrd strateg r cnvectn-dsn sng g rder cmpact dscretatn Electrnc Transactns n Nmercal Analss (997) - 7 H Sn N Kang J Zang SE Carlsn A rt-rder cmpact derence sceme n ace centered cbc grds wt mltgrd metd r slvng D cnvectn-dsn eqatn Matematcs and cmpters n Smlatn (00) - 8 M Nabav MHK Sddq J Darga A new 9-pnt st-rder accrate cmpact nte-derence metd r te Helmlt eqatn J Snd and Vbratn 07 (007) G Stmann Cmpact nte derence sceme st rder r te Helmlt eqatn J Cmpt Appl Mat 9 (007) - 0 RF Bsvert A rt-rder accrate Frer metd r te Helml eqatn n tree dmensns ACM Trans Mat Stw (987) - K Omran M Aad Fnte derence dscretatn te Benamn-Bna-Man-Brgers eqatn Nmer Metds Partal Derental Eqatn (008) 9-8 D Kaa IE Ian Eact and nmercal travellng wave sltns r nnlnear cpled eqatns sng smblc cmptatn Appl Mat Cmpt (00) WF Spt GF Care A g-rder cmpact rmlatn r te D Pssn eqatn Nmer Metds Partal Derental Eqatns (99) - JC Strwerda Fnte Derence Scemes and Partal Derental Eqatns SIAM Pladelpa nd Ed 00 8