Pattern Formation in Chemical Reactions

Transcription

1 Paer Formao Cemcal Reacos See Backreedy BSc Compg Sesso 003/004 Te caddae cofrms a e work sbmed s er ow ad e approprae cred as bee ge were referece ad bee made o e work of oers. I dersad a falre o arbe maeral wc s obaed from aoer sorce maybe be cosdered as plagarsm Sgare of sde

2 Ackowledgeme I wold lke o ak my spersor Dr. Maew Hbbard for s dspesable elp sgs ad corbo rogo e drao of s proec. I wold also lke o ak my famly for beg ere for me we I fel mos oerwelmed w e ask aead of me. Bg p o Raymod! Fally may aks o Amda ad Yaz for prodg good ared baer mes of grea sress dde.a kow.

3 Smmary Ts proec fs o e feld of scefc compao as appled cemcal kecs o esgae e arable Oregoaor processes of e Beloso-Zabosky reaco ad spaal dmesos. Frs a reew of e ecqes for eorecal modelg was derake o defy ad sfy e mercal ecqes ad approac adoped for geerag e -D ad -D Oregoaor code. Nmercal ecqes for dscresg e goerg eqaos are defed ad oe s seleced ad employed o deelop e modellg code ecessary o carry o e compaoal smlaos. Te deeloped code s frs opmsed for speed ad accracy by comparso w pblsed work e frer smlaos are carred o replcae e ose of spral waes ad e effec of aryg e dffso coeffces of e reacas. Te model deeloped ad esed ere soles e B-Z reaco w mc less me seps ad o a larger grd gg e same leel of accracy as sow by e replcao of wae paers ad cages paers for e wae ckess ad rodess ad cocerao we compared w ose obaed w oer workers [] o a fer grd ad w mc more me seps. Te ams ad obeces of e proec were sccessflly me as demosraed by s repor ad e followg delerables; -D code Malab- o dsk -D code Malab ad C- o dsk Repor wc comprses; -Resls ad Comparso of -D Malab code w C -Resls of smlaos ad comparso of deeloped C code w refereced work o es aldao accracy ad effcecy.

4 Table of Coes Ackowledgme Smmary Table of Coes.. Caper : Proec Ole.... CONTEXT. OBJECTIVES.3 MINIMUM REQUIREMENTS.4 PROJECT PLAN.5 PROJECT DELIVERABLES 4 Caper : Backgrod Researc REACTION-DIFFUSION AND THE BELOUSOV-ZHABOTINSKY REACTION 6.. Reaco-Dffso Sysems 6.. Irodco o e Beloso-Zabosky Reaco 6. THE -VARIABLE OREGONATOR MODEL 7.. F N mecasm 7.3 NUMERICAL APPROXIMATION 9.3. Aalyss of Oregoaor Eqaos 9.3. Nmercal Appromao Tecqes 0.4 SPECIFICS OF THE PROBLEM DOMAIN 0.5 MOVTIVATION FOR THEORETICAL AND COMPUTATIONAL MODELLING. 0 Caper 3 : Tecqes for Nmercal Modelg.. 3. TAYLOR S THEOREM AND NOTATION 4 3. SPATIAL DISCRETISATION TIME DISCRETISATION Eler Meods Rge-Ka Meods ACCOUNTING FOR THE REACTION TERMS REASONS FOR CHOOSING A PARTICULAR TIME AND SPATIAL

5 DISCRETISATION TECHNIQUE EXPLICIT SCHEMES 3.7 IMPLICIT SCHEMES 3.8 COMPARISON Sably Tme-sep leg 3 Caper 4 : Desg ad Implemeao REQUIREMENTS 5 4. DOMAIN CHARACTERISTICS SPATIAL STEP SIZE AND TIME STEP SIZE BOUNDARY CONDITIONS INITIAL CONDITIONS CHOICE OF EQUATION CONSTANTS MATLAB IMPLEMENTATION 30 Caper 5 : Ealao ASSESSMENT AGAINST EVALUATION CRITERIA 3 5. DIFFICULTIES ENCOUNTERED IN PROJECT 35 Caper 6 : Coclso CONCLUSIONS 4 6. FURTHER WORK 4 Refereces. 4 Apped A Persoal Refleco.. 44 Apped B Implemeao of Eplc -D ad -D Scemes 45

6 Caper : Proec Ole. Coe Ts proec fs o e feld of scefc compao as appled cemcal kecs o esgae e arable Oregoaor processes of e Beloso-Zabosky reaco ad spaal dmesos. Nmercal ecqes for dscresg e goerg eqaos are defed ad oe s seleced ad employed o deelop e modellg code ecessary o carry o e compaoal smlaos. Te code prodced cold poeally e be sed o predc e acal ocomes of e processes wo carryg o e eperme. Te deeloped code s frs opmsed for speed ad accracy by comparso w pblsed work e frer smlaos are carred o replcae e ose of spral waes ad e effec of aryg e dffso coeffces of e reacas.. Obeces Te obeces of e proec are as follows: To ealae e aros ecqes aalable o appromae e Beloso- Zabosky reaco specfcally e - arable Oregoaor model. To prodce a mercal model from oe of e ecqes defed. To prodce a compaoal model of e -D -arable Oregoaor processes. To ealae e effcecy of e algorm. To ealae e accracy of e compaoal model agas resls obaed from smlar compaoal edeaors a are dealed pblsed papers. Ts parclar obece s dffere from a saed e md-proec repor sce e al paper a was beg refereced amely [] carred o epermes b ese dealed mosly e resls of racg e moeme of e spral wae p e -D process wc s o esgaed s proec. Te ew paper beg refereced amely [] dd carry o compaoal epermes wc dealed e effec aryg e dffso raes o e robsess o spral waes..3 Mmm Reqremes Te mmm reqremes are as follows: To selec e mos sable ecqes o prodce a eorecal model of e -D ad -D - arable Oregoaor processes.

7 Ts reqreme s dffere from a saed e md-proec repor. Orgally was eded oly o dscresed e -D Oregoaor model. Howeer e logcal progresso wold be o frs dscresed e -D Oregoaor model ad deelop a code for s ad e eed e dscresao aalyss o e -D Oregoaor model ad deelop a -D code for e process. Hece s s e approac adoped ere. To prodce a code o model e arable Oregoaor processes. To eame e effec of aryg e dffso rae. I addo e followg eacemes were defed as possble eesos o e mmm reqremes saed aboe: To esgae e sses assocaed w compaoal speed. To esgae e ose of spral waes e -D Oregoaor processes..4 Proec Pla Te al proec pla below depcs a ole of proec obeces ad mescales. Te mlesoes defed were as follows: Esmaed sar dae Teae deadle Mlesoe Complee al researc ad defy mercal ecqes for appromag ordary dffereal eqaos. /0/03 5//03 Eperme w MATLAB sofware 8/0/03 4//03 Prodce Md- Proec repor ealg backgrod researc. 4//03 5//03 Prodce prooype code o model -D BZ reaco 8//03 4//03 Aalyse ecqes defed comparg ad corasg em. 4/0/04 8/0/04

