envionmen ha implemens all of he common algoihmic deails of all nodal mehods u pemis he specic mehod o e used in any concee insance o e specied y he u

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Ethel Miles
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1 Linea One-Cell Funcional Mehods fo he Two Dimensional Tanspo Equaion. Pa I. The Nodal Fomulaion y G. D. Allen and Paul Nelson Asac We develop a class of spaial appoximaions o he wo-dimensional anspo equaion in he conex of he ay-acing sweeps ha lie a he coe of he sandad souce ieaion appoach o soluion of he coesponding discee-odinaes appoximaion These nodal LOF mehods ae ased upon s applying geneal nodal decoupling echniques hen using a geneic one-dimensional LOF mehod fo each of he esuling one-dimensional anspo equaions. A vaiey of known wo-dimensional spaial appoximaions and also some new appoximaions ae shown o coespond o dieen specic concee insances of his geneal fomalism and some heoeical popeies of LOF nodal mehods ae developed. As in he one-dimensional case one advanage of his appoach is ha i pemis use of a common algoihmic appoach acoss a vey oad class of spaial anspo appoximaions wih he specic appoximaion used in any insance eing dened y a (ypically simple) use-supplied suouine. This exiiliy is demonsaed y a eseach code ha has een developed o pemi deeminaion of he ode of accuacy in seveal commonly used noms of an aiay LOF nodal mehod. Some sample esuls oained y use of his code ae pesened and discussed in he conex of pevious such esuls. Inoducion The soluion of he (discee-odinae appoximaion o he) neuon anspo equaion in geomeically complex mulidimensional seings can e exemely compuaionally inensive. Nodal mehods have poven (e.g. Baduzzaman (99)) o e well-suied o his challenge. Howeve dieen nodal mehods ae pefeed y dieen people fo dieen siuaions fo oh ojecive and sujecive easons. The sandad appoach o implemening such dieen nodal mehods is developmen of dieen codes a iniio. A novel alenaive appoach would e o povide a compuaional

2 envionmen ha implemens all of he common algoihmic deails of all nodal mehods u pemis he specic mehod o e used in any concee insance o e specied y he use. Such a common famewok has een descied in he seing of spaial appoximaions o one-dimensional anspo polems iniially fo ay acing y Nelson and Zelazny (986) and susequenly fo full souce ieaion (Nelson (987) Kelle and Nelson (988)). This famewok has susequenly een applied o povide compuaional envionmens ha pemi highly auomaed deeminaion of he ode of accuacy of a moe-o-less aiay one-dimensional spaial appoximaion (Nelson and Ek (993)) and ha faciliae he elaive compaison of dieen such appoximaions agains exising enchmaks (Javis and Nelson (994)). The pimay pupose of he pesen wok is o demonsae ha his one-dimensional famewok can e exended o povide a simila common famewok fo he algoihmic aspecs of ay acing fo a vey geneal class of nodal appoximaions o he wo-dimensional (monoenegeic) anspo equaion. The conens of his pape ae as follows. In Secion 2 he elemens of nodal mehods ae eviewed and some heoeical issues ha seem no o have een idenied peviously ae discussed. Linea one-cell funcional (LOF) nodal mehods ae asacly oulined in he following Secion 3. In Secion 4 seveal known mehods ae shown o e insances of LOF nodal mehods and some appaenly new mehods ae developed wihin his famewok. Secion 5 is devoed o some heoeical aspecs of LOF nodal mehods. A compuaional envionmen inended o pemi deeminaion of he ode of accuacy ininegal 2 and noms of an aiay LOF nodal mehod is descied in Secion 6. Some sample esuls fom applicaion of his envionmen ae pesened in Secion 7 and discussed in he conex of peviously pulished esuls egading he ode of accuacy of spaial appoximaions o he mulidimensional anspo equaion. Ou concluding Secion 8 is pimaily devoed o suggesions fo fuue elaed wok. We conclude his inoducoy secion wih a ief discussion of he advanages and also one disadvanage of implemening spaial anspo appoximaions in a common algoihmic famewok. The wo pincipal advanages ae: Only a small incemenal eo is equied o implemen an addiional spaial appoximaion. (Fo example o es is pefomance on a paicula class of polem.) 2

