NUMERICAL METHOD FOR TWO DIMENSIONAL NONLINEAR SCHRÖDIGER EQUATION WEI YU. (Under the Direction of Thiab R. Taha) ABSTRACT

Transcription

1 NUMERICAL METHOD FOR TWO DIMENSIONAL NONLINEAR SCHRÖDIGER EQUATION b WEI YU Uder he Dreco of Thab R Taha ABSTRACT The olear Schrödger euao s of remedous mporace boh heor ad applcaos The NLS pe euao s he ma goverg euao he area of opcal solos Varous regmes of pulse propagao opcal fbers are modeled b some form of he olear Schrödger euao I hs hess we roduce seueal ad parallel umercal mehods for umercal smulaos of wo dmesoal olear Schrödger euaos We mpleme he parallel mehods o he pcluser mulprocessor ssem a UGA The umercal resuls have show ha hese mehods gve good resuls ad cosderable speedup INDEX WORDS: NLS Spl-sep mehod pseudo-specral mehod Fe dfferece mehod Parallel algorhms FFTW

2 NUMERICAL METHOD FOR TWO DIMENSIONAL NONLINEAR SCHRÖDIGER EQUATION b WEI YU BE Beg Uvers of Poss & Telecommucaos Cha 8 A Thess Submed o he Graduae Facul of The Uvers of Georga Paral Fulfllme of he Reuremes for he Degree MASTER OF SCIENCE ATHENS GEORGIA

3 WEI YU All Rghs Reserved

4 NUMERICAL METHOD FOR TWO DIMENSIONAL NONLINEAR SCHRÖDIGER EQUATION b WEI YU Maor Professor: Commee: Thab R Taha Hamd R Araba Dael M Evere Elecroc Verso Approved: Mauree Grasso Dea of he Graduae School The Uvers of Georga December

5 ACKNOWLEDGEMENTS I would le o ha m maor professor Dr Thab R Taha for hs effors gudace ad suppor whch made m research beer I am so deepl graeful o he members of m advsor commee Dr Hamd R Araba ad Dr Dael M Evere for her d ad valuable help I also wa o ha all of he facul saff ad m freds he Deparme of Compuer Scece for dscussg wh hem ad learg a lo from hem especall Meg alg wh hm helps me greal Fall I would le o ha m facée Xaobo Lu for her couous suppor ad rus v

6 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS v LIST OF TABLES v LIST OF FIGURES v CHAPTER INTRODUCTION PRELIMINARIES 3 THE DISCRETE TWO DIMENSIONAL FOURIER TRANSFORM 3 THE TWO DIMENSIONAL FOURIER TRANSFORM 4 3 THE FASTEST FOURIER TRANSFORM IN THE WEST 4 4 THE SPLIT-STEP METHOD 5 5 THE PSEUDO-SPECTRAL METHOD 7 6 THE FINITE DIFFERENCE METHOD 8 7 MESSAGE PASSING INTERFACE 3 TWO DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION 3 NUMERICAL METHOD 3 NUMERICAL EXPERIMENTS 5 33 PSEUDO-SPECTRAL METHOD AND EXPERIMENTS 34 FINITE DIFFERENCE METHOD AND EXPERIMENTS 4 35 PARALLEL IMPLEMENTATION AND EXPERIMENTS 9 v

7 4 RANKING 35 5 CONCLUSION 36 REFERENCES 37 v

8 LIST OF TABLES Page Table 3: Covergece raes me for he frs-order splg mehod 7 Table 3: Covergece raes space for he frs-order splg mehod 8 Table 33: Covergece raes me for he eplc mehod 5 Table 34: Covergece raes space for he eplc mehod 5 Table 35: Resuls of parallel mplemeao for he frs-order splg mehod 3 Table 36: Resuls of parallel mplemeao for he secod-order splg mehod 3 Table 37: Resuls of parallel mplemeao for he fourh-order splg mehod 3 Table 38: Resuls of parallel mplemeao for he pseudo-specral mehod 3 Table 39: Resuls of parallel mplemeao for he modfed pseudo-specral mehod 33 Table 3: Resuls of parallel mplemeao for he eplc mehod 33 Table 3: Resuls of parallel mplemeao for he mplc mehod 34 v

