Order of Accuracy of Spatial Discretization of Method of Characteristics. Jipu Wang and William Martin

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1 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Ordr of Auray of Spaial Diriaion of hod of Chararii Jipu Wang and Willia arin Dparn of Nular nginring and Radiologial Sin Univriy of ihigan, Ann Arbor, I, 89, USA jipuwang@uih.du, wr@uih.du Bnjain Collin Oak Ridg Naional Laboraory, Oak Ridg, TN, 3783, USA ollinb@ornl.gov Abra - Th hod of hararii (oc) ha bo an apd ool for lai phyi alulaion. oc ha any advanag uh a aura rprnaion h of h lai gory and boundary ondiion. Th fla our (FS) approxiaion i o oonly ud and h linar our (LS) approxiaion an iprov h auray by prrving highr ordr paial on of h nuron our. Howvr, a a non-andard paial diriaion hod, h ordr of auray of h paial diriaion i or diffiul o obain, pially for analying linar our approxiaion, bau h oc hod uili wo of paial h h FS h and h of hararii ray ha ingra h ranpor quaion ovr h FS h. In hi work, w analy h ordr of auray wih paial roluion in oc in planar gory for boh FS and LS approxiaion and vrify our prdiion wih h hod of anufaurd Soluion (S). I i hown ha h fla our approxiaion i ond ordr aura and ha h linar our approxiaion ha a fourh ordr auray. Kyword: hod of hararii, ordr of auray, fla our approxiaion, linar our approxiaion I. INTRODUCTION Th hod of hararii (oc) ha bo an apd ool for lai phyi alulaion. oc ha any advanag uh a aura rprnaion h of lai gory and boundary ondiion. Th fla our (FS) approxiaion i o oonly ud. [,,3] Whn iprovd auray i ndd, on an ploy h linar our (LS) approxiaion. [] Howvr, h oc hod uili a non-andard paial diriaion hod wih wo of paial h h FS h and h of hararii ray ha ingra h ranpor quaion ovr h FS h. Th inraion bwn h hararii ray and h paial h i aking h rror analyi of oc oluion uh or opliad. Conqunly, h ordr of auray of h paial diriaion of oc i no wll known. In hi work, w analy h ordr of auray wih paial roluion in planar gory for boh FS and LS approxiaion and vrify our prdiion wih h hod of anufaurd Soluion (S). On dinional gory bypa h oplxiy of ray paing, nabling u o look a h rror onvrgn ra ovr paial grid rfinn alon. Bing abl o obain h rror onvrgn ra, or h ordr of auray, wih h paial roluion for boh FS approxiaion and LS approxiaion, i uful for undranding h rror inrodud in oc, whih an, in urn, infor h hoi of FS h i and h ray paing. Bing abl o loa and quanify rror an alo hlp dvlop or aura oc h. orovr, knowldg of h horial ordr of auray an b ud o vrify raor phyi od wih od vrifiaion hod, uh a S. Sion II fou on h horial prdiion of h ordr of auray. Diribud our and aring our ar analyd paraly. Wha diffrnia hi work fro prviou analyi on paial diriaion [5,] i ha w ar fro h xa oluion along h hararii and quaniaivly rak rror propagaion, aking i raighforward o gnrali h analyi fro FS o LS approxiaion. Sion III giv h nurial rul ha u S o vrify h prdiion a wll a h oc od, inluding polynoial and nonpolynoial funion for. Sion IV brifly uari h onluion. I i hown ha h fla our approxiaion i ond ordr aura and ha h linar our approxiaion ha a fourh ordr auray. II. THORY In oc, h angular flux along any hararii an b ingrad analyially wih an aud for of h righhand-id our. W will rprn h hory in ondinional gory for ipliiy. x, x, q x, () x Ingraing h abov quaion ovr a anonial paial ll j, whr x j/ x x j/, will giv u h nuron balan