3 The eec of dieen implemenaions is minimized which faciliaes compaisons of mehods (i.e. i minimizes he impac of pogam implemenaion issues). The second of hese is somewha of a doule-edged swod in ha a common algoihmic famewok canno povide he ulimae compuaional eciency fo all mehods. Similaly he ounday eween he common envionmen and he code dening a paicula mehod may no pemi he umos in guads agains oundo eo fo ha paicula mehod. The es implemenaion of any specic mehod always will ake he fom of code ailoed o ha paicula mehod. Howeve hee ae seings (e.g. compaaive esing of mehods o any siuaion in which one wans o pemi maximum exiiliy in admiing new o addiional mehois woh minimal eo) in which he advanages of a common envionmen ouweigh he disadvanages. 2 Nodal Mehods We wie he monoenegeic wo-dimensional anspo equaion in = Q: () Hee is he p angula ux in he diecions having diecion cosines and := 2 2 o moe iey in he diecion (; ). Fuhe is he oal coss secion and Q is he oal souce funcion in he diecion (; ). In he pesen wok he focus is upon he monodiecional vesion of his equaion as appopiae o execuion of he ay-acing calculaions ha consiue he compuaions inside he innemos loops of any soluion of he discee-odinaes appoximaion o (). Tha means and will e aken as xed usually as nonnegaive in ode o x he ideas sujec o 2 2 and Q will e eaed as known. Conside he ask of oaining an appoximae soluion of Eq. () on a given ecangula cell say C sujec o ounday condiions consising of known along he edges on which he veco (; ) poins ino C. I is convenien o escale his cell o he uni squae R := f(x; y)jx; y g: If he oiginal x and y dimensions of C wee h and k especively he oiginal anspo equaion ove he uni squae ecomes 2 2 = Q; x; y : 3

4 which we hencefoh = Q; x; y : (2) wih he deniions := 2 h and := 2 k. Addiionally we assume ha ove each cell is consan and all efeences o efes o is value ove he cell in quesion denoed C aove. In he ypical nodal mehod ansvese inegaions of (2) each muliplied y some se of asis funcions ae pefomed in each diecion. We selec as asis funcions he Legende polynomials R nomalized so ha P n is he unique polynomial of degee n such ha P i(z)p j (z)dz = 2 2i ij whee ij is he Konecke dela. We hen oain he momen funcions (x) and (y) dened especively y (x) := (x; y)p ny (y)dy and (y) := (x; y)p mx (x)dx: Now muliply (2) y P n (x);n =;:::;N y (P n (y);n =;:::;M x ) and inegae on x (especively y) fom o ooain he especive sysems of equaions d dx = Q ny (x) T ny (x); =;:::;N y ; (3 x ) d dy = Q mx (y) T mx (y); =;:::;M x ; (3 y ) whee he Q ny and Q mx ae dened analogously o he coesponding and excep wih eplaced y Q. The ems T ny (x) (esp. T mx (y)) ae he ansvese inegals which aise fom he inegaion (esp. ) Tny(x) := T mx (y) (x; y) P ny (x; y) P mx (x) In he nally esuling nodal appoximaions hese ansvese inegals seve o couple he wo sysems of equaions as 4

5 The ansvese inegal appeaing in Eq. (3 x ) can e expessed as T P (y)dy = (x; y)p ny (y) j = (x; ) (x; )() ny (x; y) d dy P (y)dy (x; y)[(2 )P ny (y)p 2(y)]dy = (x; ) (x; )() ny (2 ) (x) = (x; ) (x; )() ny X ()=2c = [2( 2) ] ny2(x); (4) whee xc is he geaes inege no exceeding x. Similaly ha appeaing in (3 y )isgive T P (x)dx = (x; y)p mx (x) j = (;y) (;y)() mx (x; y) d dx P (x)dx (x; y)[(2 )P mx(x)p 2(x)]dx = (;y) (;y)() mx (2 ) (y) = (;y) (;y)() ny X [()=2] = [2( 2) ] 2(y); (5) In he moe inuiive lef-igh op-oom cell edge noaion hese ead especively as and T (x) = (x; y ) (x; y )() ny (2 ) (x) ; (6 x ) T (y) = (x ;y) (x;y)() mx (2 ) (y) : (6 y ) 5

6 Noe ha he ansvese inegal ems involve ansvese cell edge daa and also momen funcions of lowe ode. As o he second in he even ha he momen index is negaive he momen is aken o e zeo. As o he s oseve ha he cell edge daa is pescied fo he soluion o he oiginal anspo equaion (2) no fo he momen funcions. Tha is some appoximaion is equied in ode o educe he sysem of M x N y 2 equaions oained fom susiuing (4) and (5) ino (3) o a sysem conaining he same nume of unknowns. The nodal closue assumpions ha ae sandadly used fo his pupose ae and (x; ) = (;y)= XM x = N X y = mx ()P mx (x) (7 x ) My ()P ny (y); (7 y ) whee he x-channels (x) (y-channels (y)) ae o e he ulimae appoximaions o (x) (especively (y)). These appoximaions ae puaively dened y susiuing (7 x ) ino (4) (7 y ) ino (5) hen susiuing he esuling expessions ino (3) and nally eplacing he ( ) eveywhee y ( ). The esuls of hese manipulaions ae he sysems of equaions d dx = XM x = X ()=2c = P mx (x) () [2( 2) ] 2 XM x = P mx (x) () Q ny ; =;:::;N y (7) and d dy = N X y = X ()=2c = P ny (y) () [2( 2) ] 2 N X y = P ny (y) () Q mx ; =;:::;M x : (8) 6