9 LIST OF FIGURES Page Fgure 3: The al codos 9 Fgure 3: Two dmesoal SSF 9 Fgure 33: Two dmesoal SSF Fgure 34: Two dmesoal SSF4 Fgure 35: Two dmesoal pseudo-specral mehods 3 Fgure 36: Modfed wo dmesoal pseudo-specral mehods 3 Fgure 37: Two dmesoal eplc mehods 6 Fgure 38: Two dmesoal mplc mehods 9 v

10 CHAPTER INTRODUCTION The olear Schrödger euao NLS s beg used o descrbe a wde class of phscal pheomea eg hea pulses solds modulao of deep waer waves ad helcal moo of a ver h vore flame [] The research o opcal solos has bee gog o for ma decades ad he wo dmesoal olear Schrödger NLS euao s oe of he ma goverg euaos hs area of sud [9] [6] [7] I hs hess we wll sud he wo dmesoal NLS euao whch s gve b [9] where s a wo dmesoal comple-valued fuco There are ma popular umercal mehods o solve he NLS euao Oe of hem s he spl-sep Fourer SSF mehod proposed b R H Hard ad F D Tapper [5] I s oe of he mos popular umercal mehods for solvg he NLS euao [] Varous versos of he spl-sep mehod have bee developed o solve he NLS euao [3] Taha ad Xu have developed spl-sep mehod for he NLS ad CNLS euaos as well as parallel mplemeaos for hese mehods [3] S Zold e al [3] has mplemeed a parallel spl-sep Fourer mehod for large-scale smulaos of he NLS euao whch are reured for ma phscal problems M S Ismal

11 ad T R Taha have roduced a fe dfferece mehod o umercall smulae he CNLSE [] I hs hess we mpleme hree schemes of he wo dmesoal spl-sep Fourer mehod ad wo schemes of he pseudo-specral mehod for he umercal smulao of he wo dmesoal NLS euao We also mplemeed he parallel spl-sep Fourer mehod ad pseudo-specral mehod wh he Fases Fourer Trasform he Wes FFTW whch s developed b M Frgo ad S G Johso [4] Moreover we have mplemeed wo schemes of he wo dmesoal fe dfferece mehods usg eplc ad mplc mehods The frs chaper of hs hess s a roduco of he NLS euao The secod chaper descrbes he Fourer rasform he spl-sep mehod he pseudo-specral mehod ad he fe dfferece mehods Chaper 3 preses he umercal mehods o solve he wo dmesoal NLS euao ad epermes resuls The fourh chaper shows he ra for all he mehods ha have bee ulzed o solve he wo dmesoal NLS euao Chaper 5 provdes a summar of m curre research wor ad gves he cocluso

12 CHAPTER PRELIMINARIES THE DISCRETE TWO DIMENSIONAL FOURIER TRANSFORM If {f [m ]} s a seuece of sze M N obaed b ag samples of a couous fuco f wh eual ervals a he dreco of m ad respecvel he s dscree Fourer rasform DFT s gve b F N M MN π M N [ l] f [ m ] e < M l < N m m l where M ad N are he umbers of samples ad drecos boh spaal ad freuec domas respecvel Ad F [ l] s he wo dmesoal dscree specrum of [ m ] f The verse wo dmesoal DFT flps he sg of he epoe whch s defed as f π M N [ m ] F[ l] e m < M < N MN N l M m l I s he verse of he forward wo dmesoal DFT he sese ha compug he verse rasfer afer he forward rasform of a gve seuece would eld he orgal seuece respecvel Boh F [ l] ad [ m ] f could be cosdered as elemes of wo M N marces ad F 3

