Time-Spectral Solution of Initial-Value Problems Subdomain Approach

Transcription

1 American Journal of Comuaional Mahemaic, 01,, 7-81 h://dx.doi.org/10.436/ajcm Publihed Online June 01 (h:// Time-Secral Soluion of Iniial-Value Problem Subdomain Aroach Jan Scheffel, Ahmed A. Mirza Diviion of Fuion Plama Phyic, Aociaion EURATOM-VR, Alfvén Laboraory, School of Elecrical Engineering, KTH Royal Iniue of Technology, Sockholm, Sweden Received March 7, 01; revied Aril 6, 01; acceed Aril 1, 01 ABSTRACT Temoral and aial ubdomain echnique are rooed for a ime-ecral mehod for oluion of iniial-value roblem. The ecral mehod, called he generalied weighed reidual mehod (GWRM), i a generaliaion of weighed reidual mehod o he ime and arameer domain [1]. A emi-analyical Chebyhev olynomial anaz i emloyed, and he roblem reduce o deermine he coefficien of he anaz from linear or nonlinear algebraic yem of equaion. In order o avoid large memory orage and comuaional co, i i referable o ubdivide he emoral and aial domain ino ubdomain. Mehod and examle of hi aricle demonrae how hi can be achieved. Keyword: Iniial-Value Problem; Mulile Time Scale; Time-Secral; Secral Mehod; Weighed Reidual Mehod; Subdomain; Domain Decomoiion 1. Inroducion The generalied weighed reidual mehod (GWRM) i a fully ecral mehod, deigned o olve arial differenial equaion in he form of iniial-value roblem. Alhough variou alicaion of ime-ecral mehod have aeared in he a, we have ried in [1] o demonrae a comrehenive view of he wide alicabiliy of ecral mehod for he ime domain. The mehod alie o boh arabolic and hyerbolic de. Examle, briefly dicued in hi aricle, are he Burger equaion, a forced wave equaion and a yem of 14 couled magneohydrodynamic equaion. Thee are all hown o be uccefully olved by he GWRM. The baic inenion of he mehod i o allow for efficien numerical imulaion of roblem wih everal ime cale, where one i rimarily inereed in he long ime cale behaviour. Thi i, for examle, ofen he cae for roblem of abiliy, flucuaion and confinemen in fuion lama hyic. However, alicaion of he mehod how ha i can alo be ued for very accurae comuaion on hor ime cale. The rial bai funcion ued for all emoral, aial and hyical domain in he GWRM are Chebyhev olynomial, owing o heir minimax roery which rovide fa convergence [] and due o he fac ha non-eriodical boundary condiion are allowed. The oluion obained by GWRM i hu emi-analyical raher han urely numerical in he ene ha i i a finie um of Chebyhev olynomial in ime, ace and, if one o chooe, arameer ace. Thi i ofen racical for alicaion. Moreover, being an acaual mehod i i no conrained by CFL or oher ime e rericion, a i he cae for exlici ime differencing mehod. In comarion wih imlici ime differencing cheme, uch a he Crank-Nicolon mehod, he GWRM ha been hown o be efficien [1]. On he heoreical ide, here remain o deermine exac condiion for convergence and accuracy, being imoran maer for fuure udy. We focu here on he inroducion of aial and emoral ubdomain in order o reduce he comuaional effor. For advanced roblem wih many aial and emoral mode, i become coly o ieraively olve he algebraic yem of GWRM equaion aociaed wih coefficien of he Chebyhev olynomial anaz. The comuaional co cale wih he cube of he oal number of coefficien. Thi hold alo if marix inverion i relaced wih efficien LU decomoiion mehod. By dividing he comuaional domain ino ubdomain, much can be gained by arallel or conecuive oluion of he correonding, reduced algebraic yem of equaion for each domain. Subdomain, or domain decomoiion, mehod have been udied ince he beginning of he 1980 [3,4]. In aricular, he o-called aching mehod, being baed on he coninuiy of he oluion and i fir-order derivaive a he ubdomain boundarie, a well a a variaional ecral-elemen mehod have been develoed. Thee aly o roblem Coyrigh 01 SciRe.

