A DISCONTINUOUS LEAST-SQUARES SPATIAL DISCRETIZATION FOR THE S N EQUATIONS. A Thesis LEI ZHU

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1 A DISCONTINUOUS LEAST-SQUARES SPATIAL DISCRETIZATION FOR THE S N EQUATIONS A Thess by LEI ZHU Subtted to the Offce of Graduate Studes of Texas A&M Unversty n partal fulfllent of the requreents for the degree of MASTER OF SCIENCE August 008 Major Subject: Nuclear Engneerng

2 A DISCONTINUOUS LEAST-SQUARES SPATIAL DISCRETIZATION FOR THE S N EQUATIONS A Thess by LEI ZHU Subtted to the Offce of Graduate Studes of Texas A&M Unversty n partal fulfllent of the requreents for the degree of MASTER OF SCIENCE Approved by: Char of Cottee Cottee Mebers Head of Departent J E. Morel Marvn L. Adas Raytcho Lazarov Jean C. Ragusa Rayond Juzats August 008 Major Subject: Nuclear Engneerng

3 ABSTRACT A Dscontnuous Least-squares Spatal Dscretzaton for the S N Equatons. (August 008) Le Zhu B. Eng. Tsnghua Unversty Char of Advsory Cottee: Dr. J E. Morel In ths thess we develop and test a fundaentally new lnear-dscontnuous least-squares (LDLS) ethod for spatal dscretzaton of the one-densonal (-D) dscrete-ordnates (S N ) equatons. Ths new schee s based upon a least-squares ethod wth a dscontnuous tral space. We pleent our new ethod as well as the lneardscontnuous Galerkn (LDG) ethod and the luped lnear-dscontnuous Galerkn (LLDG) ethod. The pleentaton s n FORTRAN. We run a seres of nuercal tests to study the robustness L accuracy and the thck dffuson lt perforance of the new LDLS ethod. By robustness we ean the resstance to negatvtes and rapd dapng of oscllatons. Coputatonal results ndcate that the LDLS ethod yelds a unfor second-order error. It s ore robust than the LDG ethod and ore accurate than the LLDG ethod. However t fals to preserve the thck dffuson lt. Consequently t s vable for neutroncs but not for radatve transfer snce radatve transfer probles can be hghly dffusve.

4 v ACKNOWLEDGEMENTS I would frst lke to thank y cottee char Dr. Morel for hs gudance and support throughout ths research. He frst ntroduced e to the coputatonal research feld and has always been an excellent and knd advsor. I thank Dr. Lazarov for hs valuable advce on fnte eleent theory. I also thank the other two ebers of y cottee Dr. Adas and Dr. Ragusa who ade any suggestons whch were nvaluable contrbutons to y research. I would also lke to thank y frends for ther encourageent and the faculty and staff n the Departent of Nuclear Engneerng for ther support. Fnally I thank y other y father and y grl frend Shanshan Gong for ther love.

5 v TABLE OF CONTENTS Page ABSTRACT... ACKNOWLEDGEMENTS... TABLE OF CONTENTS... LIST OF FIGURES... LIST OF TABLES... v v v x CHAPTER I INTRODUCTION... Descrpton of the proble... Basc concepts of the transport equaton and dscrete-ordntes (S N ) equatons... Introducton to teratve ethods... 4 Overvew of chapters... 9 II SPATIAL DISCRETIZATION FOR S N EQUATIONS... 0 Lnear-dscontnuous Galerkn (LDG) ethods... 0 Incopatblty of tradtonal least-squares ethods wth a dscontnuous tral space... 4 The new lnear-dscontnuous least-squares (LDLS) ethod... 6 Robustness and accuracy analyss for dfferent ethods wth a splfed pure absorber proble... 0 SSA spatal dscretzaton for the LDG ethod... 5 SSA spatal dscretzaton for the LDLS ethod... 9 Suary III FOURIER ANALYSIS FOR SPECTRAL RADIUS Dervaton of Fourer analyss for the LDG ethod Dervaton of Fourer analyss for the LDLS ethod Suary... 53

6 v CHAPTER Page IV ASYMPTOTIC ANALYSIS Introducton A suary of the asyptotc analyss for the LLDG ethod Asyptotc analyss for the LDLS ethod Suary V COMPUTATIONAL RESULTS L easureents for pontwse and contnuous errors Coputatonal results for spectral rad and coparson wth Fourer analyss... 8 Coputatonal results for the thck dffuson lt Suary VI SUMMARY AND FUTURE WORK REFERENCES VITA... 97

7 v LIST OF FIGURES FIGURE Page. Spatal dscretzaton and the spatal shape of the angular flux.... Coparson of pure absorber solutons Local spectral rad fro LDG wth c =.0 and = Local spectral rad fro LDG wth c = 0.98 and = t h 3.3 Global spectral rad fro LDG wth c = Global spectral rad fro LDG wth c = Local spectral rad fro LLDG wth c =.0 and = t h 3.6 Local spectral rad fro LLDG wth c = 0.98 and = t h 3.7 Global spectral rad fro LLDG wth c = Global spectral rad fro LLDG wth c = Local spectral rad fro LDLS wth c =.0 and = t h t h 3.0 Local spectral rad fro LDLS wth c = 0.98 and = Global spectral rad fro LDLS wth c = Global spectral rad fro LDLS wth c = Coparson of S pontwse convergence rate of cell-averaged scalar flux wth = =.0 and a unt half-range ncdent current... 7 t a c 5. Coparson of S contnuous L convergence rate of scalar flux wth = =.0 and a unt half-range ncdent current... 7 t a c t h

