A 2D Benchmark for the Verification of the PEBBED Code

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1 INL/CON PREPRINT A D Bechma fo the Vefcato of the PEBBED Code Iteatoal Cofeece o Reacto Physcs, Nuclea Powe: A Sustaable Resouce Bay D. Gaapol Has D. Gouga Abdeaf M. Ougouag Septembe 8 Ths s a pept of a pape teded fo publcato a joual o poceedgs. Sce chages may be made befoe publcato, ths pept should ot be cted o epoduced wthout pemsso of the autho. Ths documet was pepaed as a accout of wo sposoed by a agecy of the Uted States Govemet. Nethe the Uted States Govemet o ay agecy theeof, o ay of the employees, maes ay waaty, expessed o mpled, o assumes ay legal lablty o esposblty fo ay thd paty s use, o the esults of such use, of ay fomato, appaatus, poduct o pocess dsclosed ths epot, o epesets that ts use by such thd paty would ot fge pvately owed ghts. The vews expessed ths pape ae ot ecessaly those of the Uted States Govemet o the sposog agecy.

2 Iteatoal Cofeece o the Physcs of Reactos Nuclea Powe: A Sustaable Resouce Caso-Kusaal Cofeece Cete, Itelae, Swtzelad, Septembe -19, 8 A D bechma fo the vefcato of the PEBBED code Bay D. Gaapol a*, Has D. Gouga b ad Abdeaf M. Ougouag b a Uvesty of Azoa, Tucso, AZ, USA b Idaho Natoal Laboatoy, Idaho Falls, ID, USA Abstact A ew bechmag cocept s peseted fo vefyg the PEBBED 3D multgoup fte dffeece/odal dffuso code wth applcato to pebble bed modula eactos (PBMRs). The ey dea s to pefom covegece acceleato, also called extapolato to zeo dscetzato, of a basc fte dffeece umecal algothm to gve extemely hgh accuacy. The method s fst demostated o a 1D cyldcal shell ad the o a, wedge whee the ode of the secod ode fte dffeece scheme s cofmed to fou places. 1. Itoducto Recetly poposed desgs fo the pebble bed modula eacto (PBMR) copoate azmuthally placed cotol od bas that eque specal pupose eacto physcs methods fo pope aalyss. Theefoe, appopate soluto stadads wll be equed to esue that ew, z algothms pefom popely ad accuately. The tet of ths pesetato s to povde a algothm fo the establshmet of multgoup bechmas fo the D-(,) dffuso equato. These bechmas ae to seve as stadads of compaso fo appoxmate umecal soluto algothms, such as cotaed the PEBBED [Tey, ] fte dffeece/odal dffuso code. Ou appoach wll be to efomulate a fte dffeece (FD) scheme to povde exteme accuacy. The FD fomulato, ofte thought to be a cude soluto at best, s usually ot cosdeed fo a bechma. Its smplcty lacs the elegace of say, a odal method; howeve, at the same tme ts smplcty maes t the most wdely used of all umecal solutos algothms fo PDEs. I ou poposed soluto stategy, we assume fte dffeece appoxmatos fo the two spatal devatve opeatos of the D euto dffuso equato yeldg a multpot ecuece elato whose umbe of tems depeds o the dffeecg stecl. We ca esolve the ecuece though matx veso usg baded ad/o cojugate gadet solves. If the dffeecg scheme s cosstet, the by efg the mesh spacg, the soluto should become ceasgly moe accuate. The questo ases howeve, as to just how accuate, sce t s vtually mpossble to specfy accuacy by just oe dscete ealzato. Theefoe, a dscetzato sestvty study usually accompaes a fte dffeece fomulato to estmate how accuate a soluto actually s.--but, ca we establsh a tue bechma wth such a smple FD sestvty study? The obvous appeal s-- the fte dffeece fomulato s the most basc scheme ad theefoe the easest to teach, lea ad pogam. I addto, t epesets the oto that smple s best, thus avodg the complcato ad expese of hghe ode schemes. Coespodg autho, gaapol@cowboy.ame.azoa.edu Tel: 50/61-478; Fax: 50/

