
paarariyal

prarambha moolyaprashnangalude parihaaratthinaayi upayogikkunna samkhyaa vishakalanathil ninnulla oru samanthara algorithamaanu paarariyal.
2001ll simhangal, maade, turinisi ennivar chernnaanu ithu avatharippichathu. athinusesham, ithu ettavum vyaapakamaayi padicha samanthara-inn-time samyojana reethikalil onnaayi maari.
samanthara-inn-time samyojana reethikal.
iii ennathinu vipareethamaayi.jii. runch-kutta allengil multy-steppu reethikal, paarariyalile chila kanakkukoottalukal samaantharamaayi nadatthaam, athinaal paarariyal samanthara-inn-time samyojana reethiyude oru udaaharanamaanu. praadeshika vivechanaadhikaarathil shradha kendreekarichulla bhagika deferential samavaakyangalude samkhyaa parihaarathe samaantharamaakkaanulla charitraparamaaya mikka sramangalum exascale computingil ninnulla velluvilikal kanakkileduthu, samkhyaa softweyarile ekathaanatha vardhippikkunnathinulla saadhyamaaya maargamaayi temporal vivechanaadhikaarathinulla samanthara reethikal thiricharinjittundu.
onniladhikam samaya ghattangalkk samaantharamaayi samkhyaa pariharam kanakkakkunnathinaal, athine "ghattangalkk kuruke samaantharamaayi" aayi tharamthirikkunnu.
samanthara runch-kutta allengil extrapolation reethikal polulla "reethiyiludaneelam samaantharatha" upayogikkunna sameepanagalil ninnu ithu vyathyasthamaan, avide tharangaroopa vishramam polulla swatantra gattangal samaantharamaayi allengil "systethiludaneelam samaantharamaayi" kanakkakkaam.
charithram.
samayakramathil oru multigrid reethiyaayo samaya achuthandiloode onniladhikam shootingaayo paarariyal uruthirinjekkaam.
samayathil multigrid, samaya samyojanathinaayi onniladhikam shooting enniva 1980 kalilum 1990 kalilum thirichupokunnu.
parareeya reethi vyaapakamaayi padicha oru reethiyaanu, ithu vividha aaplikkeshanukalkkaayi upayogikkukayum parishkkarikkukayum cheythittundu.
prarambha moolya prashnangalude parihaarathinu samaantharamaakkaanulla ashayangal kooduthal pinnilekku pookunnu samanthara-inn-time samyojana reethi nirddeshikkunna aadya prabandham 1964 ll prathyakshappettu.
algoritam.
prashnam.
roopathinte prarambha moolyaprashnam pariharikkuka ennathaanu lakshyam.
soothravaakyam _ 1
valathuvashathe formula _ 2 oru minusamarnna (orupakshe nonleeniyar) functionaanennu karuthappedunnu. lines sameepanathinte oru reethiyilulla oru bhagika deferential samavaakyathinte speshyal discretisationumaayi ithu poruthappedaam. formula _ 3 thulya akalamulla pooyint formula _ 4-nte oru temporal meshil eee prashnam pariharikkan njangal aagrahikkunnu, avide formula _ 5 umm formula _ 6 umm. eee vivechanam nadappilaakkumbol formula _ 8-nulla time slies formula _ 7 adangunna oru partitioned time interwell njangalkk labikkum.
seeriyal time stepping reethi upayogichu krithyamaaya parihara soothravaakyathilekku samkhyaa ekadesha formula _ 9 kanakkaakkuka ennathaanu lakshyam (udha.jii. uyarnna samkhyaa krithyathayulla (athinaal uyarnna computational chelavu) runch-kutta). nammal eee reethiye phine solver formula _ 11 ennu vilikkunnu, ithu time formula _ 13-ll oru prarambha moolya formula _ 9-neey time formula _ 15-ll oru terminal moolya formula _ 14-lake pracharippikkunnu. namukk labhikkuna formula _ 16 upayogichu pariharam (uyarnna samkhyaa krithyathayode) kanakkaakkuka ennathaanu lakshyam. 
soothravaakyam _ 17
ithinte prashnam (aadyathethu samaantharamaayi pariharikkan sramikkunnathinulla kaaranam) thalsamayam kanakkukoottunnath computational reethiyil asaadyamaanu ennathaanu.
athu engane pravartthikkunnu.