13 THE TWO DIMENSIONAL FOURIER TRANSFORM The Fas Fourer rasform FFT s a effce algorhm o compue he dscree Fourer rasform DFT ad s verse A DFT decomposes a seuece of values o compoes of dffere freueces I s useful ad beg used ma felds However calculag DFT drecl from s defo s ofe oo slow Isead FFT s a beer wa o calculae he same resul more ucl Compug a DFT of N pos reures O N However a FFT ca calculae he same resul ol O Nlog N operaos [9] The dfferece speed could be subsaal especall for large daa ses where N ma be ver huge I hs case FFTs are of grea mporaace o a large umber of applcaos le dgal sgal processg ad solvg paral dffereal euaos A wo dmesoal FFT s acheved b frs rasformg each row replacg each row wh s oe dmesoal rasform FFT ad he rasformg each colum replacg each colum wh s rasform A wo dmesoal FFT of sze M N reures M N oe dmesoal FFT 3 THE FASTEST FOURIER TRANSFORM IN THE WEST The Fases Fourer Trasform he Wes FFTW s a sofware lbrar used o calculag DFTs developed b M Frgo ad S G Johso MIT FFTW s a comprehesve colleco of fas C roues for calculag he DFT oe or more dmesos of boh real ad comple daa ad of arbrar pu sze [3] I has gaed a wde accepace boh academa ad dusr because provdes ecelle performace o a vare of maches eve compeve wh or faser ha euvale lbrares suppled b vedors [] 4

14 FFTW auomacall adaps he DFT algorhm o deals of he uderlg hardware cache sze memor sze regsers ec The er loop of FFTW s geeraed auomacall b a specal-purpose compler The FFTW begs b geerag codeles A codele s a fragme of C code ha compues a Fourer rasform of a fed small sze eg 6 or 9 A composo of codeles s called a pla whch depeds o he sze of he pu ad he uderle hardware A rume he FFTW s plaer fds he opmal decomposo for rasforms of a specfed sze o our mache ad produces a pla ha coas hs formao The resulg pla ca be reused as ma mes as eeded Ths maes he FFTW s relavel epesve alzao accepable FFTW also cludes a shared-memor mplemeao o op of POSIX hreads ad a dsrbued-memor mplemeao based o MPI Message Passg Ierface 4 THE SPLIT-STEP METHOD The spl-sep Fourer mehod s a pseudo-specral umercal mehod used o solve olear paral dffereal euaos [] For eample cosder he followg euao L N 4 where L ad N are lear ad olear operaors respecvel I geeral he operaors L ad N do o commue wh each oher The wo dmesoal NLS euao wh a real umber ca be rewre as L N 5

15 where L N The soluo of euao 4 could be advaced from oe me-level o he e b usg he followg formula [3] [ L N ] ep 4 where deoes he me sep I s frs order accurae However would be eac f operaors L ad N are me-depede [8] Now he me-splg procedure cludes replacg he rgh-had sde of 4 b a approprae combao of producs of he epoeal operaor ep L ad ep N We ca fd oe aswer b usg he Baer-Campbell-Hausdorf BCH formula [] for wo operaors A ad B as followg ep ep ep λ A λb λ Z 43 Where λ s he coeffce of A ad B ad Z A B Ad he remag operaors Z are commuaors of A ad B commuaors of commuaors of A ad B ec The epressos for Z are acuall raher complcaed eg Z [ A B] Where [A B] AB BA s he commuaor of A ad B ad Z 3 [ A [ A B ] [ A B] B] From hs resul we ca easl ge he frs-order appromao of he epoeal operaor 4 as follows [3] 6

16 L ep A ep N 44 Noe ha hs epresso s eac wheever L ad N commue I would be more covee o vew he scheme 44 as frs solvg he olear par N he advacg he soluo b solvg he lear par L b emplog he soluo of he former as he al codo of he laer Tha s he advaceme me s carred ou wo seps he so called spl-sep mehod The secod-order appromao of he epoeal operaor 4 s gve b A ep N ep L ep N 45 The fourh-order appromao of he epoeal operaor 4 whch preserves he smmer could also be cosruced [3] [3] eg A A A [ ω ] A ω 4 ω 46 where 3 3 ω 47 3 Noe ha he operaors L ad N ma be erchaged whou affecg he order of he mehod [4] 5 THE PSEUDO-SPECTRAL METHOD Pseudo-specral mehods are a class of umercal mehods whch are used appled mahemacs ad scefc compug for he soluo of paral dffereal euaos [5] 7