2 J. SCHEFFEL, A. A. MIRZA 73 where ecral mehod in ace and finie difference mehod in ime have been ued. Here, we concenrae on he ue of aial and emoral ubdomain for he GWRM, which emloy a ime-ecral mehod. The aer i organized a follow. In Secion, a brief overview of he GWRM i given and ome reul on comarion wih exlici and imlici finie difference mehod are reened. In Secion 3, aial ubdomain are inroduced. Afer a dicuion of oible comuaional gain, differen GWRM imlemenaion of aial ubdomain are dicued. A weighing mehod for imroved convergence i inroduced. In Secion 4, emoral ubdomain are inroduced. A hown, hi i comaraively raighforward. An examle alicaion of emoral ubdomain i alo udied. In Secion 5, he hree differen imlemenaion of aial ubdomain are alied o he Burger equaion and o a large yem of reiive magneohydrodynamic (MHD) equaion. Thi i followed by a dicuion in Secion 6 and concluion in Secion 7.. The Generalized Weighed Reidual Mehod (GWRM).1. Mehod in Brief Conider a yem of arabolic or hyerbolic arial differenial equaion u Du f (1) where u u, x; i he oluion vecor, D i a linear or nonlinear marix oeraor and f f, x; i an exlicily given ource (or forcing) erm. Noe ha D may deend on boh hyical variable (, x and u) and hyical arameer (denoed ) and ha f i aumed arbirary bu non-deenden on u. Iniial u( 0,x;) a well a (Dirichle, Neumann or Robin) boundary u(,x B ;) condiion are aumed known. To avoid invering a marix oluion vecor, aociaed wih he ime derivaive, Equaion (1) i inegraed in ime; u, x; u0, x; Du, x; f, x; d 0 () The oluion vecor u u, x; i aroximaed uing fir kind, mulivariae Chebyhev olynomial erie a boh rial and weigh funcion. Thee olynomial, defined by T n (x) = co(nco 1 x), are orhogonal wihin he inerval [ 1,1] over a weigh funcion (1 x ) 1/. Chebyhev olynomial may be defined aroriaely o any given finie range [a,b] of x (hifed Chebyhev olynomial) by a linear ranformaion []. Confining u here o one aial variable x and one arameer, we hu have K L M, ; (3) u x a T T T P k0 l0 m0 klm k l m uing he ranformaion A A xa x,, P B B B wih A z z, B z z z 1 0 z 1 0 and where indice 0 and 1 denoe lower and uer comuaional domain boundarie, reecively. Prime on ummaion ign in Equaion (3) indicae ha each occurrence of a zero coefficien index render a mulilicaive facor of 1/. Ju a for andard weighed reidual mehod (WRM), he unknown coefficien a klm are deermined by requiring ha he inegral of he weighed reidual over he comuaional domain hould be zero. Performing he inegraion by ar, he reul i [1] qr q0 r qr qr x (4) a b A F (5) Here A qr and F qr correond o Chebyhev exanion of he ime inegral on he righ hand ide of Equaion (). Since A qr uually include u ielf, i uually i a olynomial funcion of he unknown coefficien a klm. For nonlinear equaion, each elemen of relaion (5) may hu be a comlex funcion of he unknown. The coefficien b r correond o he Chebyhev exanion of he iniial condiion. Equaion (5), ogeher wih aroriae boundary condiion, i ued o deermine he GWRM coefficien a qr ha coniue he oluion of he roblem. Equaion (5) can be linear or nonlinear deending uon he ye of he roblem. If linear, he coefficien can be found uing radiional mehod like Gau eliminaion. For nonlinear roblem, a emi imlici roo olver (SIR) ha been develoed [5]. SIR ha roven o be very robu for GWRM alicaion... The GWRM and Finie Time Se Mehod In order o inveigae he alicabiliy of he GWRM, ome e roblem have been olved uing he mehod [1]. The roblem were alo olved uing he finie difference ime e Lax-Wendroff and Crank-Nicolon mehod. The former mehod i exlici and i ubjec o CFL (or imilar) ime e limiing condiion. The laer mehod allow for arbirarily large ime e by uing an imlici aroach where he funcional value are deermined boh a reen and fuure ime e. For udying accuracy, he nonlinear Burger equaion u u u u (6) x x ha been olved wih iniial condiion u0, x x1 x and boundary condiion u,0u,10 for differen value of. The reul have been comared wih he exac oluion. Alhough he GWRM i rimarily Coyrigh 01 SciRe.

3 74 J. SCHEFFEL, A. A. MIRZA inended for comuing long ime behaviour of comlex roblem wih everal ime cale, i can be ued for accurae oluion of iff roblem. For he cae of Burger equaion, he GWRM wa hown o rovide efficiency cloe o ha of he Lax-Wendroff and Crank-Nicolon cheme for given accuracy. Imroved GWRM efficiency i execed for roblem wih eriodic boundary condiion. The GWRM ha he addiional advanage of roviding aroximae, analyic oluion. For udying efficiency, a forced wave equaion wa olved: u u f x, (7) x wih u,0 u,1 0, u 0, x in nπx, 0, in u x A x. Here A, n, α, β and v are free arameer, and f(,x) = A(vβ α )in(α)in(βx) i he forcing funcion. The exac oluion i 0.5, co π in π in in u x n n x A x, (8) for β = mπ, wih m an ineger. Thi roblem ha he earae yem and forcing funcion ime cale n and π. Uing he arameer value v = 1, A = 10, α= π/15, β = 3π and n = 3, he raio of hee ime cale become R nπ 145. Thu he forcing erm in (7) ha here inroduced a ime cale much longer han ha of he unerurbed yem. I wa found ha he GWRM, a inended, i well uied for long ime cale oluion of hi roblem. For uiable mode arameer, i race he lower dynamic uing ubanially le comuaional ime han he Lax-Wendroff and Crank-Nicolon cheme. See Figure 1 for he cae K = 6, L = 8, where ingle emoral and aial ubdomain are ued. If reul are ough for longer ime, emoral ubdomain are referably ued for he GRWM, in order o guaranee conan comuaional effor er roblem ime uni. For roblem wih wider earaion of he ime cale, he GWRM will be an increaingly advanageou mehod a comared o he Lax-Wendroff cheme ince he laer mu follow he faer ime cale. I may alo be noed ha he GWRM Figure 1. GWRM oluion (mooh curve) of forced wave equaion, a comared o exac oluion (ocillaing curve). average more accuraely over he fa ime cale ocillaion han he finie difference mehod. Thi i an inereing ubjec which deerve furher aenion. The comuaion have been carried ou uing Male. Alhough faer comuaional environmen exi, exac comarion of efficiency are no eenial a hi age. The examle we have given how ha he efficiency and accuracy of he GWRM i comarable o ha of boh exlici and imlici finie difference cheme in a given environmen. Furher oimiaion of boh GWRM and finie difference code could increae efficiency, bu our examle indicae ha ime-ecral mehod for oluion of iniial-value de are of inere for general ue and for comuaion of roblem in magneohydrodynamic and fluid mechanic in aricular. 