prarambha moolya prashnam pariharikkan orotta prosessor upayogikkunathinupakaram (classical time-stepping reethikal pole), paarariyal formula _ 18 prosessorukal upayogikkunnu. 18 enna soothravaakyam samaantharamaayi (oro samaya sliceilum onnu) cheria prarambha moolya prasnangal pariharikkunnathin formula 19 prosessorukal upayogikkuka ennathaanu lakshyam. udaaharanathinu, oru emm. pii. aii adisthaanamaakkiyulla codil, formula _ 19 ennathu prakriyakalude ennamaayirikkum, athesamayam oru ooppanembi adisthaanamaakkiyulla codil, formula _ 22 thredukalude ennathinu thulyamayirikkum.
parukkan parihara soothravaakyam _ 23 ennu vilikkunna eee prarambha moolya prashnam samaantharamaayi pariharikkunnathin paarariyal randaamathe time-stepping reethi upayogikkunnu.
parukkan parihaarakan phine solvarinte athe reethiyil pravartthikkunnu, dairkhya soothravaakyam _ 24 nte samaya edavelayil oru prarambha moolyam pracharippikkunnu, ennirunnaalum ithu formula _ 11 nekkaal valare kuranja samkhyaa krithyathayilaanu (athinaal valare kuranja computational chelavil) cheiyunnathu. 
phine solvarinekkaal valare kuranja computational chelavil oru parukkan solver undaayirikkunnathu parariyalumaayi samanthara vegatha kaivarikkunnathinulla thacolaanu. 
inii muthal, nammal samaya soothravaakyathil paarariyal parihaaratheyum formula 28 upayogichu aavarthana soothravaakyathil 27 umm soochippikkum.
poojyamaaya aavarthanam.
onnamathaayi, parihaaratthinte ekadesha prarambha ooham kanakkakkaan muzhuvan samaya edavela formula parukkan parihaarakan kramanugathamaayi pravartthippikkuka
soothravaakyam _ 30
thudarnnulla aavarthanangal.
aduthathaayi, ettavum kaalikamaaya parihara moolyangalil ninnu samaantharamaayi oro time sliceukalilum mikacha parihaarakarthaavu pravartthippikkuka
soothravaakyam _ 31
ippol predictor-corrector upayogichu paarariyal solution moolyangal thudarchayaayi update cheyyuka
soothravaakyam _ 32
eee ghattathil, parihara moolyangal oro aavarthanathilum maarunnille ennu nirnnayikkaan oraalkk oru nirtthal maanadandam upayogikkaam. udaaharanathinu, undo ennu parishodhichu oraalkk ithu parisodhikkaam. 
soothravaakyam _ 33
koodaathe chila sahishnutha soothravaakyam _ 34. eee maanadandam thripthippettillenkil, thudarnnulla aavarthanangal samaantharamaayi phine solver thudarnnu predictor-corrector prayogichu pravarthippikkaan kazhiyum. ennirunnaalum, maanadandam thripthippettukazhinjaal, algoritam formula _ 35 aavarthanangalil othuchernnathaayi parayappedunnu. mattu nirtthal maanadandangal nilavilundennum avaa parariyalil vijayakaramaayi pareekshichittundennum shradhikkuka.
paraamarshangal.
phine solvarinte seeriyal aaplikkeshan vazhi labhikkuna pariharam paarariyal punarnirmmikkukayum paramavadhi formula _ 22 aavarthanangalil othucherukayum venam.
ennirunnaalum, vegatha nalkunnathinu, ithu time sliceukalude ennathekkaal valare cheria niravadhi aavarthanangalil othucheranam, athaayath.iii. formula _ 37.
paarariyal aavarthanathil, formula _ 38-nte computational chelaveria vilayiruthal formula _ 22 prosessing yoonittukalil samaantharamaayi nadatthaam.
ithinu vipareethamaayi, formula _ 41-neey aashrayichirikkunna formula _ 40-nte ardham parukkan thiruthal seeriyal kramatthil kanakkaakkanam ennaanu.
saadhaaranayaayi, parukkan, phine integrettarukalkkaayi eathenkilum tharatthilulla runch-kutta reethi thiranjedukkunnu, avide formula _ 23 thaazhnna kramathilaayirikkaam, koodaathe formula _ 11 nekkaal valia samaya ghattam upayogikkunnu.