8 v FIGURE Page 5.3 Coparson of S pontwse convergence rate of cell-averaged scalar flux wth t =.0 c c = 0.5 and a unt half-range ncdent current Coparson of S contnuous L convergence rate of scalar flux wth t =.0 c c = 0.5 and a unt half-range ncdent current Coparson of S 4 pontwse convergence rate of cell-averaged scalar flux wth = =.0 and a unt half-range ncdent current t a c 5.6 Coparson of S 4 contnuous L convergence rate of scalar flux wth = =.0 and a unt half-range ncdent current t a c 5.7 Coparson of S 4 pontwse convergence rate of cell-averaged scalar flux wth = =.0 and a unt half-range ncdent current t a c 5.8 Coparson of S 8 contnuous L convergence rate of scalar flux wth = =.0 and a unt half-range ncdent current t a c 5.9 Coparson of S 6 pontwse convergence rate of cell-averaged scalar flux wth = =.0 and a unt half-range ncdent current t a c 5.0 Coparson of S 6 contnuous L convergence rate of scalar flux wth = =.0 and a unt half-range ncdent current t a c 5. Coparson of S 4 pontwse convergence rate of cell-averaged scalar flux wth t =.0 c c = 0.5 and the ethod of anufactured solutons Coparson of S 4 contnuous L convergence rate of scalar flux wth t =.0 c c = 0.5 and the ethod of anufactured solutons Coparson of S 8 pontwse convergence rate of cell-averaged scalar flux wth t =.0 c c = 0.5 and the ethod of anufactured solutons Coparson of S 8 contnuous L convergence rate of scalar flux wth t =.0 c c = 0.5 and the ethod of anufactured solutons... 80

9 x FIGURE Page 5.5 Coparson of S 6 pontwse convergence rate of cell-averaged scalar flux wth t =.0 c c = 0.5 and the ethod of anufactured solutons Coparson of S 6 contnuous L convergence rate of scalar flux wth t =.0 c c = 0.5 and the ethod of anufactured solutons Dffuson lt S solutons wth ε = Dffuson lt S solutons wth ε = Dffuson lt S solutons wth ε = Dffuson lt S solutons wth ε = Dffuson lt S solutons wth ε = Dffuson lt S solutons wth ε = Dffuson lt S 8 solutons wth ε = Dffuson lt S 8 solutons wth ε = Dffuson lt S 8 solutons wth ε = Dffuson lt S 8 solutons wth ε = Dffuson lt S 8 solutons wth ε = Dffuson lt S 8 solutons wth ε =

10 x LIST OF TABLES TABLE Page. Sngle-cell solutons fro dfferent schees.... Taylor-seres expanson for the solutons about = Local error for the fluxes Global error for the fluxes LDLS spectral rad for a pure scatterng proble (S 8 ) LDLS spectral rad wth c = 0.98 (S 8 )... 83

11 CHAPTER I INTRODUCTION Descrpton of the proble One of the ost portant nnovatons n transport spatal dscretzaton over the last 30 years was the ntroducton of dscontnuous Galerkn (DG) ethods. Such ethods are known to be hghly accurate and far ore robust than contnuous fnteeleent ethods. Nonetheless DG ethods ust stll be luped to acheve adequate robustness for deandng applcatons such as theral radaton transport n the hgh energy densty rege. Lupng can be very dffcult on unstructured eshes. In addton DG ethods gve non-unfor errors. For nstance the average and outflow fluxes obtaned wth the lnear-dscontnuous Galerkn (LDG) ethod n -D are thrdorder accurate whle the nteror nflow value s only second-order accurate. If the error s easured wth the L nor the LDG ethod gves second-order accuracy n both -D and ultdensonal calculatons. Least-squares ethods are generally hghly accurate and have certan advantages over other ethods for a posteror error estaton 3. However the standard leastsquares ethods are generally not conservatve and conservaton s portant for spatal dscretzaton technques n the nuclear engneerng county. Thus a ethod that would be nherently ore robust than the DG ethod and yeld second-order accuracy n both -D and ultdensonal calculatons would be Ths thess follows the style of Nuclear Scence and Engneerng.

12 hghly desrable. In ths thess we propose pleent and test a new lneardscontnuous least-squares (LDLS) ethod for spatal dscretzaton of the -D dscreteordnates equatons. Ths new schee s based upon a least-squares ethod wth a dscontnuous tral space. In the followng chapters we show a detaled dervaton of the LDLS ethod and the spatal dscretzaton schees. The S N equatons are solved teratvely va source teraton. Fourer analyss s perfored to study the teratve convergence behavor. We have pleented our new ethod n FORTRAN and we have run a seres of nuercal tests to study accuracy and robustness teratve convergence propertes and the thck dffuson lt perforance of the new LDLS ethod. Basc concepts of the transport equaton and dscrete-ordnates (S N ) equatons We start wth the general for of contnuous transport equaton wth both scatterng source and nhoogeneous source: ψ ( re Ω t) Ω ψ( re Ω t) t( ret ) ψ( re Ω t) = ve ( ) t ( re ' E Ω' Ω t) ψ( re ' Ω' tded ) ' Ω ' QrE ( Ω t) 4π 0 s (.) where r s the Cartesan coordnates of the partcle poston Ω s a unt Cartesan vector representng the drecton of partcle flow E s the partcle energy t s the te ve ( ) s the partcle speed ψ ( re Ω t) s the angular flux QrE ( Ω t) s the nhoogeneous source ( ret ) s the total acroscopc cross-secton and t