3. Saial Subdomain The number of oeraion in GWRM comuaion can be ignificanly reduced by uing ubdomain for he aial, emoral and hyical variable [1]. I i he rimary objecive of hi aer o qualify hi claim. Thu we begin hi ecion wih ome numerical conideraion. Ieraive oluion of he GWRM coefficien Equaion (5) will lead o aroximaely = (K + 1) 3 (L + 1) 3 (M + 1) 3 oeraion for each ieraion owing o he cubic deendence on he number of unknown for comuaion involving marix inverion. Uing LU decomoiion mehod raher han marix inverion, he number of oeraion may be reduced o 3 [6]. Thi may be an acceable amoun of work. For more comlex calculaion, however, high efficiency ofen require he emoral and aial domain o be earaed ino ubdomain. Thi would in rincile enable a linear raher han a cubic deendence of efficiency on, for examle, he number of aial mode alied o he enire domain, given ha he number of ubdomain i roorional o L. Now aume ha he emoral and aial domain are divided ino N and N x ubdomain, reecively. The reul i ha only x 1 1 x 3 NN x NN K N L N M 1 3 oeraion are hen needed for a aricular roblem, auming ha he ame oal number of mode (degree of freedom) are ufficien in boh cae. A an examle, for K = L = 11, M = and N = N x = 3 here would be a reducion from abou o oeraion. In hi ecion, we will dicu and ugge cheme for inroducing aial ubdomain. We begin by conidering he queion of how inernal boundarie oimally hould be conruced and how hey deend on he aial order of he differenial equaion. Nex we urn o dicu how he ubdomain condiion may be ieraed a he Coyrigh 01 SciRe.

4 J. SCHEFFEL, A. A. MIRZA 75 oluion i roduced Inernal Boundarie and Their Modelling Wherea emoral ubdomain are fairly eay o imlemen, aial ubdomain mu be carefully handled. The reaon i ha, for aial ubdomain, he informaion ha hould be aed on beween each ubdomain i exernally unknown and globally deenden on all oher ubdomain boundarie. For emoral ubdomain he informaion a one boundary (he iniial ime) i comleely known, hu numerically more able behaviour may be execed. A queion arie. Given he aial order of he combined yem of arial differenial equaion, wha i he order of conac required beween he inernal aial boundarie in order o avoid underdeermining he yem? Clearly, coninuiy of he funcion ielf and of i derivaive are naural condiion. We may reformulae he roblem by inead aking he queion: given ha he oin of conac beween he ubdomain are given by coninuiy of he funcion and i aial derivaive only (econd order conac), wha requiremen need be aified o avoid underdeerminaion of he yem? In he Aendix we how ha he anwer may be imly exreed a he crierion V (9) where V i he number of deenden variable ha are ued in he arial differenial equaion. Noe ha he number N of aial domain ha no influence. A an examle, conider a fourh order equaion ( = 4). If econd order conac i ued a inernal boundarie, he equaion ha o be broken down ino a lea wo econd order equaion o aify he requiremen on informaion (9), ha i V. Thi i, of coure, eaily faciliaed. Imlemenaion of econd order conac beween he ubdomain i no raighforward, however. Derivaive of Chebyhev exanion are ecrally rereened a L L1 du d GklmTl gklmtl (10) dx dx g klm l0 l0 L 1 G B x l 1 lodd km Even if he ecral coefficien G klm aociaed wih he funcion u converge, he convergence of he derivaive i weaker becaue of he mulilying facor λ ha add exra weigh o higher order G klm coefficien. In order o avoid conequenial numerical inabiliy, we have found ha an overlaing ( handhaking ) rocedure i referable. The Chebyhev exanion of he deenden variable of each ubdomain are allowed o exend a diance x ino he neighbouring domain. For imliciy we reric u here o dicu a one-dimenional aial domain. Auming N aial domain, exernal and inernal boundary condiion relaing o he deenden variable u are hen imlemened hrough u 1 x u, u 1 x u, u 1 x u, n1 n u xn x u xn x n1 n u xn x u xn x,, u x u u x u u x u, (11) N N 1 10 N for 1 n N 1 wih x n denoing he oiion of he righ-hand boundary of he nh ubdomain u n. The oal number of exernal boundary condiion, ha hould be alied a he boundarie x 0 and x 1, i. The e {x n } of inernal boundarie need no be equidianly aced, and he ize of x may alo be adaed o each ubdomain, deending on, for examle, he local iffne of he roblem. We will here conrain u o equidianly aced ubdomain boundarie having he ame value of x, he ize of which hould be oimized. We have udied differen imlemenaion of aial ubdomain of he form (11). Thee are now briefly decribed. 3.. Mehod I Deenden Subdomain In hi aroach, he comlee yem of all coefficien n a qr given by Equaion (5) for all domain, including inernal and exernal boundary condiion are olved elfconienly a each ieraion. For he inernal boundarie i hu hold ha, afer each ieraion, n, n1, u xn xu xn x (1) n, n1, u x x u x x n Thi aroach require only a few ieraion (yically 4-10 for nonlinear roblem) for good accuracy and converge wifly bu on he exene of comuaional memory, a hi involve marix inverion of a large marix, imulaneouly rereening all ubdomain. A reaon for emloying everal deenden ubdomain in reference for a ingle aial domain i ha i may be economic o localize one or more ubdomain in a region of rong gradien for an elewhere mooh oluion Mehod II Indeenden, Conecuively Udaed Subdomain Here Equaion (5) i ieraed earaely for each ubdomain, wih inernal boundary condiion a oin xn x obained from he reviou ieraion, bu wih inernal boundary condiion a oin xn x obained from he neighbouring ubdomain a he reen ieraion. Formally, n Coyrigh 01 SciRe.