7 These equaions ae o e inepeed as follows. In he equaions (7)fo he x-channels one of he M x -vecos () is known eihe fom ounday condiions o as he esul of he ay acing compuaions fom some pevious cell and similaly fo he y-channel equaions (8). (Fo example if and ae posiive hen () and () ae known.) Unde hese cicumsances he x-channel equaions (7) have he supecial appeaance of an iniial-value polem fo he x-channels and similaly fo he y-channel equaions. Howeve hese equaions do no consiue a sandad iniialvalue polem fo a sysem of odinay dieenial equaions in ha hey have he mahemaically nonsandad feaue ha each is coupled o he ohe hough he iniially unknown eminal values of he asic unknowns of he ohe sysem of equaions. Because of his hee is a fundamenal issue of whehe in he genealiy wehave posed i his polem consiues a mahemaicaly well-posed polem in he sense of exisence and uniqueness of soluions. In fac nodal appoximaions ae well-posed in his genealiy and he following esul is poved in he Appendix. Theoem. The sysem of equaions (7) and (8) has a unique (coninuous) soluion (x); = ;:::N y and (y); = ;:::M x sujec o aiay specied values of one each of he N y -vecos () and he M x -vecos (). 3 Linea One-Cell Mehods. We eun o he sysems of odinay dieenial equaions (.3) and (.4) knowing ha he ems T ny and T nx involve Legende expansions as peviously descied. Fo >and given (x) possily appoximae an LOF mehod fo he ay-acing polem (3 x ) is an appoximaion having he fom (x ) (x;h; (x)) := (; ) (x) m (; )L m (Q j C; x) m (; ) Lm (T ny j C; x) (4 x ) 7

8 Hee m ; m =;:::;p and m ; m =;:::;p ae poin funcions. The linea funcionals L m ( ; x) and Lm (; x) ae emed he asic linea funcionals of he paicula mehod. The noaion f j C means f esiced o C'. If < a fomula simila o (4 x ) holds wih x and x inechanged. Hee we have eained he x; x (lef and igh) eminology fo convenience hough fo he uni squae x = and x =. Suppessing all funcional agumens in he LOF (4 x ) and using fo (x ) and fo (x) we oain he equivalen u somewha simple appeaing fom = m L m (Q) m Lm (T ny ) (5 x ) A simila appaaus fo he nodal equaions fo he (y) follows. Fis he LOF has he fom (x ) (x ;h; (x )) := ^(; ) (x ) ^m (; )^Lm (Q j C; x ) ^ m (; )^ Lm (T mx j C; x ) (4 y ) whee ems have analogous meanings fo he op () and oom () ems as fo he lef and igh ems and hen he simplied fom =^ ^m ^Lm (Q) ^ m ^ Lm (T mx ): (5 y ) When fully susiued hese equaions may well involve volume momens of he ux (x; y). Thus he LOF equaions assume as given he incoming edge ux momens and ceain volume uxes o geneae he exiing edge uxes. Fom all hese he desied momen uxes ae compued. In all u he simples case his canno e accomplished wihou a se of companion equaons ha elae volume and edge uxes. These equaions can e geneaed in a leas wo ways (e.g. momens alance equaions) as we will see. The limis p can e dieen fo he Q and T funcionals. 8

9 In moe asac ems le us denoe y E I V and S he vecos of exiing edge uxes eneing edge uxes volume momens and souces fo a given cell. A LOF mehod hen has he fom E = T I F E G V HS The companio equaions have he fom A V B E C I DS = and we assume ha his sysem is uniquely solvale fo aive a he sysem E = T I F E HS V. In his way we The soluion fo he exiing uxes is give E =(IF) (T I HS) Specic examples will claify his. Ou s examples ely on no companion equaions ecause V is no involved. Example. (Diamond-dieence mehod.) In his example we selec a paicula se of linea funcionals ha will show ou famewok conains his vey familia example. Fis selec M x = N y =. Le L := L = L e he sole linea funcionals give he aveaging inegal L(f) := 2 f(x)dx Then he diamond-dieence ype fomula akes he fom = 2 (L(Q )L(T y )) Since L(T y )= ( 2 ) we oain he s of he equaions fo he wo dimensional diamond-dieence mehod = 2 L(Q ) ( x The second equaion deived fo he h x momen is give = 2 L(Q ) ( y 9 ): ):