13 3 s( re ' E Ω' Ω t) s the scatterng kernel. The purpose of ths work s to nvestgate ethods for the spatal dscretzaton of the S N equatons so t s suffcent to consder the proble wth the followng assuptons: ψ ( re Ω t) Steady state ( = 0); t Mono-energetc; Isotropc scatterng and sotropc external sources; One-densonal (-D) slab geoetry; Constant cross sectons. Under these condtons the transport equaton can be wrtten as: ψ s Qx ( ) µ ψ t ( µ x) = φ( x) x (.) where the scalar flux s: φ( x) = ψ ( µ xd ) µ. (.3) Eq. (.) reans contnuous n the angular and spatal varables. We frst dscretze the angles to get the S N equatons: dψ ( x) µ ψ ( x) w ψ ( x) Q ( x) (.4) N s t = dx = where =... N. In ths study we apply a syetrc Gauss quadrature set of order N where N s even. The drectons are ordered such that for =... N / µ < 0. For = N /... N µ > 0. The quadrature weghts are noralzed to.0

14 4 N.e. w =. 0. The detaled spatal dscretzaton schees for Eq. (.4) wll be = ntroduced n the next chapter. Introducton to teratve ethods Source Iteraton (SI) The S N equatons are solved va source teraton. Source teraton s based upon the followng facts: All couplng between drectons occurs on the rght sde of Eq. (.4); A frst-order advecton-reoval equaton exsts for each drecton on the left sde of the equaton; Each such equaton can be solved by perforng a sweep. By sweep the angular flux s frst calculated for the frst cell on the ncong boundary. Its outflow becoes the ncong boundary for the next spatal cell. The SI algorth for Eq. (.4) can be atheatcally represented as follows: dψ ( x) s l µ ψ t ( x) = φ ( x) Q( x) dx =... N (.5) where l s the teraton ndex and the scalar flux s gven by: N l l φ ( x) = w ψ ( x). (.6) = Assung an nfnte hoogeneous edu Fourer analyss 4 for the spatally contnuous for of the S N equatons shows that the spectral radus of SI s equvalent to the scatterng rato c:

15 5 ρ SI s = c =. (.7) t The spectral radus s the agntude of the largest teraton egenvalue. Thus t s the asyptotc rate of error reducton after any teratons when only the slowest convergng ode reans n the error. For optcally thn or hghly absorbng systes n whch partcles scatter just a few tes on the average before beng absorbed or escapng the syste the SI process wll converge quckly. Whle for optcally thck dffusve systes n whch partcles can scatter an arbtrary nuber of tes on the average before beng absorbed or escapng the syste the SI process wll converge slowly and wll be qute costly. In the latter case SI s not a practcal teratve schee. Dffuson Synthetc Acceleraton (DSA) In optcally thck dffusve systes n whch partcles can scatter any tes on the average before beng absorbed or escapng the syste the SI process ust be accelerated. Durng the past several decades uch effort has been ade n acceleraton algorths. One of the ost wdely appled ethods s Dffuson Synthetc-Acceleraton (DSA). The prncple of the DSA s to use a dffuson approxaton to estate the error of the scalar flux obtaned fro SI so that the accuracy can be proved and the teratve convergence accelerated. We next derve the DSA schee. The frst step s to solve the S N equatons by SI: / dψ ( x) / s l µ ψ t ( x) = φ ( x) Q( x) (.8) dx

16 6 where =... N. The errors at ths step n the angular and scalar fluxes are respectvely gven by: δψ ( x) = ψ ( x) ψ ( x) (.9) / / and / / δφ ( x) = φ( x) φ ( x) (.0) where ψ (x) and φ (x) are the exact solutons of the angular flux and scalar flux to the S N equatons. We subtract the SI equaton fro the contnuous transport equaton to get: / δψ ( x) / s l µ tδψ ( µ x) = δφ ( x). x (.) s Subtract the quantty δφ / ( x) fro both sdes of the above equaton to obtan the exact equaton forδψ : / / δψ ( x) / s / s l / µ tδψ ( µ x) δφ ( x) = ( δφ ( x) δφ ( x)). x (.) Re-expressng the rght hand sde usng Eq. (.0) gves: / δψ ( x) / s / s / l µ tδψ ( µ x) δφ ( x) = ( φ ( x) φ ( x)). x (.3) Thus the error of the angular flux at teraton step l / satsfes the transport equaton wth a source equal to: s / ( φ l ( x) φ ( x)). (.4)

17 7 However t s obvous that ths equaton for δψ / ( x) s as dffcult to solve as that for ψ (x). We need to substtute an approxaton to Eq. (.3) that sple enough to solve but accurate n a certan sense. The central thee of the DSA schee s to substtute the dffuson equaton for the exact transport equaton for the error. The followng s the DSA schee: / dψ ( x) / s l µ ψ t ( x) = φ ( x) Q( x) (.5) dx N l l φ ( x) = w ψ ( x) (.6) = / φ ( x) / / l a φ ( x) = s( φ ( x) φ ( x)) x 3 x t (.7) / / φ ( x) = φ ( x) φ ( x) (.8) where φ(x) denotes the estated scalar flux error fro the dffuson equaton as opposed to δφ (x) the true error. An nfnte-edu Fourer analyss 4 shows that DSA attenuates the errors of the low frequency odes whch are ost poorly attenuated by the SI. At the sae te DSA also underestates the hgh frequency odes whch are strongly attenuated by the SI. Thus DSA can decrease the spectral radus and effcently accelerate the teraton process. Fourer analyss gves the spectral radus of DSA as: where c s the scatterng rato. ρ 0.47 c (.9) DSA

18 8 S synthetc acceleraton (SSA) In ths study S synthetc acceleraton s appled nstead of DSA. Ths eans that the S equatons are substtuted for the dffuson equaton. In -D slab geoetry the S equatons are analytcally equvalent to the dffuson equaton. The reason we choose the SSA schee n the code s that we can use the sae spatal dscretzaton schee for the acceleraton as for the SI tself. Consstency of the spatal dscretzaton for the acceleraton equaton wth that of the transport equaton s essental for uncondtonal stablty and effectveness 5. The SSA schee s: / dψ ( x) / s l µ ψ t ( x) = φ ( x) Q( x) (.0) dx N l l φ ( x) = w ψ ( x) (.) = / df± ( x) / / / / ( ) s s l ± t f± x ( f ( x) f ( x)) = ( φ ( x) φ ( x)) (.) 3 dx / / φ ( x) = φ ( x) f ( x) (.3) where / / / f x f x f x ( ) = ( ) ( ) (.4) where f ± (x) are the flux of partcles travelng n the postve and negatve drectons. We subtract Eq. (.3) fro the trval equaton φ( x) = φ( x) to obtan: / / δφ ( x) = δφ ( x) f ( x) (.5) The scalar flux error at step l s equal to the scalar flux error at step l / nus the scalar flux error estate fro the S equatons.