5 76 J. SCHEFFEL, A. A. MIRZA n n n, n1, u x x u x x u x x u x x n, n1, 1 n n (13) Thi rocedure decoule he marice o be invered a each ieraion (one for each ubdomain), and ubanial comuaional eed i gained. A diadvanage i ha a larger number of ieraion i needed han for he cae of deenden ubdomain Mehod III Indeenden, Lae Udae of Subdomain Thi rocedure differ from ha of indeenden, conecuively udaed ubdomain only by ha he inernal boundarie are no aigned unil all ubdomain are comued a each ieraion level: n, n1, 1 u xn xu xn x (14) n, n1, 1 u x xu x x n 3.5. Rearding Udae of Inernal Boundary Condiion Wherea exlici heoreical crieria for convergence may be formulaed for he ingle domain roblem, correonding o oluion of a fixed yem of nonlinear equaion, he iuaion i differen for indeenden domain. The reaon i ha hoe coefficien of Equaion (5) ha correond o he inernal boundary condiion are non-conan hroughou he ieraion. If hee were conan, convergence of each ubdomain would, of coure, be guaraneed once he convergence crieria were aified. In hi cae, however, no informaion i aed beween he inernal boundarie and a global oluion i no aained. On he oher hand, if boundary informaion change oo raidly during he ieraion of Equaion (5), he ieraion cheme may no lead oward a oluion and convergence i endangered. Inuiively, a oible remedy would be o reard he informaion exchange a he inernal boundarie by only allowing for a cerain rae of change. The boundary daa ha come ino ieraion are hu relaced by a weighed combinaion of boundary value daa from boh ieraion 1 and. Thi rocedure i likely o be ucceful, ince in he limi ha all boundary daa are obained from he reviou ieraion (ha i, he boundary condiion are unalered hroughou he ieraion) well known convergence crieria exi. The inernal boundary Equaion (11) are conequenly relaced wih he following e of relaion: n, n1, u x x wu x x n n n1, 1 1wu xn x n n n1, 1wu x x n, n1, 1 u x x wu x x n n (15) The arameer w i ued o conrol he weigh of new boundary informaion in relaion o old boundary informaion. For w = 1, he cheme degenerae o he cae of indeenden, conecuively udaed ubdomain, and for w = 0 he condiion a he inernal boundarie remain fixed. In he following, we will refer o hi rocedure a he w mehod. 4. Temoral Subdomain A decribed in Secion 3, emoral ubdomain can enhance comuaional efficiency of he GWRM. Similarly a for aial ubdomain, he major role i layed by he reducion in oeraion when invering he marix for each domain, required for oluion of he coefficien Equaion (5). For inance, a emoral domain uing K mode could imly be li u ino N ubdomain wih K N mode in each. Siff differenial equaion, however, may of coure require omewha more han K N mode er ubdomain for adequae accuracy, reducing he gain in efficiency. A wa alo hown in Secion 3, oimal efficiency i likely obained when aial and emoral ubdomain are emloyed joinly. There are eenially wo differen ah o imlemen emoral ubdomain, uing ingle or mulile order conac a he emoral boundary. For ingle order conac, he reul from he reviou emoral ubdomain a ime = 1 i imly ued a iniial value a ime = 0 for he ubequen ime domain. Thi i no alway allowed: imilarly a for aial ubdomain, he number of exernal condiion o be imoed on each variable deend on he order of he yem. Single order conac require ha here are a lea a many variable a he emoral order of he combined differenial equaion. Since he GWRM yem of equaion are alway ca in he form of Equaion (1), hi i alway guaraneed for GWRM roblem. Alicaion o a large yem of differenial equaion will be reened in he nex ecion. For econd order conac in he emoral domain, he condiion V (16) mu be aified. Thi reul i found in a imilar manner a he condiion (9) for aial ubdomain. Here i he order of he ime derivaive for he yem. Second, or higher, order conac i no neceary for GWRM alicaion, bu may imrove convergence, in aricular in reence of hock. Thi i a maer for fuure udy. Adaive, emoral ubdomain can enhance accuracy and efficiency. To inroduce he adaive emoral ubdomain mehod for nonlinear roblem, a iff ordinary differenial equaion i here udied. The following equaion model he roagaion of he flame when lighing a mach: Coyrigh 01 SciRe.