8 Prodce prooype code o model -D BZ reaco 3/0/04 3/03/04 Compare code ops w epermeal resls pblsed papers. 7/03/04 /04/04 Prodce fal repor 3/04/04 /04/04 Table.: Ial proec pla As e proec progressed e al pla as sow aboe ad o be modfed as scedlg dffcles were ecoered bewee s proec ad corsework. A e me leadg p o e sbmsso of e md-proec repor sable ecqes for appromag e eqaos were defed. As saed e md-proec repor epermeg w e MATLAB sofware oled aempg o prodce a compaoal model of e -D Brger eqao w esg kow-ow acqred mercal appromao ecqes from e secod year Scefc Compg modle. By e reqred deadle s was o complee oweer work ad bee sared o prodcg e code o model e -D Oregoaor processes. A rge rp ome mea o work was doe drg e perod of 6//03 o 3/0/04 so e reqred code for e -D Oregoaor processes was fsed a e sar of Semeser. I a aemp o brg e proec pla back o rack aalysg e mercal ecqes defed ad e remader of e -D code were doe smlaeosly oer e perod 4/0/04 o 8/0/04. Frermore work o e -D code of e Oregoaor processes was psed back l 9/0/04 ad was compleed by 3/03/04 me for e progress meeg o 6/03/04. Some era me was dedcaed o wrg p e delayed draf caper ad was compleed o 4/03/04. Te remader of e me was dedcaed o e ealao of e solo compleed o /04/04 ad prodcg e ecessary capers a wold comprse e fal repor. Te followg able sows e resed proec pla akg o acco e cages descrbed aboe: 3

9 Mlesoe Esmaed sar dae Deadle Complee al researc ad defy mercal ecqes for appromag ordary dffereal eqaos. /0/03 5//03 Eperme w MATLAB sofware. 8/0/03 4//03 Prodce Md- Proec repor ealg backgrod researc. 4//03 5//03 Prodce prooype code o model -D BZ reaco. 8//03 6//03 Aalyse ecqes defed comparg ad corasg em. Complee e remader of e -D code. 4/0/04 8/0/04 Prodce prooype code o model -D BZ reaco. 9/0/04 3/03/04 Wre p of draf caper 7/03/04 4/03/04 Compare resls agas daa pblsed papers 5/03/04 /04/04 Prodce fal repor 09/04/04 3/04/04 Table.: Resed proec pla.5 Proec delerables Te ams ad obeces of e proec were sccessflly me as demosraed by s repor ad e followg delerables; 4

10 -D code Malab- o dsk -D code Malab ad C- o dsk Repor wc comprses; -Reew ad defy ecqes for eorecal ad mercal modellg ad sfy mercal approac ad ecqes adoped. -Resls ad Comparso of -D Malab code w C -Resls of smlaos ad comparso of deeloped C code w refereced work o es aldao accracy ad effcecy. 5

11 Caper : Backgrod Researc. Reaco- Dffso ad e Beloso-Zabosky Reaco... Reaco-Dffso Sysems Reaco-Dffso sysems are sysems wc wo or more cemcal speces are allowed o reac w eac oer ad smlaeosly dffse across a spaal doma [3]. Te reaco bewee e cemcal speces s sally a fas process a akes place local o e speces. Dffso oweer s a mc slower process a aemps o fd a sable sae bewee e speces. I reaco-dffso sysems ere are acaor ad bor cemcals prese. Te acaor cemcal s aocaalyc meag s cocerao s creased de o e eraco of self w oe of s prodcs. Te bor cemcal reacs w e acaor a aemp o slow dow e prodco of e acaor so as o fd a sae of eqlbrm bewee em [4]. Te dffso aspec of e sysem appled by are sres o make e coceraos of e cemcals eqal across e spaal doma ee og e reaco process cosaly cages e coceraos of e cemcals. I s e eraco bewee ese cemcal speces locally ad globally a are resposble for e formao of paers... Irodco o e Beloso-Zabosky Reaco Te classc Beloso-Zabosky Reaco s a prme eample of a reaco-dffso sysem. I a geerc sese e descrpo of reaco-dffso sysems Seco.. wold be appled o also descrbe e Beloso-Zabosky reaco. I a more specfc sese oe recpe a defes e reaco ecompasses e caalysed caalys sed ypcally s cerm III ad IV odao of a orgac speces ypcally maloc acd by acd bromae o [5] wle beg allowed o dffse across a spaal doma e.g. a per ds. Alog w abrp ad slow oscllaory color cages wc come abo from e aros odao ad redco processes a occr drg e reaco paer formaos may also be see sc as spral waeforms alog wc e color cages propagae [6]. Fgre defes e ma reacos oled e BZ reaco-dffso sysem. 6

12 Process B: HBrO compees w Br as a redcg age for BrO 3. CeIV s prodced from e aocaalyc seqece. Color cage: red ble Process A: Redco of BrO 3 o Br a e redcg age Br BrMA s prodced as a resl [Br ] Process C: Te BrMA ad CeIV reac casg e Cocrre odao of e orgac speces Color cage: ble red Fgre.: Basc reacos a comprse e BZ reaco [6] Te fll Beloso-Zabosky reaco specfes abo dffere reersble reacos amog 5 cemcal speces [7]. I a aemp o mercally model ese reacos smpler models were dered a eample beg e -arable Oregoaor model.. Te -arable Oregoaor Model Te - arable Oregoaor model of e Beloso-Zabosky reaco s a redco o a more maageable form wc s specfed by fe reacos a seem o play a mpora role e reaco-dffso paers formed... F N mecasm Tere are fe basc sb- reacos a play a pere role o e reaco process of e ere model [7]. Applyg e Law of Mass Aco o ese reaco eqaos s possble o oba a 3-arable sysem of ordary dffereal eqaos [8]: U T W T = k AW k UW k3 AU k 4U = k AW kuw fk V 5.. V T = k AU k V

13 were: k.. k 5 are rae cosas a defe e reaco raes f s e socomerc facor a s a measre of e rao bewee e saaeos ales of e coceraos U W V A are coceraos of HbrO acaor cemcalbr CeIVbor - cemcal BrO 3 respecely ad T represes me. Frermore applyg Tyso s sggesos [5] e eqaos are made dmesoless reslg ree dmesoless rae eqaos: ε = qw w.4 / w ε = qw w f.5 =.6 were: ε ε / ad q are ew dmesoless parameers erms of e cosas a defe e reaco raes. w are dmesoless coceraos of HbrO acaor cemcal Br - CeIVbor cemcal respecely ad s dmesoless me. Te calclaos wc defe ypcally cose as sow [5] sgges a Hece.5 becomes: / ε erms of e ales of e reaco raes a are / ε s parclarly small ee close o zero. f w =.7 q As a resl.5 s elmaed ad e sbsg.7.4 ad.6 ges e eqaos for e arable Oregoaor model are: = = q f q ε.8.9 were keepg w e descrpo of a reaco-dffso sysems ge Seco.. ad also represe e acaor ad bor cemcals respecely [9]. 8

14 Eqaos.8 ad.9 represe e cage cocerao of e cemcal speces a local sese. To make e sysem of eqaos coform o e reaco-dffso model descrbed Seco.. dffso erms ms be added. Eac of e cemcal speces as dffere a rae a wc wll spread across e spaal doma ge by er dffso coeffces. As a resl e eqaos of e -arable Oregoaor model w dffso across a spaal doma are []: q f q D ε =.0 D =. were e coordae sysem s ge by y z ad D ad D are e dffso coeffces of e reacas ad respecely = y s proec. sce a mamm oly wo spaal dmesos are cosdered.3 Nmercal Appromao Dffereal eqaos are wdely ad eesely sed o model pyscal peomea sc as sock predcos fld dyamcs ad cemcal kecs. Wle some of ese eqaos ca be modelled eacly oers are mc arder ad somemes ofe mpossble ad a appromao o e eac solo ms sffce; mercal appromao ecqes allow for s..3. Aalyss of Oregoaor Eqaos Te eqaos a descrbe e -arable Oregoaor processes represeed by a par of parabolc paral dffereal eqaos as ey e mmal -D case ole depede arables; e spaal arable ad e emporal arable. W s md e solo o e eqaos wll ae e form ad wc represes e coceraos of e reacas a all ales of for all me. I spaal dmesos e fcos wold be y ad y. 9