19 9 Overvew of chapters In ths chapter we brefly descrbed several spatal dscretzaton ethods for the S N equatons. We gave the -D S N equatons wth soe assuptons. The teraton technques ncludng SI DSA and SSA were ntroduced. In Chapter II we propose the new LDLS ethod and show the detaled spatal dscretzaton of the S N equatons wth the LDLS LDG and LLDG ethods. The spatal dscretzaton of the SSA schee s also presented. The robustness and accuracy of the LDLS ethod s nvestgated through a splfed pure absorber transport equaton. Chapter III presents a sngle ode Fourer analyss for the spectral radus fro both the LDG and LLDG ethods n order to ake a coparson wth that for the LDLS ethod. Both SI and SSA teraton technques are analyzed and pleented n MATLAB. Chapter IV gves a detaled asyptotc analyss for the LDLS ethod to study ts perforance n the thck dffuson lt. Chapter V presents the coputatonal results for the new LDLS ethod and s dvded nto three parts. The frst part gves the coputatonal results for the accuracy easureents of the LDLS ethod copared wth the other two ethods. The second part shows the coputatonal results for the spectral rad n soe probles. The result s copared wth the spectral radus obtaned fro the Fourer analyss. The thrd part s a study of the perforance of the LDLS ethod n thck dffuson lt. Chapter VI suarzes the conclusons fro the prevous chapters and gves soe suggestons for future work.

20 0 CHAPTER II SPATIAL DISCRETIZATION FOR S N EQUATIONS In ths chapter we frst revew the spatal dscretzaton of the lneardscontnuous Galerkn (LDG) ethods for the S N equatons. Then we propose the new lnear-dscontnuous least-squares (LDLS) ethod. The accuracy of the ethod s nvestgated for a sple pure absorber proble and copared wth analogous results for the LDG and LLDG ethods. Algorths for SSA acceleraton are derved. Lnear-dscontnuous Galerkn (LDG) ethods We begn the dervaton of the LDG ethod by ntroducng the weghted resdual ethod. We frstly gve the ndexng for the spatal dscretzaton n Fg... In ths study we focus on -D slab geoetry probles wth unfor spatal eshes. Note fro the fgure that half-ntegral ndces ply the cell-edge quanttes.e. ψ = ψ for µ > 0 / R = ψ for µ < 0. L (.) Integral ndces ply the cell-average quanttes.e. ψ = ( ψl ψr ). (.)

21 Postve drecton Negatve drecton ψ L ψ L ψ L ψ R ψ R ψ L ψ L ψ L ψ R ψ R ψ R ψ R x 3/ x x / x x / x x 3/ Fg... Spatal dscretzaton and the spatal shape of the angular flux. Here we consder the followng -D for: Aψ ( x) = Q( x) where A s a lnear operator ψ (x ) s the soluton and Q(x ) s the source functon. The set of the tral space bass functon for representng the soluton s{ B( )} L x =. Assung the tral-space functons are chosen so that ψ (x ) naturally eets the boundary condtons the approxate tral-space expanson for ψ (x ) s: (.3) L ψ ( x) = ψ B( x). (.4) = Defne the resdual as follows: R( x ) = Q( x ) Aψ ( x ). (.5)

22 The error n the soluton s proportonal to the sze of the resdual. We want to choose the expanson coeffcents { ψ } L = so that the resdual s sall. There are several typcal approaches. The weghted Resdual ethod s defned by choosng the expanson coeffcents { ψ } L = so that the resdual s orthogonal over the proble doan to the weghtng functons{ W( )} L x = that s: RxW ( ) ( xdx= ) 0 x (.6) where =... L.The set of N lnearly-ndependent functons { W( )} L x = for an L- densonal space of functons called the weghtng space. If the tral space s dentcal to the weghtng space.e. B ( x) = W( x) (.7) the weghted resdual ethod s naed the Galerkn ethod. The lnear-dscontnuous Galerkn ethod s wdely appled n the spatal dscretzaton for the transport equaton because of ts characterstcs of robustness and accuracy. We start the Galerkn ethod by choosng cardnal weght and bass functons whch are unty at a gven support pont and zero at other support ponts. For the -D LDG proble the weght and bass functons vary lnearly fro one to zero across the cell x x x ]: [ / / x / x L ( ) L ( ) h B x = W x = (.8) x x B x = W x = (.9) / R ( ) R ( ). h To solve Eq. (.4) we defne the angular flux as follows on cell :