6 J. SCHEFFEL, A. A. MIRZA 77 du d u u 3 (17) I i aumed ha u(0) = δ and ha 0. Thi i a iff differenial equaion for mall value of he arameer δ becaue of he reence of a ram a 1. We have olved hi roblem by uing rouine for ranforming nonlineariie o Chebyhev ecral ace and by formulaing an equaion of he form (5); ee Ref. [1]. A rongly ramed oluion wih δ = i comued. We have imoed an accuracy of ε = by comaring he GWRM oluion wih he exac oluion u 1 Waexa 1, where a 1 1 and W i he Lamber W funcion. Boh he comued and exac oluion are hown in Figure. A hi accuracy, hey are indiinguihable from each oher. Alo, he mallne of δ make he ram very diinc and numerically difficul o reolve. Conequenly, exlici finie difference mehod need exremely mall ime e o olve he roblem. An oimied Malab oluion ue imlici mehod ha reduce he comuaional effor o abou 100 ime e, aking a few econd on a ableo comuer. The GWRM oluion ue 69 ime domain and ake ju abou he ame amoun of comuaional ime. The emoral domain lengh ha been auomaically adaed a follow. Since i hold ha Tn 1, a ueful accuracy crierion i ak ak ak ak, where a n i he nh coefficien in he Chebyhev ecral exanion of u. In erforming he adaive comuaion, a defaul of 10 ime ubdomain i aumed and K = 6 i ued. If he accuracy crierion i aified, he ubdomain lengh i doubled a he nex domain, and if no i i halved. In he laer cae, he calculaion i reeaed for he ame ubdomain unil he accuracy crierion i aified. Thi goe on a he calculaion roceed in ime unil near he endoin, where he ubdomain lengh i adjued o land exacly on he redefined uer ime limi. Due o he iffne of he roblem, he ubdomain are concenraed near = where he ubdomain lengh may be a mall a abou ime uni. The auomaic exenion of he ubdomain lengh in mooher region ave coniderable comuaional ime; a he end of he calculaion he ubdomain lengh i more han houand ime uni Figure. GWRM and exac oluion of Equaion (17) for δ = Reul Accuracy and comuaional efficiency of wo examle roblem will now be udied, emloying he above dicued hree GWRM aial ubdomain mehod. Alo, he effec on convergence by ue of he w mehod (Secion 3.5) will be udied Burger Equaion Saial Subdomain The Burger Equaion (6) i a fundamenal hyerbolicarabolic arial differenial equaion from fluid mechanic, conaining boh convecion and diffuion erm. Thu wo earae ime cale govern he ime evoluion of an iniial ae. We howed in Secion ha GWRM accuracy i comarable o ha of exlici and imlici cheme, for a imilar number of floaing oeraion. For high accuracy, ha i a high mode number, i i of inere o deermine wheher aial ubdomain may reduce he memory requiremen and he number of oeraion. Emloying he mehod decribed earlier we comare he reul wih he analyical oluion given in Ref. [1], uing 50 erm of he exanion. The reul for Mehod I are dilayed in Table 1 for a uiable elecion of arameer. Iniial and boundary condiion are hoe aed below Equaion (6) and we have choen = By Ier i mean he number of ieraion required, Time denoe he normalized ime required o olve he GWRM yem of coefficien Equaion (5) including boundary condiion, Memory denoe normalized memory requiremen and Max error i he maximum abolue error a comared o he exac oluion. The maximum mode number L for ubdomain i obained from he relaion L10 N 1. The roo olver SIR i e o ha i olve yem (5) uing Newon mehod, wih ar daa given by he iniial condiion. I i een ha convergence i raid for all cae, even when 5 ubdomain are emloyed. The rimary informaion from Table 1 i ha alhough comuaional ime and memory requiremen decreae when emloying he relaion L10 N 1 for he aial ubdomain mode, accuracy i arially lo. Due o he hock-like rucure of he oluion near r = 1, many mode are needed for high reoluion. We will reurn o hi iue. Nex, he ame roblem i olved uing Mehod II. The reul are dilayed in Table. Clearly, le memory i ued han in Mehod I bu more ieraion are required, hu enhancing comuaional ime. The convergence of all N = 5 cae wa oo low o reach a SIR olver accuracy of wihin 100 ieraion. For he convergen cae, he abolue error i eenially he ame a for Mehod I. I become of inere o deermine wheher he w Coyrigh 01 SciRe.