15 W eqao. as a referece e eqaos ae a reaco erm ge by - ad dffso erm ge by D wc coform o dffere me scales sce reaco process bewee e speces ad dffso occr a dffere raes..3. Nmercal Appromao Tecqes Applyg appromao ecqes o dscresg.0 ad. eals dscresg eac of e paral deraes z. ad y For e me deraes ecqes clded e Eler Meods Rge- Ka Meods wereas for e spaal deraes ecqes clded e Fe Dfferece Meods all of wc ae aryg degree of accracy. I addo o e ecqes defed aboe oers sc as Fe Eleme Meods ad Gree Eleme Meods are also sed o dscrese paral dffereal eqaos of s are. Mc more deal wll be ge o e some of e appromao ecqes defed aboe laer Caper Specfcs of Problem Doma Te geeral coe of e proec s o eame e eorecal modellg ecqes ad compaoal mplemeao of a solo o reaco-dffso processes more specfcally e -arable Oregoaor model of e Beloso-Zabosky reaco. Specfcally e proec wold aemp o esgae e effecs of aryg e dffso raes e -D processes o e oscllaory beaor ad spral wae paer occrreces wc form drg e reaco..5 Moao for Teorecal ad Compaoal Modellg Te am of s proec s o prodce a robs ad accrae compaoal solo of e - arable Oregoaor model. Ts sofware ca e be sed o predc e beaor of e - arable Oregoaor we aros codos are mposed sc as dffere al codos ad cages o e eqaos cosas f ε q D D. I ca also be sed o aldae e eracy of epermeal daa ad ce ersa. Were e wo dffer dscrepaces ca be looked o wereer ey may le; e epermeal daa or mercal model. Te case mg ee es we sg a compaoal model s e oly aalable solo o e problem as somemes epermes ca be cosly o perform. 0

16 I a more geeral are compaoal modellg s sed maly for predco prposes. Some epermes e.g. cemcal reacos may o oly be cosly b also azardos. Teorecal modellg elmaes em bo s sace.

17 Caper 3 : Tecqes for Nmercal Modellg Te eqaos goerg e -arable Oregoaor reaco-dffso model of e Beloso- Zabosky reaco for wo cemcals reacg across a spaal doma are ge by: = q f q D ε.0 D = were e coordae sysem s ge by y z ad. f s e socomerc facor a s a measre of e rao bewee e saaeos ales of e coceraos ad q ad ε are cosas a defes e reaco rae = = oe spaal dmeso y wo spaal dmesos D ad D are e dffso coeffces of e reacas ad respecely. Eqaos.0 ad. as saed before are a par of parabolc dffereal eqaos wc order o be modelled compaoally frs eed o be modelled eorecally by sg sable appromao ecqes. Appromag e solo o e eqaos reqres dscresao of e s order me deraes ad as well as e d order spaal derae y Afer carryg o a lerare searc was fod a meros ecqes esed a was ormally appled o dscrese eac of e deraes sally dered from basc Taylor seres epasos of a fco abo pos space ad me. Frermore e mplemeaos of e meods ge rse o eplc or mplc scemes of appromaos.. Ts based o er compley ad mplemeaos smlar problems ad my famlary of sg em for corsework copled w e reasos ge aboe e followg appromao ecqes were cosdered as able solos o be mplemeed ad eac wold be cosdered dep.

18 Tme dscresao Eler Meods Rge-Ka Meods Spaal dscresao Fe Dfferece Meods Table 3.: Nmercal appromao ecqes Fe Dfferece Meods are relaely smple o mpleme ad are relaely compaoally epese [0] we compared agas e Fe Eleme Meods ad Gree Eleme Meods. Tey possess e adaage of beg able o effcely adle reglarly saped domas e.g. recaglar a are represeed by a se of pos wc s cosdered s work. More comple ecqes clde Fe Eleme Meods ad Gree Eleme Meods. Researc o e Fe Eleme Meods sowed a e maemacs oled was ery comple ad mplemeg em wold ae bee oo ambos ge e mescale of s work ad as a resl ey were o cosdered for s proec. Howeer s mpora o meo sce e doma of eres cold be of ay sape Fe Eleme Meods ae e adaage oer Fe Dfferece Meods sce ey are beer able o cope w rreglarly saped domas []. Sce domas a ae rreglar saped bodares are o cosdered s proec for reasos defed Caper 4 was o wor spedg era me esgag ese meods w e coe of s proec. Te dsadaage e Fe Eleme Meods ae a ey were deeloped specfcally for egeerg problems ad sally ake p mc sorage ad reqre as compg me [0]. Gree Eleme Meods was smbled po drg e wre p of e draf caper ad a s lae sage was decded a a dep aalyss wold ake p oo mc me sce e maemacs bed proed ery oled. I s oweer wor meog as ese meods are ormally appled o sole reaco-dffso problems []. Wa follows s a smmary of e ecqes Table 3. as appled we appromag eqao.. Smlar reasog s e appled o appromae eqao.3. As saed Caper e fco wc s a represeao of e cocerao of e reaca CeIV s depede o space ad me. Wa we am o fd s ow e cocerao of e reaca ares space as me progresses. 3

19 3. Taylor s Teorem ad Noao Cosderg e -D case ad applyg Taylor s seres epaso o e fco abo e pos k ad k leads o [3]: 3. k = k k k k = k k k... were k s e fed dsace bewee wo cosece saces me ad s e dsace bewee e pos! ad as depced Fgre 3.. For reasos defed laer Caper 4 e spaal pos are eqally spaced o e doma. Eqally spaced me seps are also sed sce ey are more coee o se. Ueqally spaced me seps are oweer o commo ad occr we cosderg adape appromao ecqes wc are o cosdered s proec. W a sad le a parclar po space ad me S be represeed by ad are egers ad e ale of a s po be deoed as or below s represeae of e grd pos me ad -D space [4]: = ; k were = e Fgre k k - S k -k - k - Fgre 3.: Lace of pos -D space ad me 4

20 I -D we eed o clde e era spaal arable y. Te po S s e represeed as = ; y = p; k ad e ale of e fco a s po s deoed as = y or y p p - S p -p -- p - Fgre 3. Lace of pos D space me as s o of page W e Fgres 3. ad 3. as a gde appromag eqao. wll resl fdg e ale of e cocerao a e spaal po S a e e me sace wc s deoed as y or we cosderg dmesos. I e parclar case we k = 0 3. ad 3. meas a e Taylor seres epaso of as bee appled e spaal doma oly ad a a parclar sace me s gg: 3 = Ο = Ο 3.4 Applyg smlar reasog we = 0 yelds: 3 k = f k k Ο k k = k k Ο k 3.6 5

21 6 3. Spaal Dscresao Te secod order spaal derae ge be dscresed by a mber of fe dfferece formlas wc are readly dered from e Taylor s seres epaso [5] ad ece from 3.3 ad 3.4 of e fco abo pos space a saces me. Usg e oao rodced Seco 3. ad e represeae po S a few of ese formlas are preseed below: Cosderg frs e appromao of : Addg 3.3 ad 3.4 leads o 3.7: 4 Ο = 3.7 were 4 Ο represes erms coag 4 ad ger powers of. Assmg s sffcely small ese ger powers ca be gored. Ts rearragg eqao 3.7 leads o 3.8: 3.8 wc as a local rcao error of order. Here e local rcao error represes e amo by wc e eac solo of e cocerao ale say V y a a me sep fals o sasfy e eqao ge for e appromao []. I s mpora sce ges a dea of e accracy of e appromao ecqe beg sed. Oer scemes ca be obaed smlarly by sg e Taylor s seres epaso of a dffere saces me e.g. ceerg e Taylor s seres epaso a e me sace yelds: 3.9 wc also as local rcao error of order. For e -D case y eqaos 3.8 ad 3.9 ca be rasformed respecely o: y y y y 3.0 y y y y 3.