23 3 For µ > 0 ψ ( x) = ψ for x = x R / ψ ( x) = ψ B ( x) ψ B ( x) otherwse. L L R R (.0) For µ < 0 ψ ( x) = ψ for x = x L / ψ ( x) = ψ B ( x) ψ B ( x) otherwse. L L R R (.) An analogous spatal representaton s assued for the nhoogeneous source except that there s no need to unquely defne Q on the cell nterfaces. Thus Q ( x) = Q B ( x) Q B ( x) for x [ x x ]. (.) L L R R / / Multplyng Eq. (.4) by the weght functons and ntegratng over the volue of the th cell we can obtan the standard LDG schee: For µ > 0 t h β β µ [ ( ψ L ψ R ) ψ R ] [( ) ψ L ( ) ψ R ] = s h β β h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ] 4 t h β β µ [ ψr ( ψl ψr )] [( ) ψl ( ) ψr ] = s h β β h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ]. 4 (.3) (.4) For µ < 0 t h β β µ [ ( ψ L ψ R ) ψ L ] [( ) ψ L ( ) ψ R ] = s h β β h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ] 4 (.5)

24 4 t h β β µ [ ψ L ( ψ L ψ R )] [( ) ψ L ( ) ψ R ] = s h β β h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ] 4 (.6) where φ N = w ψ (.7) L L = φ N = w ψ (.8) R R = and β = / 3. For each drecton there are two unknowns and two equatons so the syste s closed. The paraeter β s the ass lupng paraeter. If β =.0 we obtan the luped lnear-dscontnuous Galerkn (LLDG) equatons. By lupng the robustness n the thck dffuson lt s proved but t also reduces the accuracy. Incopatblty of tradtonal least-squares ethods wth a dscontnuous tral space The tradtonal least-squares ethod s based upon choosng the expanson coeffcents to nze the functonal Γ the ntegral of the square of the resdual: Γ= R ( xdx ). x (.9) However these knds of schees are not copatble wth a dscontnuous tral space. Ths s due to the fact that the dervatve of a dscontnuous functon takes the for of a delta-functon at the pont of dscontnuty. If the resdual contans a delta-functon the

25 5 ntegrand wll contan the square of a delta-functon the ntegral of whch s undefned. These concepts can be deonstrated atheatcally as follows. The tral space s the sae for both the LDG ethod and LDLS ethod on the doan of x x x ] as n Eqs. (.8) and (.9). The defntons of the angular flux [ / / and the nhoogeneous source for the LDLS ethod are the sae as those of the LDG ethod n Eqs. (.0)-(.). The dervatve of the angular flux s gven by: For µ > 0 dψ ( x) = ( ψ ψ ) δ( x x ) ( ψ ψ ). (.0) dx L R / R L h For µ < 0 dψ ( x) = ( ψ ψ ) δ( x x ) ( ψ ψ ) (.) dx L R / R L h Thus the resdual s gven by: For µ > 0 R ( x) = S B ( x) S B ( x) ( ψ B ψ B ) L L R R t L L R R µ ( ψ R ψ L ) µ ( ψ L ψ R ) δ ( x x /). h (.) For µ < 0 R ( x) = S B ( x) S B ( x) ( ψ B ψ B ) L L R R t L L R R µ ( ψ R ψ L ) µ ( ψ L ψ R ) δ ( x x /) h (.3) where the source ter S (x) ncludes both the sotropc external source Q (x) and the sotropc scatterng source

26 6 S ( x) w ( x) Q ( x). (.4) N = s = ψ Based on the tradtonal least-squares ethod the ntegral of the square of the resdual Γ need to be nzed over the nterval x x ] where [ / / x / ± = R± x / Γ ( xdx ). (.5) If substtute Eqs. (.) and (.3) nto Eq. (.5) t s clear that the ntegrand contans the square of a delta-functon and ths knd of ntegral s undefned. The new lnear-dscontnuous least-squares (LDLS) ethod We want to fnd a ethod that would be nherently ore robust than the DG ethod and yeld a unfor level of second-order accuracy n both -D and ultdensonal calculatons. The ncopatblty of the tradtonal least squares ethod wth a dscontnuous tral space s that the ntegrand contans the square of a delta-functon and the ntegral of the square of a delta-functon does not exst. However one can avod the delta-functon proble by nzng the resdual over the se-open nterval x x ] for µ > 0 and x x ) for µ < 0. But n ths case the ( / / [ / / equatons have no knowledge of the boundary value of the angular flux and the trval soluton ( ψ = 0 ) s obtaned. The central thee of our approach s to frst avod the delta-functon dffculty by nzng the square of the resdual over the se-open nterval and then partng both conservaton and knowledge of the boundary value to

27 7 the equatons by constranng the soluton to satsfy the balance equaton. The schee s deonstrated atheatcally as followng: We use the sae defnton of the resdual and the sae Cardnal bass functons. However we defne: x / x / ψ L ψ R λ R x dx λ R x dx µ x x (.6) Γ ( ) = ( ) ( ) for > 0 / / x / x / ψ L ψ R λ R x dx λ R x dx µ x / x / (.7) Γ ( ) = ( ) ( ) for < 0 where λ s a Lagrange ultpler and n whch the balance equaton for the closed doan x x ] s: [ / / x / R ( x) dx = 0 for µ > 0 (.8) x / x / R ( x) dx = 0 for µ < 0. (.9) x / Expandng Eqs. (.8) and (.9) we get: ψ L ψ R SL SR µ ( ψ R ψ R ) t h = h for µ > 0 (.30) ψ L ψ R SL SR µ ( ψ L ψ L ) t h = h for µ < 0 (.3) where the total source ter contans the scatterng source and the nhoogeneous source: S = φ (.3) s L L QL

28 8 respectvely: For µ > 0 S = φ (.33) s. R R QR Equvalently we solve the followng three equatons forψ L ψ R and λ Γ ( ψ L ψ R λ) = 0 ψ L (.34) Γ ( ψ L ψ R λ) = 0 ψ R (.35) Γ ( ψ L ψ R λ) = 0. λ (.36) For µ < 0 Γ ( ψ L ψ R λ) = 0 ψ L (.37) Γ ( ψ L ψ R λ) = 0 ψ R (.38) Γ ( ψ L ψ R λ) = 0. λ (.39) Wrtng these equatons n ore detal for µ > 0 we get: x / [ R ( x) dx] x / R ( x) x / R ( x) dx λ = 0 ψ x L ψl / (.40)