oru pii. dii. iii. yude vivechanaadhikaarathil ninnaan prarambha moolya prashnam undengil, formula _ 23 nuu oru parukkan speshyal vivechanaadhikaaravum upayogikkaam, ennaal uyarnna oordar interpolation upayogikkunnillenkil ithu samyojanathe prathikoolamaayi badhikkum.
vegatha koottuka.
chila anumanangal prakaram, paarariyalinte vegatha vardhippikkunnathinulla oru lalithamaaya saidhaanthika maathruka uruthirinjekkaam.
prayogangalil eee anumanangal valare niyanthrithamaakaamenkilum, paarariyal upayogichu speed upp nedunnathil ulppedunna trade oofukal chithreekarikkaan eee model ippozhum upayogapradamaanu.
aadyam, oro thavanayum slies formula _ 45-lum phine integratorinte krithyamaayi formula _ 46 steppukalum parukkan integratorinte formula _ 47 steppukalum adangiyirikkunnuvennu karuthuka.
ellah samaya sliceukalkkum ore neelamundennum parukkan, mikacha integrator poornna simuleshanu mukalil sthiramaaya steppu sais upayogikkunnuvennum ithil prathyekichum ulppedunnu.
randaamathethu, formula _ 48, formula _ 49 enniva upayogichu yathaakramam phine, parukkan reethikalude orotta ghattathinu aavashyamaaya computing samayathe soochippikkuka, koodaathe randum sthiramaanennu anumaanikkuka.
oru avyaktha reethi upayogikkumbol ithu saadhaaranayaayi shariyalla, kaaranam aavarthana parihaarathinu aavashyamaaya aavarthanangalude ennathe aashrayich rantaim vyathyaasappedunnu.
eee rund anumanangalkkum keezhil, formula _ 50 time sliceukalil samyojippikkunna phine reethiyude rantaim ingane maathrkayaakkaam
soothravaakyam _ 51
formula _ 50 prosessing unitukal upayogikkukayum formula _ 53 aavarthanangal nadathukayum cheiyunna paarariyalinte rantaim
formula _ 54
appol paarariyalinte vegatha vardhippikkunnu
soothravaakyam _ 55
parukkan reethi thiranjedukkunnathil undaakkenda edapaadine eee rund athirukalum vyakthamaakkunnu oru vashath, athu vilakuranjathaayirikkanam koodaathe/allengil aadya boundu kazhiyunnathra valuthakkaan valare valia samaya ghattam upayogikkanam, maruvashathu randaam boundu valuthakkaan aavarthana soothravaakyam _ 53 kuraykkendathundu.
prathyekichum, paarariyalinte samanthara karyakshamatha ithinaal parimithappedutthiyirikku
soothravaakyam _ 57
athaayath aavashyamaaya aavarthanangalude ennathinte vipareethamaanu.
saangalppika eejan moolyangalude asthiratha.
paarariyalinte vaanila pathippil saangalppika eejan moolyangalile prasnangal undu. ithu saadhaaranayaayi avasaanathe aavarthanangalilekku maathrame othucherunnulloo, athaayath formula _ 53 formula _ 22 neey sameepikkunnathinaal, speed upp formula _ 60 ellaayppozhum onninekkaal cheruthayirikkum. athinaal onnukil aavarthanangalude ennam cheruthaanu, paarariyal asthiramaanu allengil, formula _ 53 paarariyal sthiratha varuthaan paryaapthamaanenkil, speed upp saadhyamalla. hyperbolic samavaakyangalkk paarariyal saadhaaranayaayi asthiramaanennum ithinartham. ganderinteyum wandewaleyudeyum aupachaarika vishakalanam sthiramaaya gunakangalulla rekheeya prasnangal maathrame ulkkollunnulloovenkilu, viscosity gunakam valare cheruthaakukayum reynoldukalude ennam valare valuthaakukayum cheyyumbol nonleeniyar navier-stokku samavakyangalil paarariyal prayogikkumbozhum prashnam uyarnnuvarunnu. paarariyal sthirappeduthunnathinu vyathyastha sameepanagal nilavilundu, onnu krilov-subspace mechappeduthiya paarariyal aanu.
vakabhedangal.
yadhaartha paarariyal algorithathil ninnu nerittu allengil kuranjathu prachodanam ulkkonda onniladhikam algorithangalundu.