22 7 Smlar reasog ca e be appled order o appromae y o ge: p y y y y yy 3. p y y y y yy Tme Dscresao Te dscresao of e frs order me derae as w e spaal derae ca be obaed drecly from e Taylor seres epaso of abo saces me a pos space aga w aryg degrees of accracy. Aga sg e same oao as rodced Seco 3. ad e represeae po S a few sc appromaos are dealed below: 3.3. Eler Meods Re-wrg eqao 3.5 ges 3.4: k k Ο = 3.4 were k Ο s represeae of erms coag secod ad ger powers of k; splag a k s sffcely small e ese ca be gored. Now rearragg eqao 3.4 yelds: I -D k ad -D 3.5 k y y y Ts s ermed e Forward Eler meod ad as a local rcao error of order k. Sbracg eqaos 3.5 ad 3.6 yeld 3.6: 3 k k Ο = 3.6 were 3 k Ο represes e erms a coa rd ad ger powers of k. Oce aga splag a k s aga sffcely small ese erms ca be gored.

23 Rearragg eqao 3.6 ges: I -D k ad -D 3.7 y y k y Ts s ermed e ceral dfferece ad as local rcao error of order k Rge-Ka Meods Te Rge-Ka meods am o prode a ger order local rcao error [] a e meods descrbed seco.3. ad s beer accracy. Rge-Ka meods acee s by calclag more formao abo e fco we mog bewee cosece me saces. Usg ese meods for dscresg e me deraes s parclar problem meas frs applyg oe of e meods dscssed for appromag e spaal derae e reag e ere rg ad sde of e eqao as a ew fco P f s prodcg: = P f 3.8 Recosrcg more formao abo e fco drg a me-sep oles akg a weged aerage of e slope of e fco a dffere pos e fco beg P f as dcaed by Eqao 3.8 Ts k k = P f = P f k were k k represe ales of e slope of f drg a me-sep. Te e fco ale a e po sace me ca be calclaed as follows []: I -D = kk 3.9 represeg e cocerao ale a e ew ad -D 3.0 y = kk 8

24 Ts s ermed e Mdpo Meod ad as a local rcao error of order k. Ee more g-order rcao error ca be aaed ad ece beer accracy by calclag ee more formao abo e fco mog from oe me sep o e e: k k k k 3 4 = P = P f = P f = P f f k 3 k k 3. were k Κ Κ k 4 aga represe ales of e slope of f drg a me-sep. Oce more e ale of e fco a e ew sace me s ge by: k 6 = k k k k 3 ad -D 3. k y = k k k k 3 Ts s ermed e Rge-Ka Order For Meod ad as a error of order k. 3.4 Accog for e reaco erms Te reaco erm eqao. eeds o be accoed for appropraely [6] order o complee e appromao process. Tere are may ways o do s b sce we are dealg w eqally spaced me seps ad grd pos oly parclar ways are preseed. If we le G = e s possble o ae e reaco erms ealaed as G or G -D ad -D. 3.5 Reasos for coosg a parclar me ad spaal dscresao ecqe As sow e proec scedle e early sages work ad sared o e mplemeao of e Forward Eler meod for me dscresao ag ecoered e secod year adaced scefc compg modle. I was o l afer e sccessfl mplemeao of e -D code sg e forward Eler for me dscresao wc was doe 3/03/04 a a aemp was made a mplemeg oe of e oer meods lsed Seco.3. Afer carefl 9

25 aalyss of wa eac meod reqred was decded a was sffce o keep e forward Eler me for e fal code. Ts decso was made based o e followg se of crera: Speed Implemeao smlar problems by oer researcers Te ceral dfferece meod was elmaed compleely before ee beg assessed agas e se of crera saed aboe. Ts was doe becase reqres e ealao of y. Ee og ad a accracy of order k proed awkward o se we ryg o sar gs off ge e al codos of e sysem. So for eample e al codos specfy e cocerao of e reaca all e spaal pos o e doma a me say. Te formla for e ceral dfferece allows for fdg e cocerao ale a ese pos a me erms of er ales a ad e laer of wc does o es a e begg of e smlao. Te Rge-Ka meods proed o be e bes of e ecqes defed we cosderg e accracy; e mdpo meod ad Rge-Ka Order For meod are of order k ad 4 k accracy respecely. I was o l afer e progress meeg a mplemeg e Rge-Ka meods was aemped. Howeer s mplemeao proed o be qe dffcl especally we ryg o keep rack of e k ales drg eac me sep. Ts fac copled w me resrcos ad codg dffcly mea mplemeao a s sage e proec proed dffcl ad order o keep o scedle s meod was abadoed. I s oed [7] a mplemeg e Rge-Ka s qe effce. I reqres more fco ealaos per me sep fco ealaos w e Modfed Eler ad 4 fco ealaos w e Rge- Ka meod. Ts cold ae aderse effec o compao me depedg o e are of e fco o be ealaed. W e aboe reasos md ad e fac a e forward Eler meod was mplemeed sccessflly smlar problems carred o [8] frer reforced my decso sg e forward Eler oer e Rge-Ka meod. Te ma dsadaage of e Forward Eler meod as see from e aboe aalyses o rcao errors dcae s e leas accrae of e ree ecqes defed. Te Forward Eler Meod copled w approprae represeao of e reaco erms ad dscresao of e spaal deraes ge by Eqaos 3.8 ad 3.9 prodces eplc or mplc scemes. 0

26 3.6 Eplc Scemes W referece o Fgre 3.3 below eplc scemes work by sg e preosly deermed cocerao ales a me sace o calclae e cocerao ales a e e me sace 9. Hece oly oe kow.e. cocerao ale a e ew me sace qay appears e appromao sceme wc ca be ealaed erms of kow qaes. As a resl we appromag eqaos.0 ad. eplc scemes am o fd e ew coceraos of e reacas ad a parclar pos space a e e sace me based solely o er ales space a e preos sace me. k A A Ukow cocerao ales Kow cocerao ales - Fgre 3.3 : Eplc scemes ealae ew ales based o old ales Based o e coces of sg e forward Eler for me dscresao ad eqao 3.8 for e spaal dscresao e followg s e reslg eplc sceme for appromag eqao.: I -D ad -D kd = k 3.3 kd = k for = 3... N ad = 3... M ad = 3... T were N s e mber of spaal pos sampled e coordae dreco M s e mber of spaal pos sampled e y coordae dreco ad T s e oal mber of me seps.