29 9 x / [ R ( x) dx] x / R ( x) x / R ( x) dx λ = 0 ψ x R ψr / (.4) x / R ( ) 0. x dx = (.4) x / For µ < 0 we get: x / [ R ( x) dx] x / R ( x) x / R ( x) dx λ = 0 ψ x L ψl / (.43) x / [ R ( x) dx] x / R ( x) x / R ( x) dx λ = 0 ψ x R ψr / (.44) x / R ( ) 0. x dx = (.45) x / Substtutng the expresson for the resdual n Eqs. (.) and (.3) nto Eqs. (.40)- (.45) we obtan: For µ > 0 {[ S B ( x) S B ( x) ( ψ B x L L R R t L L / t h µ µ dx λ ( ) = 0 (.46) ψ R BR ) ( ψ R ψ L )]( t BL )} x / h h {[ S B ( x) S B ( x) ( ψ B x / L L R R t L L t h µ ( ) 0 µ dx λ µ = (.47) ψ R BR ) ( ψ R ψ L )]( t BR )} x / h h

30 0 ψ L ψ R SL SR µ ( ψ R ψ R ) t h = h. (.48) For µ < 0 {[ S B ( x) S B ( x) ( ψ B x / L L R R t L L t h µ } ( ) 0 µ dx λ µ = (.49) ψ R BR ) ( ψ R ψ L )]( t BL ) x / h h {[ S B ( x) S B ( x) ( ψ B x / L L R R t L L t h µ ( ) 0 µ dx λ = (.50) ψr BR ) ( ψr ψl )]( t BR )} x / h h ψ L ψ R SL SR µ ( ψ L ψ L ) t h = h. (.5) We have three equatons and three unknowns for each drecton so the syste s closed and we are able to obtan ψ L ψ R and λ (although λ s not physcally sgnfcant) for each spatal cell. Ths copletes our descrpton of the LDLS ethod for the -D slab-geoetry S N equatons. Robustness and accuracy analyss for dfferent ethods wth a splfed pure absorber proble To deonstrate the concepts and derve results for the basc ethods wth a nu of coplexty here we frst consder the followng splfed transport equaton: ψ ψ = 0 x (.5)

31 whch s defned over the nterval [0 x 0] wth the left boundary condton ψ exact L ( 0) =. The exact soluton for ψ ( x0) (the outflow) s: The Taylor-seres expanson about = 0 s: exact ψ R ( x ) = exp( x ). (.53) 0 0 exact ψ R ( ) = O( ) (.54) 6 4 where = x0 s the total ean-free-paths of the cell gven. The soluton of the average flux s: 0 x exp( ) exact ψ avg ( x) = exp( x) dx. x = (.55) 0 0 The Taylor-seres expanson about = 0 s: exact ψ avg ( ) = O( ). (.56) Solvng ths splfed pure absorber proble wth LDG ethod and LLDG ethod fro Eqs. (.3)-(.6) and wth LDLS ethod fro Eqs. (.46)-(.5) we can get the followng solutons n Table.:

32 Table. Sngle-cell solutons fro dfferent schees. Interor nflow Outflow Average flux Exact ) exp( ) exp( LDLS LDG LLDG Fro Table. the outflow fro LDG becoes negatve after three (3) eanfree-paths whle that fro both the LLDG and LDLS ethods reans postve over the whole doan. In order to nvestgate the accuracy of each ethod the Taylor-seres expanson about 0 = s appled to the expressons n Table. and the results are shown n Table..

33 3 Table. Taylor-seres expanson for the solutons about 0 =. Interor nflow Outflow Average flux Exact ) ( O ) ( O LDLS ) ( 5 3 O ) ( O ) ( O LDG ) ( O ) ( O ) ( O LLDG ) ( O ) ( O ) ( O The error obtaned fro a sngle step s called the local error. The local error s coputed assung the nflow s exact. The error obtaned n a cell that s suffcently far fro an outer boundary s called the global error. In general the global error s one order lower than the local error because of the error buld up as outflow errors becoe nflow errors for adjacent cells. The local error for the fluxes s gven n Table.3 based on the result n Table.. Also the global error for the fluxes s gven n Table.4.

34 4 Fro Table.4 we can see that both the LDLS and the LLDG ethods yeld a unfor second order global error. Table.3 Local error for the fluxes. Interor nflow Outflow Average Flux LDLS 3 rd 3 rd nd LDG nd 4 th 3 rd LLDG nd 3 rd nd Table.4 Global error for the fluxes. Interor nflow Outflow Average Flux LDLS nd nd nd LDG nd 3 rd 3 rd LLDG nd nd nd The outflow wth dfferent ethods s plotted n the Fg... It can be observed that the LDG ethod yelds a negatve soluton for thcknesses greater than 3 ean-freepaths. Both the LLDG and LDLS solutons are postve and onotone. Therefore both the LLDG and LDLS ethods are ore robust than the LDG ethod. The LDLS soluton s ore accurate than the LLDG soluton for sall whle for large the

35 5 LLDG ethod vares as so the LLDG ethod s the ost accurate for large. whereas the LDG ethod and the LDLS ethod vary as 0.8 LDLS LDG LLDG Exact 0.6 Scalar Flux Mean-Free-Paths Fg... Coparson of pure absorber solutons. SSA spatal dscretzaton for the LDG ethod The basc concept and schee of SSA acceleraton has been ntroduced n the last chapter. In ths secton we gve three detaled spatal dscretzatons for the SSA schee appled wth the LDG LLDG and LDLS schees respectvely. As entoned before the advantage of SSA s that we can use the sae spatal dscretzaton schee for both the low-order S transport equatons and the hgh-order S N equatons thereby