krilov-subspace mechappeduthiya paarariyal.
neriya prashnangalkk phine methed formula _ 62 srishtikkunna vivarangal parukkan reethi formula _ 63-nte krithyatha mechappeduthaan upayogikkaamenn neratthe thiricharinjirunnu. yadhaarthathil, parariyalumaayi adutha bandhamullathum ennaal thiruthal engane cheiyunnu ennathil cheria vyathyasangalullathumaaya samanthara antharleenamaaya time-integrator pittaykkaanu eee ashayam roopappeduthiyathu. oro aavarthana soothravaakyathilum _ 53 phala soothravaakyam _ 65 formula _ 67-nulla moolya soothravaakyam _ 66-naayi kanakkakkunnu. eee vivarangalude adisthaanathil, subspace
soothravaakyam _ 68
oro paarariyal aavarthanathinum sesham nirvachikkukayum update cheyyukayum cheiyunnu. formula _ 70 muthal formula _ 71 vareyulla oorthoogoonal projection formula _ 69 aayi soochippikkuka. thudarnnu, parukkan reethikk pakaram mechappetta integrator formula _ 72 upayogikkuka.
aavarthanangalude ennam vardhikkumbol, space formula _ 71 valarukayum parishkkaricha propagettar formula _ 74 kooduthal krithyatha kaivarikkukayum cheyyum. ithu vegathil othucheralilekku nayikkum. paarariyalinte eee pathippin leanier hyperbolic paarshyal deferential samavaakyangal sthiramaayi samyojippikkaanum kazhiyum. kuranja adisthaana reethiye adisthaanamaakkiyulla nonleeniyar prashnangalilekkulla vipuleekaranavum nilavilundu.
hibrid paarariyal spectral maattivacha thiruthalukal.
spectral deferred corrections (ezz. dii. sii) yumayullah paarariyalinte samyojanathe adisthaanamaakkiyulla mechappetta samanthara kaaryakshamathayulla oru reethi emm nirddeshichittundu. miniyoon. ithu parukkan mikachathumaaya integrator ezz. dii. sii. yilek thiranjedukkunnathu parimithappedutthunnu, mechappetta samanthara kaaryakshamathaykkaayi vazhakkam thyajikkunnu. soothravaakyam _ 75-nte paridikk pakaram, hibrid reethiyile samanthara kaaryakshamathayude paridhi maarunnu.
soothravaakyam _ 76
soothravaakyam _ 77 ennathu seeriyal ezz. dii. sii adisthaana reethiyude aavarthanangalude ennavum soothravaakyam _ 78 aanu, ithu saadhaaranayaayi samanthara hibrid reethiyude aavarthanangalude ennamaanu. nonleeniyar multigriddle upayogikkunnathupole oru "poornna ekadesha padhathi" cherthukondu paarariyal-ezz. dii. sii hibrid kooduthal mechappeduthi. ithu "sthalathilum samayathilum samanthara poornna ekadesha padhathi" (pfasst) vikasippikkunnathilekku nayichu. joolichu suuppar computing sentaril vikasippichedutha barnus-hutt trea coode adisthaanamaakkiyulla kanika pariharamaaya pepsicaayi piphaasttinte prakadanam padikkappettittund. ibm bluejeen/pii sisttam jujeenil 262,144 korukalum upayogicha simulationukal kaanikkunnath speshyal trea paaraalalaiseshante sachureshanuppuram adhika speed upp srishtikkaan piphaastinu kazhiyumennu.
samayathil multigrid kuravu (mgrit).
multigrid reduction inn time reethi (mgrit) vyathyastha smootharukal upayogichu onniladhikam thalangalilekku multigrid-inn-time algorithangal enna nilayil paarariyalinte vyaakhyaanathe saamaanyavalkkarikkunnu. ithu kooduthal pothuvaaya sameepanamaan, ennaal oru prathyeka paaraamettarukal thiranjedukkunnathinu ithu paarariyalin thulyamaanu. emm. jii. aarr. tii nadappaakkunna exbride library laurence livermore naashanal laborattariyaan vikasippikkunnath.
paraxp.
pariexpi parariyalinullil exponential integratorukal upayogikkunnu. rekheeya prasnangalil parimithappedutthumbol, ithinu mikkavarum opttimal samanthara vegatha srishtikkaan kazhiyum.