27 Te accracy of e sceme s of order k ad I ae cose o ae for reasos ge Caper 4 e spaal sep sze ad p o ake o eqal ales ad were I ae cose o appromae e reaco erm by G. Ts was de o e fac a sg G o represe e dffso erm wold ae e erm wc clearly egaes e defo of a eplc sceme. o e rg ad sde of e eqao 3.7 Implc Scemes Implc scemes oweer are dsgsable becase sce ey work by deermg cocerao ales a e ew sace of me based o er ales a e preos me sace as well as a 9. Hece or more kow cocerao ales a e ew me leel may appear e appromao s prodcg a sysem of smlaeos eqaos. As a resl we appromag eqaos.0 ad. mplc scemes am o fd e ew coceraos of e reacas ad a parclar pos space a e e sace me based solely o er ales a e preos sace me as well as e ales a e crre sace me. k B B B Ukow cocerao ales B Kow cocerao ales - Fgre 3.4 : Implc scemes calclae ew ales by sg ese ew ales as well as old ales. Oce aga based o e coce made o mpleme Forward Eler for me dscresao ad eqao 3.9 me ad spaal dscresao e followg s e reslg mplc sceme: I -D λ D λ D λ D = k 3.5

28 ad -D for 4λ D λ D = k 3.6 = 3... N ad = 3... M ad = 3... T were N s e mber of spaal pos sampled e coordae dreco M s e mber of spaal pos sampled e y coordae dreco ad T s e oal mber of me seps. Te accracy s of order k ad e spaal sep szes eac dreco are eqal. Oce aga I ae cose o appromae e reaco erm by G. Usg e represeao of e reaco erm ge by G complcaes e solo process sce meas yo wll ae erm o e rg ad sde of e eqao wc clearly poses a problem sce we are ryg o fd e cocerao ales of a e e me sace we appromag s eqao ad o e cocerao ale of. So a sysem of smlaeos eqaos s prodced wc eeds o be soled a eac me sep. 3.8Comparso 3.8. Sably De o e fe precso calclaos eforced o e appromao scemes errors are rodced we ryg o appromae e deraes wc propagae as e calclaos proceed. Ts e ma cocer a caracerses e sably of a mercal sceme s weer ese errors are amplfed or damped dow as e calclao progresses [5]. A codoally sable mercal sceme s oe were resrce bodg lms are eforced o ales a e spaal sep p ad me sep k ca ae; aboe ese resold ales e sceme s ermed sable. Ucodoally sables scemes oweer are caracersed by e codo a o sc bods es. Hag sad a we eglecg e reaco erm e eplc sceme descrbed aboe as e sably crero o e me sep s ge by k = α were 0 < α < -D ad 0 < α < -D. Wereas e mplc sceme s codoally sable wα < Tme-sep leg From e cepo of sg a parclar sceme a bodg lm s eforced o e me sep leg. Te me sep leg becomes a lmg facor especally we aempg o mproe e accracy e represeao e spaal dreco. 3

29 Cosderg e sably crero mposed for e eplc sceme splaes a k < ad k < -D ad -D respecely wc clearly sows a redcg e 4 ale of order o ge beer accracy meas sbsaally redcg e ale of k wc eals large compao me. By coras e mplc sceme mples a k < reealg a mc larger seps me are allowed we ryg o crease e accracy sag o compg me. Howeer e me seps sold o be oo large as correc solos may be obaed. Wa follows s a bref smmary as o wy e eplc meod was mplemeed. Despe ag a resrce sably ad me sep leg crero was relaely smple o mpleme ad compaoally epese [4]. W e mplc ecqe becomes ecessary o mpleme e sparse mar a wold resl we sg s sceme for e -D case. Hag sad a sg e mplc meod meas a a eac me sep a mar wc represes e sysem of smlaeos eqaos eeds o be ered wc s cosly erms of compg me especally f e doma o be represeed s ery large. Frermore preos researcers as [ 8] ae sed e eplc sceme ad fod s o be adeqae ece e eplc ecqe s adoped for s work. Fll deals of e mplemeao of e eplc meod meoed are ge Apped B. 4

30 Caper 4 : Desg ad Implemeao 4. Reqremes We modellg e -D ad -D Oregoaor processes a few ecesses eed o be place. To sar gs off al codos a represe e al dsrbo of e cocerao ales of e reacas across e doma eed o be sppled. Lkewse e cocerao ales a e bodares of e doma also eed o be accoed for appropraely for e drao of e eperme. As dscssed Caper 3 ere s a resrce sably crero assocaed w e eplc sceme of appromao beg sed s proec. Ts we coosg e spaal sze a approprae me sep sold be cose o esre sably. Te frs order me derae ad secod order spaal deraes Eqaos.0 ad. are e replaced by er respece appromaos as descrbed by e eplc sceme Caper 3 as well as sbsg ales for e aros eqao cosas f ε q D D. W algebrac maplao e reslg eqaos ca e be rearraged order o ae e kow qay.e. e cocerao ales a e e me sace beg o e lef ad sde ad kow qaes o e rg ad sde. So for eample: W e cocerao ales kow a all spaal pos ally ge by e al codos we ca proceed o fd e cocerao ales a e e sace me by solg a sysem of algebrac eqaos a relae e kow coceraos o ose a are kow. So referrg o Caper 3 Seco 3.4 Fgre ad e eplc sceme beg se effecely e -D case afer me sep from e begg of e smlao me e ew cocerao ales ge by of e reacas a all spaal pos k ad q kd f = ε q kd = k for = 3... N Ad a more geeral coe: k q kd f = ε q kd = k 5

31 for = 3... T ad = 3... N 4. Doma Caracerscs Te doma caracerscs referred o ere are e doma sze ad sape. I -D e doma of eres s represeed by a smple geomery e form of a recagle. By dog s we esre a cera grd pos cocde w e bodares makg calclaos a e bodares easer. I ms be oed a ag e doma of eres doma be represeed by a rreglar sape s o commo. I meas oweer assgg grd pos o e bodares s o as easy as sow Fgre 4. wc frer complcaes calclaos. Ts s deal w by eer assgg e bodary ales o e spaal pos closes o or modfyg e spaal dscresao ecqe ear e bodares [5 0]. y grd pos Irreglar bodary does o cocde w grd pos y Fgre 4.: W a rreglar doma assgg grd pos o e bodares s dffcl W e sape of e compaoal doma md a sqare mes s sed wc meas sep sze bo spaal drecos ad y ake o eqal ales. Ts coce was frer reforced sce a ecqe for e spaal dscresao a as e same order of accracy bo spaal drecos was beg sed. By dog s calclaos are smplfed ad are easer o perform ad code. Te ales of ad y eed o be of e same sze. Ts mg be sefl we e spaal derae s dscresg by a ecqe a as aryg accraces e spaal dmesos. So order o make p for e arao accracy wll be ecessary o make oe of e spaal seps smaller. 6