36 6 ensurng consstancy. Recall that SI wth SSA n the contnuous spatal schee s descrbed n Eqs. (.0)-(.3). We frst show the spatal dscretzaton of the SSA schee for both the LDG and LLDG ethods. Based on Eqs. (.3)-(.8) the hgh-order stage (SI) can be wrtten as: For µ > 0 / / / t h β / β / µ [ ( ψ L ψ R ) ψ R ] [( ) ψ L ( ) ψ R ] = s h β l β l h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ] 4 / / / t h β / β / µ [ ψr ( ψl ψr )] [( ) ψl ( ) ψr ] = s h β l β l h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ]. 4 (.57) (.58) For µ < 0 / / / t h β / β / µ [ ( ψl ψr ) ψl ] [( ) ψl ( ) ψr ] = s h β l β l h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ] 4 / / / t h β / β / µ [ ψ L ( ψ L ψ R )] [( ) ψ L ( ) ψ R ] = s h β l β l h β β [( ) φl ( ) φr ] [( ) QL ( ) QR ] 4 (.59) (.60) where φ N l l L wψl = = (.6) N l l R wψr = φ = (.6)

37 7 and where the paraeter β s the ass lupng paraeter wth β = / 3 correspondng to LDG and β = correspondng to LLDG. For the low-order (S ) equatons the spatal dscretzaton and the tral space are the sae as those for the SI equaton wth S quadrature. The flux of partcles travelng n the postve ( µ = / 3) and negatve ( µ = / 3) drecton n the tral space on the doan of x [ x / x / ] can be wrtten as: For the postve drecton f ( x) = f for x = x R / f ( x) = f B ( x) f B ( x) otherwse. L L R R (.63) For the negatve drecton f ( x) = f for x = x L / f ( x) = f B ( x) f B ( x) otherwse. L L R R (.64) The followng s the schee for the low-order (S ) equatons: For the postve drecton / / / t h β / β / [ ( f L f R ) f R ] [( ) f L ( ) f R ] = 3 s h β / / β / / [( )( f L f L ) ( )( f R f R )] 4 s h β / l β / l [( )( φl φl ) ( )( φr φr )] 4 / / / t h β / β / [ f R ( f L f R )] [( ) f L ( ) f R ] = 3 s h β / / β / / [( )( f L f L ) ( )( f R f R )] 4 s h β / l β / l [( )( φl φl ) ( )( φr φr )]. 4 (.65) (.66)

38 8 For the negatve drecton / / / t h β / β / [ ( f L f R ) f L ] [( ) f L ( ) f R ] = 3 s h β / / β / / [( )( f L f L ) ( )( f R f R )] 4 s h β / l β / l [( )( φl φl ) ( )( φr φr )] 4 / / / t h β / β / [ f L ( f L f R )] [( ) f L ( ) f R ] = 3 s h β / / β / / [( )( f L f L ) ( )( f R f R )] 4 s h β l / l [( )( φ L φ L ) ( β / )( φ l R φ R )] 4 (.67) (.68) N where φ L = wψ L and φ R = w = N ψ R =. The paraeter β s the ass lupng paraeter. If β = we obtan the SSA schee for the standard LDG equatons. If 3 β =.0 we obtan the SSA schee for the luped lnear-dscontnuous Galerkn (LLDG) equatons. For the postve drecton there are two equatons and two unknowns f and / L f per cell. The boundary flux / R f can be elnated va upwndng.e. Eq. / R (.63). For the negatve drecton there are two equatons and two unknowns f and / L f per cell. The boundary flux / R f can be elnated va upwndng.e. Eq. / L (.64). If we consder the entre slab whch contans I spatal cells there are 4I equatons and 4I unknowns (the left and rght boundary fluxes can be obtaned fro the boundary

39 9 condtons). Therefore the syste s closed. We can solve the 4I coupled equatons through the followng atrx: where F 4 I s the soluton we desre.e. The coeffcent atrx A F δϕ 4I 4I 4I = 4 I F = [ f f f f... f f f f / / / / / / / / 4 I L L R R L L R R / / / / T f LI f LI f RI f RI... ]. A I 4 I (.69) (.70) 4 s a 7-dagonal 4 I 4I atrx and the source ter δ φ 4 I s obtaned fro the soluton of SI terate. After solvng the above equatons the accelerated scalar flux can be obtaned as: φ l l / l / l / L φl f L f L = (.7) φ l l / l / l / R φr f R f R. = (.7) SSA spatal dscretzaton for the LDLS ethod For the LDLS ethod the defnton of the flux representaton for the S equatons s the sae as that for LDG on the doan of x x x ] as n Eqs. [ / / (.63) and (.64). Based on Eqs. (.46)-(.5) the hgh-order stage (SI) can be wrtten as: For µ > 0 {[ S B ( x) S B ( x) ( ψ B l l / x / L L R R t L L t h / µ / / ( ) 0 µ dx λ = (.73) ψr BR ) ( ψr ψl )]( t BL )} x / h h

40 30 {[ S B ( x) S B ( x) ( ψ B l l / x / L L R R t L L t h ( ) 0 l / µ l / l / µ dx λ µ = (.74) ψr BR ) ( ψr ψl )]( t BR )} x / h h For µ < 0 / / l l / / ψl ψr SL SR µ ( ψ R ψ R ) t h = h. (.75) {[ S B ( x) S B ( x) ( ψ B l l / x / L L R R t L L t h } ( ) 0 l / µ l / l / µ dx λ µ = (.76) ψ / ) ( )]( ) x RBR ψr ψl tb L h h {[ S B ( x) S B ( x) ( ψ B l l / x / L L R R t L L t h ( ) 0 l / µ l / l / µ dx λ = (.77) ψ / ) ( )]( )} x RBR ψr ψl tb R h h / / l l / / ψ L ψ R S L S R µ ( ψ L ψ L ) t h = h (.78) where S w Q (.79) N l s l L = ψ L L = S w Q. (.80) N l s l R = ψ R R = Based on these SI equatons the resdual for S acceleraton can be wrtten as: For the postve drecton