17 Gve we wa o fd wh a small The frs sep we should ae s o compue a ermedae value u b applg he rghmos operaor he smmerc decomposo V r / h u e 5 where h s he reduced Plac cosa Vr depeds ol o poso r To solve 5 reures ol a po wse mulplcao The e sep s o appl 5 T / h u e u where T s he ec eerg To smplf he above calculao we ca ae he followg euao o compue u Φ h / m e Φ whch also reures ol a po wse mulplcao Φ could be obaed from u usg he Fas Fourer rasform FFT verse FFT The fal compuao s u could be obaed from V r / h e u We ca summarze he above seuece as [ [ ] V r / h h / m V r / h e F e F e b Φ b usg he 6 THE FINITE DIFFERENCE METHOD Fe dfferece mehods are umercal mehods for appromag he soluos o dffereal euaos usg fe dfferece euaos o appromae dervaves [8] 8

18 6 EXPLICIT METHOD I eplc mehod we calculae he sae of a ssem a a laer me from he sae of he ssem a he curre me Gve Y as he curre ssem sae ad Y as he sae of he ssem as a laer me he we have Y F Y Usg he classcal eplc mehod wh ceral dfferece me he fe dfferece represeao of would be 6 where < N ad < N To compue frs we use as he al codo ad use o 6 o ge The for T we use he whch we ge he las compuao o oba 6 IMPLICIT METHOD I mplc mehod we fd he soluo b solvg a euao volvg boh he curre sae ad he laer oe of he ssem Gve Y as he curre ssem sae ad Y as he sae of he ssem a a laer me we solve he followg euao o oba Y G Y Y 9

19 Usg he alerag dreco mplc mehod wh ceral dfferece me he fe dfferece represeao of would be / / / / / 63 ad / / / / / / / 64 where N < ad N < To compue frs we use as he al codo ad use o 63 o ge / Secod we use he / whch we ge he las compuao o 64 o oba The for T we repea las wo operaos o oba 7 MESSAGE PASSING INTERFACE Message Passg Ierface MPI s a lbrar specfcao for message-passg proposed as a sadard b a broadl based commee of vedors mplemeaos ad users Message passg s a paradgm ha has bee wdel used o cera classes of parallel maches especall hose wh dsrbued memor Processes rug o such maches commucae hrough messages [4]

20 CHAPTER 3 TWO DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION The orgal wo dmesoal olear Schrödger euao s as follows 3 where s a comple-valued fuco The eac oe-solo soluo of 3 s gve b [7] A e κ κ ω θ cosh 3 p [ B B υ ] I 3 A s he amplude of he solo B s he verse wdh he -dreco ad B s he verse wdh he -dreco υ represes he veloc of he solo κ ad κ represes he solo freuec he ad drecos respecvel whle ω represes he solar wave umber ad fall θ s he phase cosa of he solo The epoe p whch s uow a hs po would be deermed whe fdg he eac solo soluo [7] B 3 we have he followg par of relaos κ B κ υ A B p B B [ ] ω B B κ κ

21 3 NUMERICAL METHOD b Frs we sud he wo dmesoal NLS euao 3 wh he al codo gve A e κ κ θ cosh 3 [ B B υ ] where p B B κ 6 κ 8 Here we also assume ha sasfes perodc boudar codo wh perod [-P P] Afer ormalzg he spaal perod o [ π] we have π P 3 where P whch s he half legh of he perod X π P / P ad Y π P / P The we dvde he erval [ π] he -dreco o N eual subervals wh grd spacg X π / N ad deoe X X N as he spaal grd pos We also dvde he erval [ π] he -dreco o N eual subervals wh grd spacg Y π / N ad deoe Y Y N as he spaal grd pos Now we advace he soluo of 3 from me o he e me-level as follows: Frs we ol focus o he olear par o advace he soluo []: whch could be solved eacl wh 33 { X Y } X Y ˆ X Y ep 34 Secod we focus o he lear par:

22 π P For he wo dmesoal dscree Fourer rasform we have N N [ mx Y ] 35 ˆm Fm ep 36 N N N m N N N ad Accordg o 36 ad 37 we have N N ˆ ˆ m m [ mx Y ] F ep 37 N N N m N N N m N N [ mx Y ] d dˆ m ep 38 d d N N m N N m m ep[ mx Y ] d ˆ 39 d N N N N m ep[ mx Y ] d ˆ m 3 d Subsug 38 3 o 35 ad euag ever par of ems elds he followg resul Solvg 3 we have d d m ˆ m ˆm π P 3 3