2 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) quaion wihin hi ll, whih in urn an b ud o driv h xprion for h ll-avragd angular flux j. To furhr iplify algbra, h fir ll i ud o rprn h anonial ll wih orrponding o x j / and orrponding o x j /. Opraing on quaion () wih d yild q,, whr, d (3) q q, d Th xiing angular flux an alo b olvd analyially wih an ingraion faor. (),, q, d () Inring quaion () ino quaion () yild q, q, d (5) Th approxiaing ll-avragd angular flux i xprd blow, whr q approxia q and q i h avrag of hi approxiaion q. q, q, d () Th rror i dfind a h diffrn bwn h xa analyial xprion of ll avragd angular flux a in quaion (5) and i approxiaion a in quaion (). q q (7) q, d q, d Thi rror dpnd on how wll q, approxia h ru our hap. quaion (7) i valuad for boh FS and LS approxiaion and i h bai for rror analyi hroughou h papr. Whn diffrn approxiaion (.g. FS and LS approxiaion) ar ud, h following dfiniion of q i akn, wih angular dpndn droppd o avoid yboli nangln. q, fla our q (8) q q, linar our whr roh our on, q q d fir our on, q q q q d q d, ll nr loaion, No ha h dfiniion q involv a ranforaion fro a global oordina y o a loal on and i brifly illurad blow. Au our QZ i h oal our in a lab gory in a global oordina y Z Zax. Th our Zj /, Zj / i o b Q Z ovr a anonial ll linarid a q q (9), whr i h loal oordina in urrn ll j originad a h lf dg of h ll. Nx, w how how o obain h fir paial our on q fro global quanii QZ. QZ q q q Prfor opraion * d on h abov quaion. Th righ-hand-id (RHS) giv RHS q q d q Th lf-hand-id (LHS) giv

3 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Z j/ Z j/ LHS q d j/ Q Z Z Z d Z Z Z j/ Q Z Z Q Z Z dz Z j/ QZ Z Z QZ Thrfor, h fir paial our on an b xprd a q QZ Z Z QZ For a fixd our probl, h our oni of boh aring and diribud our oponn a follow. q,, d qs, () Th diribud our i dnod wih ubrip S bau S i h o oon diribud our in h drinaion of h ordr of auray and od vrifiaion prai. Thrfor, in hi papr, S will rprn h diribud our wih known ahaial for. No ha h naur of h aring our i vry diffrn fro ha of h diribud our. Th aring our hang and upda and rolv i paial dpndn ovr h our iraion, whil h diribud our ha a known ahaial for and do no hang in hi iraiv pro. No urpriingly, hi diinion ak h rror analy for h wo our yp vry diffrn. Thi i parially rfld in h praial xprion of h roh and fir our on of h wo our yp dfind in quaion (5). roh our on,, n n q q, S d q q +, a S fir our on,, n q q, q n n ˆ qs qs q = q a + S, (5) whr ˆ, d i h fir paial on of h angular flux and ˆ i i approxiaion. No ha h uprrip (n) and (n-) in h abov quaion ar h iraion indx and ha a quaniy wih uprrip (n) and i ounrpar wih uprrip (n-) will b qual up o h onvrgn riria upon onvrgn, allowing h uprrip o b droppd. In h following wo ubion, w will look a h diribud our and h aring our paraly du o h diffrn for of hir praial xprion. Wih h uprpoiion prinipl, univral for any linar y, h our hr ar addiiv o h final ordr of auray wih paial roluion will b h lowr of h wo andalon ordr of auray fro ah our yp. W look a h ordr of auray rlad o approxiaing diribud our fir.. Ordr of Auray rlad o Approxiaing Diribud Sour In hi ubion, aring our i aud o b ro. Thi happn in a purly aborbing arial. No ha whn h diribud our approxiaion q ingra q xaly, whih i uually h a, h fir wo r in quaion (7) anl ou, rduing h rror o a iplr for q d q d Now w look a FS and LS approxiaion rpivly.,, () A. Fla Sour (FS) Approxiaion FA approxiaion ipli h following q q (7) Uing quaion () o valua h rror inrodud ino h ll-avragd angular flux du o FS approxiaion q d q d, (8) Thr a wr xaind: onan, linar, and quadrai our hap. For a onan our hap, h abov rror i. Thi i vrifid in our oc D od, howing ha h oluion i aura o ahin priion rgardl of h finn of h grid. Sond, if h our i linar in pa, h fla our approxiaion inrodu an rror of ond ordr wih h h widh. Auing q, < <, h rror an b valuad wih quaion (8). d d d d (9) whr () whih i h opial hikn n by a nuron flying in h dirion hararid by μ.