32 4.3 Spaal Sep Sze ad Tme Sep Sze Te spaal sep sze ad me sep sze are mpora as ey sold be cose so a sesble ad reasoably accrae solos are obaed a reasoable amo of compao me Caper 3 rodced e dea of sably ad demosraed a a small spaal ad me seps are adaageos order o prodce solos a are sable we sg e eplc sceme as mplemeed s proec. Howeer s ecessary a ey are o oo small as ey cold adersely affec compao me becase of e crease problem sze. Decreasg e mes sze meas samplg more spaal pos s creasg e accracy space. Howeer s comes a a prce sce meas a e me sep sze eeds o be cosderably small o acee s accracy wc r meas more compaoal seps order o reac a solo. Work preseed [] sggess a mes sze of 0.5 space s w 00 eqally spaced pos ad a me sep of 0.00 me s sg a eplc mplemeao wereas work preseed [] ses mes sze of 0.05 space s w 000 eqally spaced pos ad a me sep of me s sg a mplc mplemeao. I addo [] also sggess decreasg e mber of sampled spaal pos o 80 made o dfferece o e solos. Frer proof of s was fod [] were s sow a e solos obaed were depede of e mber of spaal pos sampled ad me sep. Ts payg aeo o e ales aboe ad afer some epermeg wc oled erfyg e occrrece of spral waes w e al codos ad parameers as sggesed Secos 4.4 ad 4.5 respecely a mes sze of 0.85 space s w 80 eqally spaced pos ad a me sep me s proed o be adeqae prodcg resls smlar o ose predced [] ad []. Coosg s mes sze was based o eecg e code e fle oregoaor_d.m w aros mes szes ad comparg e op of e code o e resls ge []. Ts was also cose so as o acee relaely accrae solos wo reqrg mc compao me as e Malab mplemeao was prog ery effce for larger mes szes. Ts e correspods o sg a spaal doma of Ω = [ ] [ ] rogo s work less oerwse saed. 4.4 Bodary Codos Te bodary codos referred o ere are e ales of e cocerao of e reacas a e ery edge of e spaal doma. I praccal saos e reacas are allowed o reac ad dffse w a fe area e.g. a per ds e bodares of wc are mpermeable meag a e reacas are able o 7

33 escape e essel. To mmc s as closely as possble compaoal modellg sg omogeeos Nema codos are appled o e bodares: = 0 = 0 Tese eqaos are sed o specfy a e rae of flow of e cocerao across e bodary s zero. Frermore ese parclar ypes of codos allow e prodco of spral waes e -D case oce appled o all bodares [] so er applcao s esseal. To apply ese codos e -D case e ales of e coceraos of e reacas calclaed a poso N are assged as e bodary codos wc are represeed a poso N drg calclaos a eac me sace. N N = = N N Smlar bodary codos are appled for e -D case b s me o all for sdes of a recaglar doma. So effecely -D we are sg e bodary codos ge by: = = = 0 y y Dagrammacally represeed Fgre 4. y M M p p bodary pos le o sdes of oer recagle eror pos N- N Fgre 4. : Cocerao ales as reaed a e bodares 8

34 From e Fgre 4. ca be see a wa follows are codos as appled o e lef bodary of e doma of eres: = = M M = = M = = M M = = M Applyg smlar reasog e codos for e remag 3 bodares are easly dered. 4.5 Ial Codos I s coe al codos refer o e cocerao ales of e reacas a e ery begg of e smlao sce we eed o ae sarg ales order o esgae ow ose ales eole oer me. Uless saed oerwse e al codos sed for e -D processes as ge by [] are as follows: = = q 0.8 q f f f f θ 8π f for 0 < θ < 0.5 elsewere Howeer [] adocaes a ee og radom al codos cold be appled e al codos ge aboe resl e rapd prodco of spral waes wo gog rog wld rase saes. I s for s reaso sg ese codos are desrable sce wll beefcal o ae we esgag e effec of cagg e reaco cosas sc as e dffsy cosa D or D o e prodco of spral waes esgaed Caper Coce of eqao cosas Te effec of aryg e eqao cosas are esgaed Caper 5 ad s wll be specfed a a me. Howeer al esg as bee carred o ad e coce of eqao cosas wold be: 9

35 f =. 4 ε = 0. 0 q = D =. 0 D = 0. 6 Tese ales were cose based solely o e fac a ey esre e prodco of a spral wae []. 4.6 Malab Implemeao I ae cose o mpleme e -D ad -D Oregoaor processes Malab. Ts was de o my famlary ad comfor w e lagage comparso o ay of e oers ecoered. I addo Malab as a bg adaage oer e oer lagages sce possesses a powerfl seleco of bl meods a make salsao of e resls ery easy. Moreoer Malab was bl especally for scefc ad egeerg problems wc frer reforced s coce as e lagage o se. Te -D process was mplemeed as e fco oregoaor_d.m. I akes as p e parameers begme edme dt L R N f e q D D represeg e sar me of e smlao e ed me of e smlao e me sep sze lower coordae of doma pper coordae of doma mber of spaal pos o be sampled e dreco respecely. Te las fe represe e cosas specfed by e eqao goerg e process. Te cocerao ales are sored w a ecor of sze N oe ale for eac spaal po sampled. W ese ales e program wll deerme e ew cocerao ales of e reacas a cremes of dt l a e specfed ed of e smlao based o e eplc sceme beg sed s proec. Te -D processes were mplemeed e mc e same way. I was mplemeed as e fco oregoaor_d.m w e addo of yl yr M wc are e lower y coordae ad pper y coordae of e doma ad e mber of spaal pos o be sampled e y coordae dreco. Te cocerao ales are sored w a -D array or mar of sze N*M oe ale for eac coordae e -D space. Oce aga e cocerao ales are calclaed a cremes of dt l e ed of e smlao s reaced w e eplc sceme beg sed s proec. I order o mproe e accracy space meas more grd pos wold eed o be sampled. For eample e -D case as saed seco grd pos are sampled eac of e spaal drecos makg a oal of 6400 spaal pos. Ts erefore meas a eac me sep 30

36 6400 algebrac eqaos eed o be soled wc correspod o e cocerao ales a ese 6400 spaal pos wc ook appromaely 8s o e Wdows 000 maces e Colosss Laboraory. Icreasg e mber of grd pos o 00 eac dreco meas samplg 0000 pos wc obosly creases compao me wc was sow o be appromaely 3s. I also became appare afer some epermeg a order o ge approprae wae perods space a larger doma wold eed o be cosdered. W e -D processes we are ow samplg more grd pos e spaal doma as opposed o e -D processes. As a resl more compao me eeds o be dedcaed o calclag e cocerao ales a ese eras spaal pos. Lkewse o ge approprae wae perods me meas allowg e smlao o r for a loger me perod. Usg e me sep sze ge seco 4.5 sowed a compao me creases f we were o r e smlao for a loger perod of me sce ere are more saces me a wc calclaos eed o be performed. For eample rg e smlao for me w 80 grd pos ook 8s wle rg e smlao for me s w e same mber of grd pos ook 64s aga w e oer parameers lef eqal. Te code oregoaor_d.m dd adle ese parameer cages ery well ad sed p mc compao me. Ts was de o e fac a Malab s a erpreed lagage meag s compled o e fly. Ts operaos sg codoals e.g. f saemes ad loops bo of wc are sed e code are ery effce. Improg o s compao me s possble by sg some of Malab s bl fcos e.g. fd b becase of my famlary w ese fcos added o e effcecy. I ms be oed oweer opmsg e code sg ese bl fcos somemes meas resrcrg e algorm compleely w e loss of good readably. As a becmark o assess e algorms effcecy Malab erms of compao me e code was coered o compled lagage C. Ts C program was mplemeed e fco oregoaor_d.c.te same parameers as descrbed aboe are passed o e fco ad aga e coceraos are calclaed. Te ales are sored e fles U. ad V. wc are e read by Malab fle r_c.m order o salse e resls. 3