23 4 m p m m ep ˆ ˆ π 3 The applg 3 o he followg Y X F F Y X m ˆ We have: Y X F m p F Y X m ˆ ep π 33 Thus 33 s he spl-sep Fourer mehod for he frs-order splg appromao 44 where s he me sep F ad F - are he forward ad verse dscree Fourer rasforms respecvel To advace me from o b he spl-sep Fourer mehod wh he secodorder splg appromao 45 we should ae he followg seps []: Applg 34 o advace he soluo usg he olear par { } Y X Y X Y X m m m ep ˆ Applg 33 o advace he soluo usg he lear par Y X F m p F Y X m ˆ ep ~ π 3 Applg 34 o advace he soluo usg he olear par { } Y X Y X Y X m m m ~ ~ ep Advaceme me from o b he spl-sep Fourer mehod wh he fourhorder splg appromao 46 could be obaed wh followg seps [3]:

24 Frs advace me from o ω usg he secod-order spl-sep Fourer mehod where 3 3 ω 3 Secod advace me from ω o ω usg he secod-order spl-sep Fourer mehod Fall advace me from ω Fourer mehod ad we oba appromao o o usg he secod-order spl-sep 3 NUMERICAL EXPERIMENTS To es he umercal mehod we compued he L orm ad L orm a he ermag me T [] Also we compue he relave error for he followg coserved ua I dd 3 Euao 3 s calculaed usg he wo dmesoal Smpso s rule whch s descrbed as follows For a wo dmesoal fuco z f ad respecvel we deoe he euall spaced sample pos as ad a b ad c d wh he erval m h m where b a h ad m d c 5

25 6 The compose Smpso s rule would be as R b a d c h f D S dd f da f where { d b f c b f d a f c a f h h f D S b f b f a f a f 4 4 m m m m d f d f c f c f 8 6 m m f f } 4 8 m m f f Ad could be show ha he error erm s of he form 4 4 o h o h f E D S ha s 4 4 o h o h f D S dd f b a d c Smpso s rule has he paer of weghs: Ad wo dmesoal Smpso s rule has eeded hs paer o:

26 I our umercal epermes we used N 56 for dffere values of me seps o es he accurac of he frs-order spl-sep schemes ha are ulzed solvg The resuls are show Table 3 Table 3: Covergece raes me for he frs-order splg mehod N 56 T L L cpus 4 73E- 4576E E E- 4578E E E E E E E E E- 4575E E E E E

27 I he above ad followg ables L s he f orm L s he Euclda orm s C C he relave error ad C s he coserved ua for he eac soluo a C C s he coserved ua for he umercal soluo a ever me sep Also our umercal epermes we used 5 for dffere values of N o es he accurac of he frs-order spl-sep schemes ha are ulzed solvg The resuls are show Table 3 Table 3: Covergece raes space for he frs-order splg mehod 5 T N L L cpus E E E E E E E E E E E E E E- 9764E The umercal soluos of he wo dmesoal NLS euao 3 a wh he al codo 3 usg he above spl-sep Fourer mehods wh 5 ad N 56 are show below: 8

28 Fgure 3: The al codos The modulus of he al codo of euao 3 a Fgure 3: Two dmesoal SSF The modulus of euao 3 a usg he wo dmesoal frs-order SSF 9

29 Fgure 33: Two dmesoal SSF The modulus of euao 3 a usg he wo dmesoal secod-order SSF Fgure 34: Two dmesoal SSF4 The modulus of euao 3 a usg he wo dmesoal fourh-order SSF where N 8

30 33 PSEUDO-SPECTRAL METHOD AND EXPERIMENTS Pseudo-specral mehod s a Fourer mehod whch s rasformed o Fourer space wh respec o ad dervaves or oher operaors wh respec o are he made algebrac he rasformed varable Aga we ormalze he spaal perod o [ π] for coveece [] Wh hs scheme could be compued as m F X Y F The combed wh a leap frog me sep he wo dmesoal NLS euao 3 s appromaed b π X Y X Y F m F X Y p 4 33 Followg he deas of Forberg ad Whham we mae a modfcao appromag he wo dmesoal NLS euao 3 begg X Y X Y F s m F X Y π p 4 33 The algorhm o mpleme 33 ad 33 s descrbed as below: Frs we deoe he al codo as u whch s eual o a he ver Secod we useu o represe where a frs Thrd for o umseps whch s we do he followg:

