AN ANALYTICAL SOLUTION FOR THE ONE-DIMENSIONAL TIME- DEPENDENT S N TRANSPORT EQUATION FOR BOUNDED AND UNBOUNDED DOMAIN IN CARTESIAN GEOMETRY

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1 9 Iraoal uclar Alac Cofrc - IAC 9 Ro d JaroRJ Brazl Sbr7 o Ocobr 9 ASSOCIAÇÃO BRASILEIRA DE EERIA UCLEAR - ABE ISB: A AALYTICAL SOLUTIO FOR TE OE-DIMESIOAL TIME- DEPEDET S TRASPORT EQUATIO FOR BOUDED AD UBOUDED DOMAI I CARTESIA EOMETRY Cyha F. Sgao Marco T. Vlha ad Tífa T. oçalz Prograa d Pó-raduação Eghara Mcâca PROMEC Uvrdad Fdral do Ro rad do Sul Rua Saro L 45/ Poro Algr RS cgao@q.cq.br; vlha@q.cq.br; fa.gocalz@hoal.co Prograa d Pó raduação Maáca Alcada PPMA Uvrdad Fdral do Ro rad do Sul Av. Bo oçalv Poro Algr - RS ABSTRACT I h wor w ror a aalycal oluo for h dd S raor quao a la aralll gory for uboudd doa w a for <. For uch w aly h Lalac rafor chqu h varabl ad h LTS aroach h aal varabl. By h rocdur w co ou wh a aalycal oluo for h agular flu gral for alyg h boud of h agular flu a fy. W r urcal ulao ad alo valdao by h aaly of h ayoc bhavor of h calar flu a lab.. ITRODUCTIO Rcly aard h lraur h TLTS hod [] whch olv h -dd S aroao of h uro raor quao a lab for oorgc uro. Th rcal faur of h hodology rl o h : alcao of h Lalac rafor varabl oluo of h rulg quao by h LTS hod ad rcoruco of h agular flu -varabl ug urcal vro. W ar awar of vral wor lraur for h y of robl cocrg uboudd doa [ ] bu for boudd doa carc [8 9 ]. Sg furhr h wor w calz h alcao of h chqu for robl wh uboudd doa. For uch du h aalycal faur of h oluo rd ar for for a lab aarg a rvou wor of Sgao al. [] w u rlac h boudary codo a h far d of h lab by h boud of h agular flu a fy. To h h goal w vah h ubvcor of arbrary coa aocad o h of h ov gvalu. W alo dlay urcal ulao ad valdao wh h ayoc bhavor of h calar flu a lab.

2 . TE TLTS METOD L u codr h followg oroc -dd S raor robl: ψ ψ ψ ψ ' d ' S v for > ubc o h al codo ψ φ wh boudary codo ψ f for > >. 3 ad boud of h agular flu a fy. r w ado h adard oao for h arar. Alyg h Lalac rafor chqu h varabl quao w co ou wh h quao: ' d ' R 4 ubc o h boudary codo f for > 5 ad boud of h rafor agular flu a fy. r do h Lalac rafor of ψ ; ad R φ S. Th S aroao v v of h abov aaz rad l: d w R 6 d ubc o h boudary codo l f for > 7 ad boud of h rafor agular flu a fy a h dcr drco. r ar h roo of h h dgr Lgdr Polyoal ordrg a dcra ordr -< < < / < < / < < < h raford agular flu h dcr IAC 9 Ro d Jaro RJ Brazl.