7 78 J. SCHEFFEL, A. A. MIRZA Table 1. Mehod I oluion of Burger equaion. N L x Ier Time Memory Max error Table. Mehod II oluion of Burger equaion. N L x Ier Time Memory Max error NA NA NA mehod decribed in Secion 3.5, uing he weighing arameer w, can imrove convergence of Mehod II. Reul are hown in Table 3. The value of w given indicae he value ha reul in he lea number of ieraion of Equaion (5). We find ha convergence can be imroved uing he w mehod, in aricular for he cae wih everal aial ubdomain. The w mehod i mo effecive for large x. Finally, we reurn o he queion how many aial mode are required in each of everal ubdomain o obain he ame accuracy a ha of he ingle domain for he Burger equaion. Reul are hown in Table 4, uing Mehod II. I i een ha Mehod II offer a ah o comarable accuracy uing le memory, a he exene of comuaional ime. For all cae conidered, Mehod III converge lower han Mehod II; hu we do no reor on any deail of hee calculaion. The reul i no urriing, ince Mehod II can be regarded a a hybrid mehod in relaion o Mehod I and III in he ene ha one ubdomain boundary i inananeouly udaed a he coefficien of each ubdomain are ieraed uing Equaion (5). Le u ummarie he reul from alicaion of aial ubdomain. Regarding convergence, we conclude ha Mehod I converge fa for all cae conidered. Mehod II need more ieraion and doe no converge a Table 3. Mehod II oluion of Burger equaion. N L x Ier Time Memory Max error w Table 4. Mehod II oluion of Burger equaion, higher accuracy. N L x Ier Time Memory Max error w all for cae feauring many ubdomain. Emloying he w arameer mehod, however, Mehod II convergence i obained for nearly all cae. Mehod III generally feaure oorer convergence roerie han Mehod II. Accuracy i high for he ingle ubdomain cae, ince he Burger equaion feaure a hock near r = 1, where high ecral order are needed for good reoluion. Mehod II reache he ame accuracy boh for he cae of and 5 ubdomain when he order of he Chebyhev ecral exanion i 7 and 5, reecively. Thi how ha for hock-like roblem, he order of he ecral exanion needed in each domain i higher han LN, ha i, he oal number of degree of freedom mu be increaed. For he ame value of N and L, all mehod rovide he ame accuracy. Comuaional memory requiremen are he highe for Mehod I, ince i involve a global oluion where all Chebyhev coefficien are imulaneouly inerrelaed a each ieraion. Memory reducion by u o 40% were demonraed for Mehod II. The reduced memory requiremen i couled o he ize of he marix equaion correonding o Equaion (5), and for boh Mehod II and III a ubanial increae in efficiency in he ene ha he ieraion comuaional ime i reduced o a fracion of ha of he ingle domain cae. The number of ieraion required are higher for Mehod II and III, however, and he oal comuaional ime become comarable o, or higher han, ha of Mehod I. In concluion, uing a imle bu demanding e roblem we have hown ha aial ubdomain mehod, in combinaion wih he w mehod, ha a oenial o alleviae memory requiremen for he GWRM while reerving accuracy and, in mo cae, convergence. Efficiency i, however, reduced for hee cae. Coyrigh 01 SciRe.

8 J. SCHEFFEL, A. A. MIRZA Magneohydrodynamic Equaion Saial and Temoral Subdomain We nex urn o an advanced alicaion of GWRM ubdomain mehod. The roblem a hand i a lama abiliy roblem formulaed a a yem of couled magneohydrodynamic (MHD) equaion. The following e of equaion govern he reiive MHD model: 0 u du jb d E ub j (18) d 0 d E B B 0 j Thee macrocoic (fluid) lama equaion are he coninuiy and momenum equaion, Ohm law (including reiiviy) and he (adiabaic) energy equaion followed by Faraday and (he dilacemen curren free) Amere law, reecively. Regarding noaion, E and B denoe he lama elecric and magneic field reecively, u i he fluid velociy, j i he curren deniy, i he kineic reure, ρ i he ma deniy, η i he reiiviy, Γ = 5/3 i he raio of ecific hea and μ 0 i he vacuum ermeabiliy. The MHD abiliy roblem coni of a yem of 14 nonlinear, couled arial differenial equaion. Boundary condiion correonding o a erfecly conducing wall are rovided from he requiremen ha he radial comonen of he magneic field and fluid velociy hould vanih, and from he fac ha he arallel comonen of he elecric field a he wall hould vanih. For furher deail, ee Ref. [7]. A andard way of inveigaing lama abiliy i by linearizaion of he above equaion followed by Fourier decomoiion in he azimuhal and axial direcion (circular cylindrical geomery i aumed here). All deenden quaniie Q are conidered a he um of an equilibrium erm Q 0 and a mall erurbaion q, hu Q = Q 0 + q. Perurbaion are aumed o be roorional o ex[i(mθ + kz)] where k and m are he axial and azimuhal mode number reecively. For a given erurbaion, abiliy i comleely deermined by he equilibrium. An equilibrium i unable if i feaure a ime deendence ex(γ), where γ i a oiive number, and able (wavelike oluion) if γ i imaginary. The Equaion (18), ogeher wih iniial and boundary condiion, have been olved wih he GWRM for a number of differen equilibria. We here udy he abil- iy of he z-inch equilibrium B 0θ = r, B 0z = 0, 0 = 1 r, where r denoe he radial coordinae. Alo, i i aumed ha ρ = conan and ha η = 0. The effec of an iniial erurbaion (m,k) = (1,10) i followed for a ufficienly long ime ha a deendence of he form ex(γ) ha ime o develo. The oluion o hi roblem i known from comuaion uing oher mehod. We have alied boh Mehod I and II o hi roblem. In Figure 3(a) i dilayed he evoluion in ime and ace of he radial, erurbed fluid velociy, uing Mehod I wih 5 emoral ubdomain (N = 5) and a ingle aial domain, uing order K = 5 and L = 10 and a ime domain reaching o 10 normalized (Alfvén ime) uni. The equilibrium wa unable for hi erurbaion. Obained growh rae i γ = 1.03, o be comared wih he correc reul γ = In Figure 3(b) he ame mehod i ued for 5 emoral domain, bu now for 3 aial domain uing x = 0.05 and wih order K = 4 and L = 3. The obained growh rae wa γ = In comarion wih he ingle ubdomain cae, 81% of he comuaional ime wa (a) r (b) Figure 3. (a) GWRM ingle aial domain oluion of an MHD abiliy roblem formulaed uing Equaion (18), howing radial velociy ur. Mehod I wa ued wih N = 1, N = 5, K = 5, L = 10; (b) GWRM oluion a in Figure 3(a), bu here N = 3, x = 0.05, N = 5, K = 4, L = 3. Coyrigh 01 SciRe.