31 π u F m F X Y 3 p for 33 or u F s m F u 3 π p for 33 u 3 u u u u u3 4 u u u u Fall we oba The appromao of he lear par of 3 s he dfferece bewee 33 ad 33 A soluo of 35 would eacl sasf he lear par of 33 Ad 33 s / < / π learl sable for ucodoall sable [] However accordg o lear aalss 33 s The soluos of he wo dmesoal NLS euao 3 a wh he al codo 3 usg he pseudo-specral mehods wh 5 ad N 56 are show below:

32 mehod Fgure 35: Two dmesoal pseudo-specral mehods The modulus of euao 3 a usg he wo dmesoal pseudo-specral Fgure 36: Modfed wo dmesoal pseudo-specral mehods The modulus of euao 3 a usg he modfed wo dmesoal pseudospecral mehod 3

33 4 34 FINITE DIFFERENCE METHOD AND EXPERIMENTS 34 EXPLICIT METHOD B usg he eplc mehod a me ad a cera po X Y where X * ad Y * we have 34 To oba we ae he followg seps: Frs we deoe he al codo as u whch s eual o a he ver begg Secod we use u o represe where a frs Thrd for o umseps whch s eual o we do he followg: u u u u u u u u u 3 * u u u u 4 u u * 5 u u u Ad fall we oba To do he umercal epermes we used N 56 for dffere values of me seps o es he accurac of he eplc mehod ha are ulzed solvg The resuls are show Table 33

34 Table 33: Covergece raes me for he eplc mehod N 56 T 5 L L cpus E- 77 E E E- 75E E E- 76E E E- 787E E We also used 5 for dffere values of N o es he accurac of he eplc mehod ha are ulzed solvg The resuls are show Table 34 Table 34: Covergece raes space for he eplc mehod 5 T 5 N L L cpus E E- 4E E E- 6788E E- 76E E E- 77E E E E E- 474 The soluos of he wo dmesoal NLS euao 3 a wh he al codo 3 usg he eplc mehod wh 5 ad N 56 are show below: 5

35 6 Fgure 37: Two dmesoal eplc mehod The modulus of euao 3 a usg he wo dmesoal eplc mehod 34 IMPLICIT METHOD For he mplc par we use alerag dreco mplc ADI mehod Wh ADI mehod a me ad a cera po X Y where X * ad Y * we have / / / / / 34 ad / / / / / / / 343 The wh 34 ad 343 we do he followg o oba [9]:

36 7 Frs we mpleme he ADI mehod a loop over he -dreco: for : N for : N r r r g ed solve g A ew : ed where A r r r r r r r r r r r r r r r r Ad ew s a ermedae sage To ge A frs from 34 we have r r r r r r / / / 344 where * 4 r The for N < ad N < we pu he coeffces of he lef sde of 344 o a mar whch s A

37 Secod we mpleme he ADI mehod a loop over he -dreco: for : N for : N ed g r r r solve A : g ed where A could be obaed usg he same mehod he frs sage bu here r 4* Ad fall we oba The eecuo of he frs ad secod seps advaces he soluo wh a sep me overwrg The soluos of he wo dmesoal NLS euao 3 a wh he al codo 3 usg he ADI mehod wh 5 ad N 56 are show below: 8

38 Fgure 37: Two dmesoal mplc mehods The modulus of euao 3 a usg he wo dmesoal mplc mehod 35 PARALLEL IMPLEMENTATION AND EXPERIMENTS For he parallelzao of he frs-order spl-sep Fourer mehod we parallelze as followg [3]: Deoe A of sze N N as he appromao soluo o a me Assume ha here are processors a dsrbued-memor parallel compuer The parallelzao of 34 s sraghforward The we dsrbue A amog processors Each processor l wh arra elemes A[ ln / ] o [ l N / ] A where l wors o s ow subarras depedel whou commucag wh ohers Afer ha we emplo FFTW s MPI roues o parallelze he calculao of F ˆ X Y For ep m F ˆ X Y π m p s 9