10 The coupling is elaively simple in his case. In he case wihou scaeing namely he case when Q conains only souces all unknowns ae availale in jus hese wo ses of equaions. To aain he fom of he diamond dieence mehod say as found in [Lewis and Mille p66] eliminae s equaion and and y s eliminaed we oain fom he fom he second equaion. Wih he susciped x s = = L(Q ) L(Q ) : Recall his fomulaion is given in scaled coodinaes. Example 2. (Genealized diamond-dieence mehod.) In his example we ake M x and N y aiay (posiive ineges) p = ; ake fo L = ^L he aveaging opeao L dened aove and fo L = ^ L he funcional dened y L (f) := (f() f()): 2 Finally dene he diamond-dieence ype LOFs n =:::;N y y = Mx [ 2 L(Q ) X = [(2 )( 2 ( 4 () ny )( () mx )] )=2]: Noe hee we have used he Legende seies epesenaion of (x; ) and (x; ) as in (7a). The coesponding equaion fo is = 2 L(Q ) X Ny [ = 2 ( 4 [(2 )( () mx )( () ny )] )=2]:

11 The simpliciy of his mehod such as i is ess wih is avoidance of any unknowns ohe han he cell edge momens. When N y = M x = he mehod has he paiculaly simple fom and y = = 2 [ Qy 2 ( y 2 [ Qy 2 ( )] ) 2 ( )]: Simila fomulas fo and x can e deived. This mehod is no popely wihin he scope of nodal mehods. In a puis sense his mehod is no popely wihin he scope of LOF mehods in as much as he peliminay Legende expansion is necessay. Refeences Baduzzaman Ahmed (99) Advances in Nuclea Science and Technology 2.. Javis R. D. and Nelson P. (994) Compue Physics Communicaions Kelle H. B. and Nelson P. (988) Tanspo Theoy and Saisical Physics Nelson P. (987) Annals of Nuclea Enegy Nelson P. and Ek D. S. (993) Compue Physics Communicaions Nelson P. and Zelazny R. (986) Nuclea Science and Engineeing Appendix - Poof of Theoem By escaling he independen vaiales he iniial/eminal-value polem associaed wih he channel equaions (7) and (7) can e ewien as and u (z) =A(z)u(z)B(z)v() q u (z); v (z) =C(z)v(z)D(z)u() q v (z); whee u and v ae veco funcions of especive dimensions N y and M x he coninuous maix funcions A(z);B(z);C(z) and D(z) ae known as

12 ae he coninuous veco funcions q u (z) and q v (z) and he iniial values u() and v(). This is equivalen o he inegal sysem u(z) = v(z) = Z z Z z A(z )u(z )B(z )v() u() q u (z ) dz ; (A u ) C(z )v(z )D(z )v() v() q v (z ) dz ; (A v ) We wie his as (u; v) = T(u; v) whee he opeao T is dened y (u ; v ) = T(u; v) wih u (v ) he igh-hand side of (A u ) (esp. (A v )). We esalish he desied esul y demonsaing ha T is a conacive mapping on he space of odeed pais of coninuous veco funcions (u; v) of dimensions as aove unde he nom k(u; v)k = max z n e 4Mz max fku(z)k; kv(z)kg fo appopiaely seleced M whee kk is he usual max nom (also known as he inniy nom) on vecos. As his space is complee unde his \sanom" he desied esul follows fom he well-known conacive mapping heoem. Le (u ; v )=T(uu ;vv ) fo (u; v) and (u ; v ) aiay elemens of he space. Then u (z) = Z A(z ) u(z )u (z ) B(z )(v() v ()) dz : (A 2) Now choose M o e lage han any of he (maximum o inniy) maix noms fo he maices A(z);B(z);C(z) o D(z) fo z. I hen follows fom Eq. (A-2) ha ku (z)k Z z Z z o ; M k(u(z ) u (z )k k(v() v ()k dz 2Me 4Mz dz k (u u ; v v ) k < e4mz 2 k(u u ;v v k : The same inequaliy is similaly esalished fo kv (z)k hus k(u ; v )k (=2)k(u u ; v v )k ; which gives he desied esul. 2

On The Estimation of Two Missing Values in Randomized Complete Block Designs

On The Estimation of Two Missing Values in Randomized Complete Block Designs Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.