9 80 J. SCHEFFEL, A. A. MIRZA needed, wih a memory requiremen of 8%. I hould be noed ha relaively high value of K would be required for hi cae for ufficien accuracy if a ingle emoral domain were ued. Thi i due o he exonenial deendence of he oluion, which require high order ecral reoluion. Uing order K = 5 and L = 10, he value γ = 0.73 wa found, and a ufficienly high value of K, he inernal Male memory (on a able-o comuer, order 50 Mb) wa inufficien. Alicaion of Mehod II wa unucceful; convergence could only be obained for imracical ubdomain ime inerval (<0.1 ime uni). Thi being in ie of alicaion of he w mehod, and of variou echnique o exedie convergence. Alhough hi i a linear roblem in he unknown variable, ieraion are needed o aify he inernal boundary condiion, ince he ubdomain are arially decouled. Thu he SIR olver wa fruilely ried wih variou value of i convergence arameer. Likewie, Mehod III wa unucceful in handling hi comlex roblem. The reaon i found in ha, wherea Mehod I reain he acaualiy of he GWRM, Mehod II and III inroduce indeenden aial region ha become couled o he lengh of he ime inerval hrough a CFL-like crierion. Thu, ime domain mu be ke mall for convergence. 6. Dicuion The GWRM convergence of roblem wih a ingle aial domain i ofen adequae, bu memory requiremen may be large when many aial mode are emloyed for obaining high accuracy. Thi i mainly due o ha a marix equaion for all GWRM coefficien need be olved a each ieraion by a roo olver, a rocedure ha cale a L 3 oeraion. By emloying a e of aial ubdomain, memory requiremen may be reduced. Saial ubdomain are here imlemened in hree differen way. In Mehod I, he inernal boundary condiion of all ubdomain are imulaneouly olved for a each ieraion; ee Equaion (1). Thi ill require he ue of a ingle, global marix equaion for he GWRM coefficien. In Mehod II, he ubdomain are decouled o ha he connecing inernal boundary condiion are udaed conecuively a each ieraion; ee Equaion (13). Finally, in Mehod III he inernal boundary condiion are udaed joinly a he end of each ieraion; ee Equaion (14). I i in he wo laer cae ha he main oenial for reducion of memory requiremen exi, and hu of efficiency in olving he GWRM marix equaion. If LN ecral mode 3 are emloyed a each ubdomain, NL N oeraion would be erformed for coefficien marix oluion a each ieraion. Thi rereen a gain in efficiency by a facor N. We how ha, in order o reain accuracy, more mode are needed in each ubdomain, o he gain in efficiency i omewha reduced for he ame accuracy. Uually Mehod II and III require more ieraion han Mehod I. The caue for hi i ha he informaion of he inernal boundary condiion i changing raidly. By inroducion of a weighing arameer w, we have hown ha by conrolling he rae a which informaion ener he inernal boundary condiion, oherwie numerically unable cae may be abilized. For nearly all cae udied in hi work, Mehod II ha urned ou o be referable o Mehod III. All hree mehod have been emloyed o olve he examle magneohydrodynamic abiliy roblem. Thi i an advanced roblem, ince i involve he imulaneou oluion of 14 couled arial differenial equaion, wih a raidly (exonenially) growing oluion. Mehod I wa uccefully alied, boh for ingle and mulile aial ubdomain and uing mulile emoral domain. Wherea Mehod II and III were alicable o he Burger examle, hey were however ineffecive for hi cae in he ene ha unrealiically mall ime domain were eenial for convergence. We eculae ha hi behaviour i relaed o he caual behaviour ha i imoed by arially decouling he ubdomain (exce for boundary oin) from each oher. Saial decouling imlie a CFL-like condiion o be aified. For N = 3, i would hu eem ha he ime inerval may no exceed he raio of characeriic lengh (1/3 in normalized uni) o characeriic eed (1 in normalized uni), hu being aroximaely equal o 1/3 ime uni. The w mehod wa hown o add increaed abiliy, bu convergence remained unrealiically low for all value of w. Turning o he Burger cae, he exlici finie difference abiliy crierion i aroximaely [1] L N = for N = 5. Thi may exlain he need for he w mehod here. We have emloyed he SIR roo olver [5], wih arameer e o make i idenical o Newon mehod, for inverion of he GWRM marice of hi work. Saring oin for he ieraion wa choen o be he iniial condiion of he roblem. I hould be menioned ha furher oimizaion of convergence may be aquired by adjuing he arameer of SIR. Temoral GWRM ubdomain were udied for fir order conac alicaion ince he GWRM equaion are formulaed a a e of fir order arial differenial equaion, and are eay o imlemen. Nonehele, econd or higher order conac in ime hould be conidered in fuure work o imrove convergence for advanced roblem. 7. Concluion Imlemenaion of aial and emoral ubdomain for Coyrigh 01 SciRe.