39 parallelzao s also sraghforward Fall we use FFTW s MPI roues aga o parallelze π X Y F ep m F ˆ X Y p m Smlarl we could parallelze for he secod-order fourh-order spl-sep Fourer mehods ad he pseudo-specral mehod We mpleme he parallel algorhms of he spl-sep Fourer mehods ad he pseudospecral mehod o pcluser of UGA Ad we opmze all he codes a he same opmzao level I our smulaos speedup S p s defed as S p where ad Tme spe o ru he MPI code o sgle processor Tme spe o ru he MPI code o processors The resuls for parallel mplemeao of spl-sep mehod ad pseudo-specral mehod are show Table 35 o Table 39 From he umercal epermes resuls we could observe clearl ha S creases as he p problem sze N N creases wh a fed umber of processors ad whe N s large S p obaed o he mulprocessor compuer rug he parallel codes s cosderable 3

40 Table 35: Resuls for parallel mplemeao of frs-order splg mehod 5 p s he me o p processors S p s he speedup o p processors N 8 N 56 N 5 sec sec sec sec S / S 4 / S 8 / Table 36: Resuls for parallel mplemeao of secod-order splg mehod 5 p s he me o p processors S p s he speedup o p processors N 8 N 56 N 5 sec sec sec sec S / S 4 / S 8 /

41 Table 37: Resuls for parallel mplemeao of fourh-order splg mehod 5 Arra sze s 8 for boh ad p s he me o p processors S p s he speedup o p processors N 8 sec sec sec 87 8 sec 658 S / 47 S 4 / 4 9 S 8 / 8 55 Table 38: Resuls for parallel mplemeao of he pseudo-specral mehod 5 p s he me o p processors S p s he speedup o p processors N 8 N 56 N 5 sec sec sec sec S / S 4 / S 8 /

42 Table 39: Resuls for parallel mplemeao of he modfed pseudo-specral splg mehod 5 p s he me o p processors S p s he speedup o p processors N 8 N 56 N 5 sec sec sec sec S / S 4 / S 8 / For he eplc ad mplc mehod we do he parallelzao for he calculao of ad each for loop Resuls are show he followg ables Table 3: Resuls for parallel mplemeao of he eplc mehod 5 p s he me o p processors S p s he speedup o p processors N 8 N 56 N 5 sec sec sec sec S / 6 99 S 4 / S 8 /

43 Table 3: Resuls for parallel mplemeao of he mplc mehod 5 p s he me o p processors S p s he speedup o p processors N 8 N 56 N 5 sec sec sec sec S / S 4 / S 8 /

44 CHAPTER 4 RANKING We have descrbed seve mehods o solve he wo dmesoal olear Schrödger euao To es her performaces ad ge her rag based o her fe orm Euclda orm ad relave error we have doe he followg epermes: Le N euals o 8 ad 56 respecvel For each N we use he al codos: κ 6 κ 8 B B for all hese seve mehods record her resuls whe T 5 The for each resul we fd ou s fe orm Euclda orm ad relave error Accordg o hese crera we ge he rag: Two dmesoal eplc mehod Two dmesoal spl-sep Fourer mehod usg he frs-order splg mehod 3 Two dmesoal spl-sep Fourer mehod usg he secod-order splg mehod 4 Modfed wo dmesoal pseudo-specral mehod 5 Two dmesoal pseudo-specral mehod 6 Two dmesoal spl-sep Fourer mehod usg he fourh-order splg mehod 7 Two dmesoal mplc mehod 35

45 CHAPTER 5 CONCLUSION I hs hess we have appled he well-ow spl-sep Fourer mehod pseudo-specral mehod ad fe dfferece mehod for solvg he wo dmesoal olear Schrödger euao We have mplemeed hree spl-sep Fourer mehod schemes wo pseudo-specral mehod schemes ad wo fe dfferece mehod schemes We fd ha he hgher-order splsep scheme eeds more compuaoal me ha he lower-order spl-sep mehods For he parallel mplemeao of hose schemes wh fed umber of processors we foud ou ha as he problem sze becomes larger he speedup becomes larger We also could acheve cosderable speedups o he mulprocessor compuer b rug he parallel codes for large problem szes 36

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