4 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) A (or τ) approah ro, naly, a w rfin h paial grid, h abov rror i xpandd nar a follow, 3 5 O () 8 Thrfor, h rror i ond ordr wih h h i. Third, if h our i quadrai in pa, h fla our approxiaion will onvrg o h ru oluion wih hird ordr, whih i hown blow. Auing q, < <, h rror fro quaion (8) an b valuad d d 3 d d A τ, h rror approah ro wih hird ordr, () O (3) 8 9 Howvr, du o h ruur and opraor in h nuron ranpor quaion, only undr pial iruan will w hav a inglon quadrai our hap ha do no hav a linar oponn. Thi will au a dgradaion in ordr of auray fro hird ordr o ond ordr in probl wih a quadrai our hap, in h rror fro h linar oponn i doinaing h rror onvrgn ordr. orovr, wih induion, i an b hown ha gnrally h fla our approxiaion an a b b ond ordr aura in pa. B. Linar Sour (LS) Approxiaion LS approxiaion ipli h following q q q, () Uing quaion () o valua h rror inrodud ino h ll-avragd angular flux via LS approxiaion,, q d q d (5) q, d q q d whr, roh our on, q S q, S d fir our on, q S qs qs () I i ay o how ha linar our will b abl o rprn fla our wih no rror in q will b ro and q an rprn h onan our. Nx, w how ha linar our approxiaion an rprn linarly diribud our wih no rror. Auing q, < < in q and q dfind in quaion () giv q q d d q q q (7) 3 Th approxiaing our hap i xaly h a a h originally anufaurd our hap a hown blow q q q (8) q La, if h our hap i quadrai in pa a,, w hav q q q d d 3 q q q 3 3 (9) Conruing h linar approxiaion wih h roh and fir our on giv, q q q (3) Th rror inrodud via linar our (LS) approxiaion an b valuad wih quaion (5) d d d d 3 (3) 3 Th rror approah ro wih fourh ordr hown by Taylor xpanion nar τ=. 3 5 O (3) 3 7 A niond in h fla our ion, w norally do no hav a andalon quadrai our, rahr i uually o wih a linar oponn. Howvr, wih linar our approxiaion, h linar our an b xaly rprnd, hu having linar oponn will no dgrad h h -ordr auray.

5 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Th xpd and obrvd ordr of auray for h purly aborbing arial i uarid in Tabl, whih alo how oniny bwn xpaion and obrvaion.. Ordr of Auray rlad o Approxiaing Saring Sour Thi ubion fou on h rror inrodud in llavragd angular flux du o an rror in approxiaing aring our wih diffrn our approxiaion h, i.., FS and LS. Th following rror xprion fro quaion (7) ill hold, xp ha now q, inlud boh aring,. our q,, d and ohr diribud a our,.g., qs q q q, d q, d (33) Sin h prviou ubion ha udid h ordr of auray rlad o diribud our, hi ubion will fou on h ordr of auray rlad o approxiaing aring our ONLY. Th ovrall ordr of auray will b h lowr of h wo. Wih aring our only, h xprion for h roh and fir paial our on an b iplifid. roh our on,, n n q q a, fir our on,, n q q, q n n ˆ = q a Th FS approxiaion i lookd a fir. A. Fla Sour (FS) Approxiaion FS approxiaion ipli h following q q (35) Uing h abov quaion (33) o valua h rror inrodud ino h ll-avragd angular flux du o FS approxiaion. Th fir oponn an b valuad in h following way q q n, d d n d (3) Au ha h angular flux i ioropi and ha h ld quadraur aifi, h abov quaion (3) an b furhr rdud ino h following for upon onvrgn (iraion uprrip droppd) Thrfor, (37) (38) Nx, w look a h ond rror oponn uing h praial xprion fro quaion (3),, d d n d (39) Au ioropi angular flux, h abov quaion an b furhr rdud o h following for upon onvrgn, d d, d d () Th oal rror i olvd fro h abov quaion. d, d d () Siilarly, hr a wr xaind: onan, linar, and quadrai our hap. For a onan aring our hap, h abov rror i ro. To how hi, au a onan flux hap, whih will giv a onan aring our, () h ruling rror fro FS approxiaion an b valuad uing quaion () d d (3) d Thrfor, h xpd rror will b ro dpi how oar h h i.