3 . Multgoup dffuso equato The fudametal assumptos of the D multgoup dffuso equato cosdeed hee wll ot be ovely estctve othe tha equg steady state ad a heteogeeous medum composed of cotguous homogeeous egos. Each eegy goup ca clude fsso eactos, wth fsso eutos appeag ay goup. We allow up- ad dow- scatteg of ay stde ad each ego ca have a geeal space vayg fxed souce. The steady state multgoup dffuso equato fo homogeeous ego ad goup g s D G g g g g fg g g1 G gg g Qg g1 (1) fo 0 N, 1 g G. We shall specfy exteal bouday codtos as ecessay. I vecto fom, Eq.(1) ego, s M whee M x xq x (a) G, x G, 11, 1, 13,... 1 G, 1,, 3,... G, 31, 3,... 33, 3 G, G 1, G, GG, ad the goup flux ad souce vectos ae q x 1 x x... x x Q x/ D... Q x/ D G 1 1 G G T T.(c) (b) I Eq.(b), the uclea paametes fo homogeeous ego ae, gg gg g fg g gg Dg g fg gg, g g. D Equato (a) ow becomes g I q (d), (3) whee we have suppessed the efeece to ego..1. Matx decomposto A patcula smplfcato of Eq.(3) s possble f we dagoally decompose the teacto matx to 1 T T, whee x= x, ad dag l, l 1,..., G T x x x. By defg S 1... G 1 T 1 T q, (4) we fd the followg decoupled set of mooeegetc dffuso equatos: S, (5) 1,,..., G. Theefoe, T 1. (6)

4 The advatage of ths fomulato s obvous, sce we eed oly teat the oe-goup case fo each homogeeous ego, q B s. (7) 3. Fte dffeece (FD) dscetzatos 3.1 1D Cyldcal geomety Fo ths case, Eq.(7) s 1 d d Bs q d d wth 0., (8a) I the followg, we cosde oly flux bouday codtos. (8b) 0 0, Fo a homogeeous medum, the spatal doma [ 0, ] s coveetly pattoed to subtevals each of thcess h wth epesetg teval cetes ad h gve by 0. h 1/ Wth ths coveto, the FD appoxmato to Eqs(8) becomes c b a f (9) 1 1 wth 1/ c b 1/ a f h q. 1/ 1/ s h B Ths appoxmato povdes a test case fo the D- FD algothm establshed ext. 3.., -cyldcal geomety We show the specfc geomety cosdeed the followg fgue: Fg 1. Homogeeous wedge medum whch we dcate seveal homogeeous elemets of a heteogeeous wedge. Fo ths geomety, Eq.(7) becomes 1 1, Bsj, q, -1 j+1 j j-1 +1 (10) ad s dscetzed the usual way to gve the followg FD equato: j j 1, j 1, j g g f whee, j1, j1 j (11) 3

5 h j h Bs, j 1/ 1/ 1 h g h f h q. j j 1/ 1/ h A bloc tdagoal fom fo the coeffcet matx the emeges B1 C A B C A3 B3 C , A A B C A 1 B 1 ad we ae to solve, A S fo the flux. 4. A 1D bechma demostato It s a umecal fact that the fte dffeece fomulato, as peseted above, cotas dscetzato eo. Oe dscete ealzato theefoe caot guaatee a desed accuacy. Thus, a pope fte dffeece fomulato must clude a sestvty vestgato to justfy the chose dscetzato. Ufotuately, fomato as to the accuacy of a calculato fom a ad-hoc sestvty study s ofte coclusve. Thus, the fal dscetzato chose ca gve esults fa fom the tuth. Fo ths easo, a cosstet sestvty vestgato, leadg to bechma qualty esults, s poposed whch s othg less tha edefg what a umecal soluto to a PDE meas. Cosde the pots a dscetzato space, of ( h, h j), to defe a dscete map. I essece, the soluto (at ay,j), f cosstet, coespodg to ths ealzato, s a elemet a sequece whose lmt, as the dscetzatos appoach zeo, appoaches the tue soluto. Thus, the tue soluto s the lmt of a sees of dscete ealzatos athe tha just a sgle oe. The ey to the lmt howeve s the egulaty of the sequece ad ts ate of covegece. Thus, f we apply covegece acceleato, we could possbly ga umecal advatage. A patculaly effcet covegece acceleato s the Wy-epslo (We) acceleato [Bae, 1996] j 1 0 j 0 S j, j 0,1,..., N j j1 j1 j 1 1 1, j 0,1,..., N, 0,1,..., N 1 whee S j s the sequece of dscete ealzatos. j Oe teogates the esultg tableau alog the dagoal fo covegece D demostato of We acceleato The 1D fte dffeece (FD) fomulato, Eq.(9), wll test ou poposed bechmag cocept. We assume a 1D homogeeous medum (cyldcal shell) wth gve fluxes at the e ad oute ad ad o exteal souce. Fo ths case, the aalytcal soluto s I0 0 K0 K0 0 I0 K0 I0 0 K0 0 I0 K0 I0 I0 K0 K0 I0 0 K0 0 I0 0 whee I 0 ad K 0 ae modfed Bessel fuctos of the fst ad secod ds espectvely ad 4