3 IAC 9 Ro d Jaro RJ Brazl. drco ad R h raford ourc r. Rcag quao 6 ar for w hav: R A d d 8 whr A a ordr ar who r ar f w f w a 9 wh boudary codo: F ad boud codo aly l To h o w u oc ha h LTS oluo for 8 robl rad l: D L D ξ ξ ad alo ad ` do rcvly h / ordr vcor for h ov ad gav drco. Furhr ± D ar rcvly h ov ad gav gvalu dagoal arc of / ordr ad h gvcor ar of A aarg quao 9. O h ohr had h arcular oluo wr a: d R B d R B 3 whr B D 4 To coruc h oluo for uboudd doa w u aly h boudary ad boud codo for h agular flu. To fulfll h boud of h agular flu

4 IAC 9 Ro d Jaro RJ Brazl. w forcd h vcor ξ quao o vah. I ur ou ha h oluo for uboudd doa rad l: > ξ f D 5 ad < ξ f D 6 ow w ar oo o dr h uow vcor ξ alyg h boudary codo quao 5. Cocludg w u rar ha ad ar dfd l h l of dfd quao 3 wh L go o fy. ow w ar abl o rcoruc h agular flu for uboudd doa ug h dfo of h Lalac rafor vro for h raford agular flu w a: d γ γ π ψ. 7 W cla ha h abov aaz h aalycal oluo for h robl 6 h ha o aroao ad alog drvao. To ovrco h dffculy of olvg aalycally h l gral quao 7 for hor ad log ra w olv urcally by h aua-quadraur ch QS h avr Wy-Rho WR ad h avr-sf S algorh. I h qul w brfly dcrb h algorh. Idd h aua quadraur ch aroa coour gral for h dd agular flu wr l M a d γ γ π ψ 8 whr a ar h roo ad wgh of h aua quadraur []. O h ohr had h Lalac vro by h avr algorh [3] gv by: l l ψ 9 wh h rcurv forulao: l l M

5 I ordr o acclra h covrgc of h abov algorh w u h acclraor chqu: Sf [4] ad Wy-Rho [3]. Th rao for choog h aroach co fro hr good rforac obrvd lraur. Cocludg w ar oo o rar ha h gral archd TLTS oluo rd by quao 5 for > ad 6 for <. Fally w u rcall ha robl for uboudd doa wh < < ca b raghly dr ug h Placz La [5] for h raford robl. I fac h Placz la a ha h ady-a robl a la aralll gory for < < rlad o h o for < < hrfor our aaly for uboudd doa cold. 3. UMERICAL RESULTS AD COCLUSIO I h qul o how h a of h TLTS hod o olv raor robl laaralll gory for uboudd doa < w olv by h hodology h robl: 3.. I h fr robl w ror a udy abou h urcal aroach dcud o valua h oluo havg d h obcv of fdg ou h or arorad hod rgardg h accuracy o valua h l gral oluo rood. For uch l u codr h robl: ψ ψ ψ ψ ' d ' ψ ψ 4 > ψ wh h ug carg cro co valu. ad. 9. I abl w r h TLTS rul valuag h l gral oluo by aua Quadraur avr Wy-Rho ad avr-sf aroach. W do h calculao for all ad larg 6 w a for ragg fro o 8 c. W alo coar h rul wh h ady-a rul drd by h LTS 6 chqu rcallg ha a go o fy h -dd oluo go o h aoary oluo. W u bar d ha h LTS 6 oluo ac o h d of robl. v a clor loo o h rul rord w raghly fgur ou ha avr-sf chqu h or accura ad coquly uabl o valua h TLTS l gral oluo hr for hor ad log r ra. So far fro ow w corora h avr-sf aroach o h TLTS hod. 3.. w aalyz h ayoc bhavor of h oluo for h lab by crag h lab hc ordr o valda h rood oluo. To h d w olv h robl: IAC 9 Ro d Jaro RJ Brazl.

6 Tabl : Sudy of h TLTS 6 oluo for calar flu a for ragg fro -6 c o 8 c ug aua Quadraur avr Wy-Rho ad avr-shf urcal aroach c aua Quadraur avr Wy-Rho avr Sf LTS LTS ψ ψ ψ ψ ' d ' ψ ψ L > ψ wh carg cro co. ad. 9 Tabl : Th ayoc bhavor of h gral TLTS 6 calar flu oluo obad by crag h lab hc fro 4 o 7 f for ragg fro o 4c.9. - c. L UDTLTS c. IAC 9 Ro d Jaro RJ Brazl.