6 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Sond, if h aring our i linar in pa, h fla our approxiaion will inrodu an rror of ond ordr wih h paial h widh. Auing h angular flux in h following for, () h rror inrodud fro fla our approxiaion an b valuad fro quaion () d d d d d d (5) A τ approah ro, naly, a w rfin h paial grid, h abov rror an b xpandd nar τ=, 3 O () Thrfor, h rror i ond ordr wih h paial h widh. Third, if h aring our i quadrai in pa, h FS approxiaion o aring our will inrodu a hird ordr rror, whih i ngligibl opard o h nd ordr rror inrodud fro approxiaing diribud our. Thi i hown blow. Again, au h angular flux hap,, (7) Thn w an valua h rror inrodud by FS approxiaion o h aring our wih quaion (). d d 3 d d d 3 d 3 3 (8) A τ approah ro, naly, a w rfin h paial grid, h abov rror an b xpandd nar τ=, O (9) 8 Thrfor, h rror i hird ordr wih h h widh. B. Linar Sour (LS) Approxiaion LS approxiaion ipli h following q q q, (5) Uing quaion (33) o valua h rror inrodud ino h ll-avragd angular flux du o LS approxiaion. Th fir rror oponn i q q q q q q q Th ond rror oponn i (5) q, d q q d (5) q, d q d q d Au ha h angular flux i ioropi and ha h ployd quadraur i hon uh ha i aifi. Applying h praial dfiniion of h roh and fir paial our on dfind in quaion (3), h fir rror oponn ak h following for upon onvrgn. q q n Th ond rror oponn ak h following for upon onvrgn. d q d q d d d ˆ ˆ d (53) (5) quaion (5) i ud o olv for h oal rror in ll-avragd angular flux. ov all h r involving and o h LHS and olv for h oal rror. ˆ ˆ d d d d ˆ d (55)

7 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Whn h aring our hap i onan, auing h angular flux hap a, (5) h ruling aring our hap and h roh and fir paial on of h angular flux ar qa, d, d (57) ˆ, d d quaion (55) i valuad and w onlud ha h rror inrodud du o LS approxiaion oward aring our i ro. d d d d ˆ d (58) Nx, w how ha approxiaing linar aring our wih LS approxiaion giv ro rror a wll. Auing h angular flux hap, (59) h ruling aring our and h roh and fir paial on of h angular flux ar q, d d d d, () ˆ, d d 3 Again, quaion (55) i valuad and w onlud ha h rror inrodud du o LS aupion oward linar aring our i ro. d 3 d d d ˆ d d d d d ˆ d () Th la a valua h rror inrodud du o inauray in approxiaing quadrai aring our wih LS approxiaion. Auing,, h aring our and h roh and fir paial on of h angular flux ar q, d d, d d () 3 3 ˆ, d d On again, quaion (55) i ud o valua h rror fro LS approxiaion. No ha h raio ˆ / i droppd in h forulaion for having a highr ordr han O, whih will b hown lar. 3 d 3 d d 3 d d d d d d d (3) d d d d d Th rror inrodud by LS approxiaion oward quadrai aring our i h ordr by xpanding h abov rror nar τ=. 3 7 Thrfor, h ordr of auray i h a a ha rlad o approxiaing diribud our. 5 O () C. Aing rror in ˆ In hi ubion, w giv an rror an for ˆ, dnoing h rror in h fir paial on of h angular flux. I i hown ha i ordr i highr han ha of h oal rror. Th fir paial on of h angular flux ovr h anonial ll i dfind a ˆ, d (5) whr

8 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7),, q, d () Thrfor, ˆ, d, q, d d (7), d q, d d Th approxiaion for ˆ i ˆ, d q, d d (8) ˆ i dfind a h diffrn bwn h analyial xprion ˆ and i approxiaion ˆ, whih i drind by how wll q, approxia h ru our hap q,. ˆ ˆ ˆ, d d d q q d d whr upon onvrgn q ˆ (9) (7) q ˆ In a of a quadrai paial angular flux, (or quadrai aring our), h rror dfind in quaion (9) i valuad ˆ d d q d d q d d uliplying boh id by giv (7) d ˆ d q dd q d d d d d q d d q d d d q q 3 3. whr / Inring h praial xprion of h q and q dfind in quaion (7) ino quaion (7) and hn olv for. ˆ ˆ A, h rror i xpandd nar τ=. 5 O 8 3 ˆ O O O 5 O O (7) (73) (7) In a horr xprion, i i of h following for, ˆ O O (75)