6 B s (we assume B s s egatve). N e ufomly dstbuted edt pots wll be teogated fo covegece. By sequetally halvg the tevals betwee edts, a sequece of solutos at the N e edts wll esult. Note that the edt pots ae commo to the sequeces of all dscetzatos allowg fo the acceleato. Table 1 gves the eo fo the hghest level dscetzato (7) fo ths example whee B 1, 0.5,, s 0 wth bouday codtos 0 0, 1. O compag the We esult to the exact, we obseve ealy double pecso mache accuacy. Most mpotatly, the ogal FD flux s 6 odes of magtude less accuate. I addto, compaso to the pevous teate ( colums 3 ad 5), the We acceleato povdes a moe ealstc assessmet of how close the esult s to the actual. We ca theefoe coclude that the FD fomulato wapped a We acceleato deed gves bechma accuacy-- at least fo 1D. 4.. Demostato of We acceleato: D--(,) geomety The soluto of Eq (7) fo the (,) wedge show Fg. below has bee mplemeted as a FD. s the dex the -decto ad j the decto. The dmesos of the wedge cosdeed ae 0.5, 1, 0 deg, 0deg 0 0 0, j 0,, j 0, 0 j 0, 1, 0.,0, Table shows the last teate (l = 6), whee covegece fo the acceleated flux s to all places show. The coespodg ogal FD teate has coveged to 3-places at best. It would tae at least 5 moe levels of FD teatos fo full covegece of the ogal FD, whch would eque moe memoy tha s cuetly avalable o plaet Eath. To ga effcecy, we apply the cojugate gadet solve [Pa, 7]. Fo ths, the A-matx must be symmetc; ad the FD scheme of Eqs(11) s ot appopate. The equed symmetc fom s h 1/ 1/ j h, h Bs, j 1/, g f 1/ 1 h h h q. j j, We fd detcal esults to Table wth the CG solve less tha half the computatoal tme. 5. Fal cofmato A geeal method of geeatg D--(,) multgoup bechmas fo a heteogeeous medum has bee poposed ad demostated o a smple oe-ode wedge. A sestve fgue of met of the acceleated soluto s the ablty to deteme the ode () of the appoxmato as show Table 3. If the exact soluto wee tuly ot exact, the ode would ot be foud to the 4- dgts dsplayed. 0 wth 0 Fg. (,) Sgle wedge elemet B s 1 ad -bouday codtos 5

7 1.75E+ Table 1 Eo fo 1D applcato of We at teato 7 \eo Ogal /Ogal Ogal /Exact We/We We/Exact l = 7, = E E E E E- 1.E E E E E- 1.5E+.60E E E E- 1.50E E E E E E E E E- Table (, 5-Dgt Bechma (-Bouday Codto) l= 6, = 30 ogal FD \ E E E-.7953E E E E-.7953E E E E-.8888E E E E E E E E E- acceleated E E E-.889E E E E E E E E E E E E E E E E E- Refeeces Tey, W. K., H. D. Gouga, ad A. M. Ougouag, "Dect Detemstc Method fo Neutocs Aalyss ad Computato of Asymptotc Buup Dstbuto a Recculatg Pebble-Bed Reacto," Aals of Nuclea Eegy 9 () Bae, G ad P. Gaves-Mos (1996), Pade Appoxmats (Cambdge Uvesty Pess, NY). Pa,R., Pvate commucato, INL, May, 7. Table 3 Cofmato of the appoxmato ode usg the We Acceleato E 0.0E+.E E-.E+.E E-.E+.E+.E+.E+.7953E-.E E E+ 6