37 Caper 5 : Ealao Based o e proec obeces e followg se of crera as bee defed o ealae e fal solo: Te effcecy of e algorm sed erms of me ake o complee e compaos. Accracy of e code erms of prodcg smlar qalae resls preseed []. 5. Assessme agas ealao crera As dscssed Caper 4 e eplc sceme mplemeed Malab proed effce erms of e compao me reqred. As a becmark o assess e compao me e code was coered o C. Table preses a smmary e wo mplemeaos erms of compao me. Acal compao me/s Nmber of me seps Malab C Table 5.: Comparso of compao me o ge approprae wae perod me sg e deeloped oregoaor_d.c ad oregoaor_d.m. Doma Ω = Toal mber of grd pos = As Table sows e mplemeao C s abo 0 mes faser a e Malab mplemeao. Te code as wre Malab may be composed o r more effcely b s reqres g ed ser eperse wc was o possble o deelop e proec mescale. Ts may be beefcal erms of ser coeece as bo e compao ad pos-processg of e resls ca be derake Malab. Howeer wrg ad complg e program C ad pos processg e resls Malab aceed a ger leel of effcecy dscssed below w oly lmed mpac o ser coeece. Also we creasg e accracy space by samplg more grd pos e C erso of e code proed aga o be abo 0 mes faser a e Malab erso. 3

38 Nmber of grd pos Acal compao me/s Malab C Table 5.: Comparso of compao accracy space sg e deeloped oregoaor_d.c ad Ω = Nmber of me seps = oregoaor_d.m. Doma Frermore e ery are of e eplc sceme sed reqres a a small spaal me sep be sed. Oe oer alerae as dscssed Caper 3 seco 3.6 cold ae bee o mpleme a mplc sceme wc allows a larger me sep o be sed. Usg a mplc sceme reqres a a mar be ered a eac me sep wc s ery cosly especally f e mar represes a large mber of grd pos. Howeer ecqes for fas mar erso sc as LU facorsao ad Gassa Elmao es wc redces e compg me reqred for e erso. As eplaed Caper 3 sg e mplc meod meas a a eac me sep a mar wc represes e sysem of smlaeos eqaos eeds o be ered wc s cosly erms of compg me especally f e doma o be represeed s ery large. Frermore preos researcers as [] ae sed e eplc sceme ad fod s o be adeqae ece e eplc ecqe was adoped for s work. Te work preseed [] wc sed a mplc sceme dealed e robsess of spral waes we e parameers f ε q D D were caged. Tey clded deals o e separao ad wd of e spral waes prodced. Te resls preseed [] are sed s work for comparso as e followg seco ealaes e accracy of e code deeloped s work based o pyscal obseraos preseed []. Te work doe [] ses a grd of sze of I s work a grd sze of 0 0 was sed rogo as recommeded by [] based o grd depedecy ess codced a work. Frer s work grd depedecy was cecked by codcg smlaos o a 0 0 grd ad o measrable dfferece were obsered e resls as sow Fgres a ad b. Howeer e compaoal me for e 0 0 grd was oer wce as log as 33

39 for e doma as sow Table 5.. Hece e grd sze was adoped e followg smlaos. Eperme : For eqao cosas of f =. 4 ε = 00. q= 000. D = D = 06 [] sggess a a cker ad. roder spral wae a w parameers of f =. 4 ε = 0. q = D = D = 0 6 s prodced.. Te resls of s works smlaos are preseed Fgres ad. Fgres a ad b are e resls o a grd ad 0 0 wc demosraed e grd depedecy dscssed aboe. To a ed Fgres c ad Fgres a ad b are e resls o e grd for e eqao cosas aboe ad al codos as se o Caper 4 Seco 4.5. Ts works smlao resls for eqaos cosa as sow Fgre a are smlar o ose obaed by [] dcag a e model deeloped ere s able o replcae e wae beaor. Frer cages o e eqaos cosas from o resls smlar creases wae ckess ad rodess as ose obsered [] for bo me seps ad me seps. Terefore e comparso of s works smlaos sow Fgres a ad c ad Fgres a-b w ose of referece [] dcae a e model deeloped ere replcaes bo e wae paers ad cages ese wae paers as obsered me. Ts work wc ses a eplc sceme for e mercal solo comprsg e forward Eler me ad cered dfferece space ge solos mc qcker me seps ad o a larger grd b w e same accracy a ose obaed [] wc ses a mplc sceme w a mc fer grd Eperme : Addoally frer smlaos of e code deeloped s work were ow cagg e dffso coeffce D from o 0.0 for eqao cosas 3 f =. 4 ε = 0. 0 q = D = 0. 0 D = 0 6 ges resls as sow Fgres a-c. Tese. resls are smlar o ose obaed [] as sow Fgre a Fgre 3a ad Fgre b Fgre 3b were for small ales of D e al profle of e acaor dffses slowly ad acqres a rg lke srcre caracersed by arrow srpes were e acaor coceraor s g ad mc wder alleys were e acaor cocerao s ery small. Here aga e code deeloped s work prodces smlar resls obaed [] ee der dffere al codos b o a larger grd ad w less me seps a ose sed []. Te model deeloped ad esed ere soles e B-Z reaco w mc less me seps ad o a larger grd gg e same leel of accracy as sow by e replcao of paers ad cages 34

40 paers for e wae ckess ad rodess ad cocerao we compared w ose obaed o a fer grd ad w mc more me seps. 5. Dffcles ecoered proec Tese were prmarly relaed o maag e mlesoe scedle sc as; Learg Malabs fll fcoaly so o make Malab code more effce ook mc loger a orgally esaged. No ag eog me o mpleme oer ecqes for e me derae e.g. Rge-Ka was aemped b proed fddly we dealg w sysem of eqaos. Ts was aemped afer e progress meeg 6/03/04. Implemeg e mplc -D ad -D meods; proed o be fddly ad ery complcaed o mpleme. Dffcly erpreg pyscal meag of plos prodced by -D code. No ag daa o compare resls of -D code agas. Sccesses - Afer mor modfcaos o e obeces as oled Caper ad reewg e lerare proec ole ad mlesoe became clear a codg e Rge-Ka ad mplc sceme of wc a grea deal of me was spe o s ecessag a reew of e mlesoe o e esg -D or laer deeloped -D code was o ecessary a s sage. Ts was bore o by e sccesses of e -D C code sg e eplc sceme as dscssed aboe. 35

41 Resls for Eperme : me seps a Resls o e grd me seps me seps b Resls o 0 0 grd me seps 36

42 me seps c Resls o grd me seps Fgres Acaor cocerao for parameers of f =. 4 ε = 0. q = D = D = me seps a Resls o grd me seps 37

43 me seps b Resls o grd me seps Fgres Acaor cocerao for parameers of f =. 4 ε = 0. 0 q = D = D = 0 6. Te resls aboe depc e reds obsered by []. Fgres a ad b sow e spral waes prodced are deed cker ad roder we compared w Fgres a ad c. Resls for Eperme : me seps a a 38

44 me seps b b me seps c c Fgres 3 Resls preseed [] 39

45 me seps d Fgres 3 Ts works resls I sold be oed a e resls preseed s work appear reersed we compared o e resls []. Ts s de o Malab s coeoal oreao of s co ordae aes. 40