9 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Thrfor ˆ will b a la wo ordr highr han O. ihr way, h raio ˆ / i ngligibl or i opard o /, validaing h dropping of hi raio whn valuaing quaion () for a quadrai aring our a. III. XPRINTAL RSULT oc D od i dvlopd for ing h prdiion of h ordr of auray for FS and LS approxiaion. Four probl ar dvid for h, ah of whih oni of an aud flux hap and h orrponding anufaurd our lid blow. Ca : onan our hap,, qs, Ca : linar our hap,, q, S Ca 3: quadrai our hap,, q S Ca : non-polynoial our hap, q, S (77) (78) (79). Ting Purly Aborbing arial For purly aborbing arial, h aring ro ion i o b ro. Ca giv ahin-priion rul for boh FS and LS approxiaion for all h i. Th following Figur, Figur and Figur 3 how h grid rfinn rul for Ca, Ca 3 and Ca (wihou aring our) involving linar, quadrai and a nonpolynoial anufaurd our. Figur. Ordr of auray wih quadrai anufaurd our (FS: nd ordr, LS: h ordr) Figur 3. Ordr of auray wih non-polynoial anufaurd our (FS: nd ordr, LS: h ordr) All h xprinal rul agr wih h analyial prdiion and ar lid in Tabl. Th nubr bfor h lah i h obrvd ordr of auray and h on afr h lah i h prdid ordr of auray, i.., in a for of (obrvaion/prdiion). Tabl. Obrvd and Prdid Ordr of Auray (purly aborbing) Approx. Conan Linar Quadrai FS xa / xa nd / nd nd / 3 rd nd * LS xa / xa xa / xa h / h * Th arrow india h aforniond ordr dgradaion Angular rror fro h nurial rul i rovd fro h ovrall rror wih h rror roval hniqu dvlopd in prviou work. [7].. Ting Saring arial Th xpd ordr of auray rlad o approxiaing aring our wih FS and LS approxiaion ar abulad in Tabl. Figur. Ordr of auray wih linar anufaurd our (FS: nd ordr, LS: xa)

10 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) Tabl. Prdid Ordr of Auray (aring our only) Approx. Conan Linar Quadrai FS xa nd 3 rd LS xa xa h A niond bfor, opiling h rul fro Tabl and Tabl by aking h lowr ordr giv h xpd ordr of auray for a wih boh aring our and diribud our, whih i lid in Tabl 3. Tabl 3. Obrvd and Prdid Ordr of Auray (aring our plu diribud our) Approx. Conan Linar Quadrai FS xa / xa nd / nd nd / nd LS xa / xa xa / xa h / h Th following Figur, Figur 5 and Figur how h grid rfinn rul for Ca, Ca 3 and Ca (wih aring our) involving linar and quadrai anufaurd our. Figur. Ordr of auray wih quadrai anufaurd our (FS: nd ordr, LS: h ordr) All h xprinal rul agr wih h analyial prdiion and ar lid in Tabl 3. IV. CONCLUSIONS A yai analyi of h ordr of auray for paial diriaion of oc hod in a lab gory ha bn prford for boh fla our approxiaion and linar our approxiaion. I i hown ha inluding aring our do no dgrad h ordr of auray and ha h ordr of h rror of h fir paial on of h angular flux i wo ordr highr han ha of h roh paial on of h angular flux. Boh horial prdiion and xprinal rul how ha fla our approxiaion i ond ordr aura and ha linar our approxiaion ha a fourh ordr auray. Figur. Ordr of auray wih linar anufaurd our (FS: nd ordr, LS: xa) ACKNOWLDGNTS Thi rarh wa uppord by h Conoriu for Advand Siulaion of Ligh War Raor ( an nrgy Innovaion Hub (hp:// for odling and Siulaion of Nular Raor, undr U.S. Dparn of nrgy Conra No. D-AC5-OR75. Figur 5. Ordr of auray wih quadrai anufaurd our (FS: nd ordr, LS: h ordr)

11 &C 7 - Inrnaional Confrn on ahai & Copuaional hod Applid o Nular Sin & nginring, Jju, Kora, April -, 7, on USB (7) RFRNCS. R. ASKW, A Chararii Forulaion of h Nuron Tranpor quaion in Copliad Gori, AW-, p. 8, U.K. Aoi nrgy Auhoriy (97)... J. HALSALL, CACTUS, A Chararii Soluion of h Nuron Tranpor quaion in Copliad Gori, AW-R-9, U.K. Aoi nrgy Auhoriy (98). 3. D. KNOTT, KRA, A Lai Phyi Cod for odlling h Daild Dplion of Gadolinia Ioop fro BWR Ful Dign, Ph.D. Thi, Th Pnnylvania Sa Univriy (99).. Rodolfo. Frrr, Jol D. Rhod III, A Linar Sour Approxiaion Sh for h hod of Chararii, Nular Sin And nginring, Volu Fbruary. 5..W. Larn and W.F. illr, Jr., Convrgn ra of paial diffrn quaion for h dir-ordina nuron ranpor quaion in lab gory, Nul. Si. and ng., 73, 7-83, 98...W. Larn and P. Nlon, Fini Diffrn Approxiaion and Supronvrgn for h Dir-Ordina quaion in Slab Gory, SIA, JNA 9(), Jipu Wang, Willia arin, Bnjain Collin, Appliaion of Th hod of anufaurd Soluion To Th D Sn quaion, Pro. PHYSOR Sun Vally, Idaho, ay -5,, Arian Nular Soiy () (CD-RO).