10 J. SCHEFFEL, A. A. MIRZA 81 he generalized weighed ecral mehod (GWRM [1]) have been demonraed. In hi mehod, alo he ime domain i given a ecral rereenaion in erm of Chebyhev olynomial. Two examle alicaion have been ued; he nonlinear Burger equaion and a yem of 14 couled magneohydrodynamic equaion. Three differen mehod emloying aial ubdomain were inroduced. Wherea he fir involved imulaneou oluion of all, global Chebyhev coefficien, he oher wo have he oenial of being comuaionally le demanding becaue heir correonding Chebyhev coefficien marix equaion are ieraed earaely a each ieraion e. I wa found ha emoral ubdomain are eenial for efficiency for boh high accuracy and exended ime comuaion. Solving a iff ordinary differenial equaion i wa hown ha an adaive GWRM ime domain formulaion comared well wih commercial ofware boh regarding efficiency and accuracy. 8. Acknowledgemen We graefully acknowledge helful dicuion wih Dr. Thoma Johnon. REFERENCES [1] J. Scheffel, A Secral Mehod in Time for Iniial-Value Problem, American Journal of Comuaional Mahemaic, 01. [] J. C. Maon and D. C. Handcomb, Chebyhev Polynomial, Chaman and Hall/CRC, New York, 003. [3] R. Peyre, Secral Mehod for Incomreible Vicou Flow, Sringer-Verlag, Berlin, 00. [4] C. Canuo, M. Y. Huaini, A. Quareroni and T. A. Zang, Secral Mehod, Evoluion o Comlex Geomerie and Alicaion o Fluid Dynamic, Sringer-Verlag, Berlin, 007. [5] J. Scheffel and C. Håkanon, Soluion of Syem of Non-Linear Equaion A Semi-Imlici Aroach, Alied Numerical Mahemaic, Vol. 59, No. 10, 009, doi: /j.anum [6] W. H. Pre, S. A. Teukolky, W. T. Veerling and B. P. Flannery, Numerical Recie, Cambridge Univeriy Pre, Cambridge, 199. [7] J. Weon, Tokamak, nd Ediion, Clarendon Pre, Oxford, Aendix In hi Aendix, we derive a crierion for he minimum number H of conac oin required a he inernal boundarie in order o deermine, and no underdeermine, an aroximae oluion ha ue he full informaion from boh he yem of differenial equaion and from he exernal boundary condiion. H will be a funcion of he aial order of he yem of differenial equaion, he number of aial ubdomain N and he number of variable V. The analyi i rericed o a ingle aial dimenion bu can eaily be generalized o higher dimenion. The crierion can alo be alied o emoral domain. We aume ha he aroximae oluion can be exreed a a runcaed exanion in ome bai, for examle Chebyhev olynomial. The oal number of unknown coefficien of he yem ha mu be given informaion i VN, ince he oluion for each variable in each ubdomain need informaion in he form of relaion o be comleely deermined. The differenial equaion will rovide for informaion o olve for all bu of hee in each ubdomain. Thu, in oal, he differenial equaion will rovide VN N V 1N equaion. Thi give he oal number of inernal boundary condiion ha are needed; 1 1 VN V N N where conribuion from exernally given boundary condiion have been accouned for. The exernal boundary condiion may be alied on any variable, a any exernal boundary and for any order le han. The wo ouermo aial ubdomain relaed o each variable alo have conac wih he inner ubdomain. Thee oin add u o HV in oal. The deired crierion can now be derived from he requiremen ha he average number of inernal conac oin N 1 HV VN hould be le or equal han H. Solving hi equaion, here reul he crierion H (A1) V The crierion i indeenden of, a i hould be. For double handhaking, ha i for H =, we ee ha condiion (A1) i eaily aified for he magneohydrodynamic roblem of Secion 5., for which = 6 and V = 7. Addiional conac oin near each inernal boundary may of coure be added for comuaional reaon. N Coyrigh 01 SciRe.