7 L UDTLTS c. L UDTLTS I abl w dcd h TLTS 6 oluo aad by ragg h lab hc fro 4 o 7 f for h valu o ad 4 c ad w alo coar h rul wh h o obad by TLTS oluo for uboudd doa dubbd a UDLTS 6. A l co of h rul dcd w roly ralz h cd covrgc of h crag lab hc oluo o h uboudd oluo oc h rul for L7 f cocd wh h LTS 6 oluo for uboudd doa. Fro h rvou dcuo w ralz ha h rood oluo bd h lgac ad lcy du h aalycal characr a robu chqu o gra bchar rul for h or of robl hr for hor ad log r ra. Th ufcav for h affrav bad o h rovd covrgc of h TLTS for h ac oluo wh go o fy [6 7] ha a h rror rducd o h roud-off rror. Movad by h ucc of our ob h a of olvg h -raor robl o-doal Cara gory hr for lab [] ad uboudd doa ug h doubl Lalac rafor chqu w focu our fuur rarch o h u of archg aalycal oluo by h hodology for h wo-doal ady-a S raor quao a rcagl. 4. ACKOWLEDEMETS Th auhor ar grafully dbd o CPq Colho acoal d Dvolvo Cífco Tcológco for h aral facal uor o h wor. 5. REFERECES. B Sgao C.F. Vlha M.T. ad oçalz T.T. "A aalycal oluo for h dd S raor quao a lab" Krch Vol Flo W.L. ad aaol B.D. "T-Dd O-Doal Traor Calculao Ug h Srag Ray Mhod" uclar Scc ad Egrg Vol aaol B.D. ad McKy P.W. "Th rao of T-Dd uro Traor Soluo If Mda" uclar Scc ad Egrg Vol IAC 9 Ro d Jaro RJ Brazl.

8 4. aaol B.D. "Soluo of h -dd oorgc uro raor quao -f du" Traor Thory ad Sacal Phyc Vol aaol B.D. T Dd urfac agular flu for a -f du wh cular rflco" uclar Scc ad Egrg Vol aaol B.D. McK P.W. ad Pddcord K.L. "Th grao of -dd uro raor oluo f da" uclar Scc ad Egrg Vol Byo T.D. ad Cola M. "Drc Soluo of T-Dd uro lowg dow robl ug urcal Lalac rafor f da" uclar Scc ad Egrg Vol Kllr P.M. ad L J.C. "A -dd collo robably hod for odoal ac- uclar racor c" uclar Scc ad Egrg Vol Olvra J.P. Cardoa A.V. ad Vlha M.T.M.B "Soluo of h O-doal T-Dd Dcr Orda Probl a Slab by h Scral ad LTS Mhod" Aal of uclar Ergy Vol El WalS.A. Dghdy A.R.Sallah M. T-dd uro raor f da ug Porag-Eddgo aroao Aal uclar Ergy Vol Turc. ulcyuz M.C. ad Tzca C. " oluo of h dd lar uro raor quao for a lab ad a hr" Krch Vol P R. "aua Quadraur Forula for urcal grao of Browch' Igral ad Ivro of Lalac Trafor" Joural of Egrg Mahac Vol Való P.P. ad Aba J. "Coaro of quc acclraor for avr hod of urcal Lalac rafor vro" Iraoal Joural of Cour Mahac Vol Sfh. "urcal vro of Lalac rafor" Coucao of h ACM Vol Ca K.M. offa F. Placz. Iroduco o h hory of uro Dffuo US ovr Prg Offc Vol Pazo R.P. Thoo M. ad Vlha M.T. "Error boud for cral collocao hod for lar Bolza quao" Iraoal Joural of Couaoal ad urcal Aaly ad Alcao Vol Vlha M.T ad Pazo R.P. "Covrgc of h LTS hod: aroach of C grou" Progr uclar Ergy Vol IAC 9 Ro d Jaro RJ Brazl.