46 Caper 6 : Coclso 6. Coclsos Ts proec ealed aalysg e relea mercal appromao ecqes aalable a aemp o prodce a mercal model represeg e -arable Oregoaor model of e Beloso-Zabosky reaco. Seeral ecqes a are ormally appled o dscrese e me ad spaal derae were eamed. Te forward Eler meod ad e cered dfferece meods were sed a eplc sceme ad ally a -D code was formlaed Malab. No epermeal or compaoal daa was fod o be aalable for comparso w smlao w e -D code deeloped e frs sages of s work. Ts was scceeded by a -D Malab code ad e by a -D code programmed C w lked salsao Malab. Te laer code was fod o work more effcely a e -D Malab code we appled o smlao rs. Frer smlaos codced w s deeloped code esgaed grd depedecy. More smlaos ad resls were obaed a compared e -D Malab code w e correspodg C compled erso. I was fod a -D C code was compaoally mc faser ad prodced smlar resls o a of e Malab -D code. Ne e -D C code was compared w refereced work [] ad ese smlaos sowed a was able o prodce smlar reds afer ceckg for grd depedecy. Te deeloped -D C code sg a *80 ad less me seps was more effce prodcg smlar wae paers ad dffso reds for bo smlar eqao cosas ad we aryg e dffso coeffce we compared w refereced work [] wc sed a ad smaller me seps. Te ams ad obeces of e proec were sccessflly aceed as were all e proec delerables were me sc as -D code Malab- o dsk; -D code Malab ad C- o dsk ad s repor. 6. Frer Work: Ts wold eal eedg e deeloped -D C code o clde a mplc sceme so s eablg larger me seps o be ake. Icreasg e accracy me ad space by mplemeg ger order ecqes of appromao e.g. sg e Rge-Ka meods for me dscresao. Comparg e accracy of e -D code agas epermeal or compaoal daa. Opmsg e -D Malab code o make more effce a sg Malabs bl fcos ad ecorsg ecqes. 4

47 Refereces [] Jake W. Skaggs W.E. Wfree A.T.989 Cemcal Vore Dyamcs e Beloso- Zabosky Reaco ad e Two-Varable Oregoaor Model Joral of Pyscal Cemsry [] Ramos R.I. 004 Robsess of spral waes wo-dmesoal reace-dffse meda Appled Maemacs ad Compao [3] Trk G. Reaco-Dffso [Ole]. [Accessed 3 rd December 003]. Aalable from World Wde Web: p:// [4] A fredly rodco o reaco-dffso sysems [Ole]. [Accessed 4 December 003]. Aalable from World Wde Web: p://a-lab.cs..l/pbs/rd_vreeke_fredlyirodco.pdf [5] Gray P. Sco S.K. 994 Cemcal Oscllaos ad Isables: No-Lear Cemcal Kecs. Claredo Press: Oford. [6] A Aalyss of e Beloso-Zabosk Reaco [Ole]. [Accessed 3 rd December 003]. Aalable from World Wde Web: p:// [7] Tyso J.J.985 A Qaae Acco of Oscllaos Bsably ad Traelg Waes e Beloso-Zabosk Reaco : Oscllaos ad Traelg Waes Cemcal Sysems Feld R.J. Brger M. Eds.; p08 Wley: New York. [8] Mrray J.D. 993 Maemacal Bology. Sprger: Germay. [9] M -M -Vllar V. 998 Araco ad replso of spral waes by localzed omogeees ecable meda Pyscal Reew Leers 583. [0] Almaz K. Heda R. ad Solma M. 003 Nmercal aalyss of a reaco-dffsocoeco sysem Compers ad Cemcal Egeerg

48 [] Brde R.L. ad Fares J.D. 997 Nmercal Aalyss. Brooks/Cole Pblsg Compay: USA. [] Oyeekwe O.O. 00 Gree eleme solos of olear dffso-reaco model. Compers ad Cemcal Egeerg [3] Garabeda P.R. 964 Paral Dffereal Eqaos. New York; Lodo:Wley. [4] Sm G.D. 978 Nmercal Solo of Paral Dffereal Eqaos: Fe Dfferece Meods. Oford Uersy Press: Oford. [5] Lapds L. ad Pder G.F. 999 Nmercal Solo of Paral Dffereal Eqaos Scece ad Egeerg. Jo Wley ad Sos. [6] Ramos J.I. 983 A Reew of Some Nmercal Meods for Reaco-Dffso Sysems Maemacs ad Compers Smlao [7] Press W.H. Flaery B.P. Tekolsky S.A. Veerlg W.T. 99 Nmercal Recpes C: Te Ar of Scefc Compg. Cambrdge Uersy Press. [8] Gog Y. ad Crs. D.J. 003 Aspral Waes Reaco-Dffso Sysems. Pyscal Reew Leers 908. [9] Orega J.M. ad Poole Jr. W.G.98 Nmercal Meods for Dffereal Eqaos. Pma Pblsg Ic. [0] Caraa B. Ler H.A. ad Wlkes J.O. 969 Appled Nmercal Meods. Jo Wley ad Sos Ic. [] Pär-Eader E. ad Söberg A. 999 Te MATLAB 5 adbook. Harlow:Addso-Wesley. 43

49 Apped A : Persoal Refleco Ts seco ams o prode a descrpo o e proec eperece. Uderakg s proec as sow e mporace of proper me maageme ad ask scedlg. Tere were sages were I was ryg o complee corsework ad a e same e me mee a proec mlesoe deadle wc proed dffcl o do. I dsg sarg e corsework as soo as ey are ssed s e wse coce erms of geg e work reqred for a proec deadle o be doe o a g sadard. Howeer s dffcl o dge or deerme ow mc work ad me s eeded o lear mpleme eece ad aalyse ad erpre ew formao or code a programme wo pror learg eperece. May mes a grea deal of effor ad me was spe researcg ad learg o se ew maeral o become comforable as a eper ser or ryg o ge gs doe eacly rg fe deal. Fdg e rg compromse or balace bewee compleg obs or dersadg someg eacly grea deal o wa s ecessary o ge e ob doe accepable me ad o accepable sadards s mpora ad ca oly perfeced oer me ad rog more eperece gaed workg o proecs. Tere s a grea deal of work a eeds o be doe f e proec s o be compleed o a g sadard. Beg largely resposble for e sapg ad dreco of e work s bes o be realsc ad o opefl a aceg ambos mlesoes. Wrg p draf capers o e work doe sead of gdeles s more effece a dedcag a perod earg e ed of e proec o prodce e fal docme based o ese gdeles. Tere were perods were some Capers ook loger a epeced o complee wc resled workg for rdclosly log ors o mee e deadle or cac p. Proof readg ad formag of e fal docme akes p some amo of me ad s a eercse wc sold be ge a leas a week s bffer me o complee. Coosg a proec based o a area of persoal eres s crcal o e sccessfl ad of e proec. Ee og I dd coose a proec e feld of scefc compg wc I fd ery eresg s applcao o cemcal kecs s someg I fel mc repdao dealg w as e proec progressed. Te reaco-dffso process beg modelled s proec s qe comple ad s sll a area of growg eres. Hag sad a ag a sem-frm dersadg of e applcao area from e cepo wold elp a grea deal compleg e proec. To some degree derakg s proec as poed o e amo of work oled we derakg a proec dsry ad demosraed e eed for a really good realsc pla of ees ad mlesoes. 44