41 3 s / l / l r ( x) = [( φl φl ) BL ( x) ( φl φl ) BR ( x)] s / / / / [( f L f L ) BL ( x) ( f R f R ) BR ( x)] / [ f B ( x) f B ( x)] ( f f ) / / / t L L R R R 3h L (.8) ( / / f L f R ) δ ( x x / ). 3 For the negatve drecton s / l / l r ( x) = [( φl φl ) BL ( x) ( φl φl ) BR ( x)] s / / / / [( f L f L ) BL ( x) ( f R f R ) BR ( x)] / [ f B ( x) f B ( x)] ( f f ) / / / t L L R R R 3h L (.8) ( / / f L f R ) δ ( x x / ). 3 The least-squares functonals for the S equatons take the followng for: x / x / / / L R = x / x / (.83) γ ( f f λ ) r ( xdx ) λ r ( xdx ) x / x / / / L R = x / x / (.84) γ ( f f λ ) r ( xdx ) λ r ( xdx ) where λ s a Lagrange ultpler. The balance equaton n the closed doan x x ] s: [ / / For the postve drecton x / r ( ) 0. x dx = (.85) x /

42 3 For the negatve drecton x / r ( ) 0. x dx = (.86) x / Substtutng Eqs. (.8) and (.8) nto Eqs. (.85) and (.86) we obtan the balance equatons as follows: For the postve drecton f f ( f f ) h 3 / / / / L R R R t / / / / s h ( f L f L ) ( f R f R ) = / l / l s h ( φl φl ) ( φr φr ). (.87) For the negatve drecton f f ( f f ) h 3 / / / / L R L L t / / / / s h ( f L f L ) ( f R f R ) = / l / l s h ( φl φl ) ( φr φr ). (.88) Now we solve the followng three equatons for each drecton respectvely: For the postve drecton γ ( f f λ ) = 0 / / L R / f L (.89) γ ( f f λ ) = 0 / / L R / f R (.90)

43 33 / / γ ( f L f R λ) = 0. λ (.9) For the negatve drecton γ ( f f λ ) = 0 / / L R / f L (.9) γ ( f f λ ) = 0 / / L R / f R (.93) / / γ ( f L f R λ) = 0. λ (.94) For the postve drecton there are three equatons and three unknowns per cell.e. f / L / f R and the Lagrange ultpler λ. λ s not physcally sgnfcant and can be elnated. Thus now we have two equatons and two unknowns per cell so the syste s close. The boundary flux f can be elnated va upwndng.e. Eq. / R (.63). For the negatve drecton there are three equatons and three unknowns f / L / f R and λ per cell. After elnatngλ we have two equatons and two unknowns so the syste s close. The boundary flux f can be elnated va upwndng.e. / L Eq. (.64). If we consder the entre slab whch contans I spatal cells there are 4I equatons and 4I unknowns (the left and rght boundary fluxes can be obtaned fro the boundary condtons). Therefore the syste s closed. We solve the 4I coupled equatons also through Eq. (.69). The coeffcent atrx 4 s a 7-dagonal 4 I 4I atrx and the A I 4 I

44 34 source ter δ φ 4 I s obtaned fro the soluton of SI terate. After solvng the above equatons the scalar flux wth the corrected error can be obtaned as: φ l l / l / l / L φl f L f L = (.95) φ l l / l / l / R φr f R f R. = (.96) Suary In ths chapter we brefly descrbed the ost coonly appled spatal dscretzaton ethod: lnear-dscontnuous Galerkn (LDG) ethod. We proposed our new LDLS ethod. We also presented the spatally dscretzed for of the SSA schee. The equatons obtaned were pleented n a FORTRAN code and the nuercal results wll be presented n Chapter V. We explored the robustness of the LDLS ethod through a splfed pure absorber transport equaton and copared the robustness of the solutons fro LDG and LLDG. We found that the LDLS ethod s ore robust than the LDG ethod and ore accurate than the LLDG ethod n the -D slab geoetry.

45 35 CHAPTER III FOURIER ANALYSIS FOR SPECTRAL RADIUS In ths chapter we perfor a Fourer analyss for the LDG LLDG and LDLS ethods to nvestgate the teratve convergence behavor of both the source teraton and SSA process. We consder an nfnte hoogenous edu and unfor esh proble. All the calculatons are pleented by MATLAB. Dervaton of Fourer analyss for the LDG ethod The contnuous for of the SSA schee s gven n Eqs. (.0)-(.3). In ths secton we want to perfor the Fourer analyss for the LDG ethod. Our frst task s to obtan the scalar flux error at step l / n ters of the scalar flux error at stepl. The exact LDG schee s gven n Eqs. (.3)-(.8) and ts SI schee s gven n Eqs. (.57)-(.6) n the prevous chapter. Based on these equatons we obtan the exact equatons for the error: For µ > 0 / / / t h β / β / µ [ ( δψl δψr ) δψr ] [( ) δψl ( ) δψr ] s h β l β l = [( ) δφl ( ) δφr ] 4 / / / t h β / β / µ [ δψr ( δψl δψr )] [( ) δψl ( ) δψr ] = s h β l β l [( ) δφl ( ) δφr ]. 4 (3.) (3.)

46 36 For µ < 0 / / / t h β / β / µ [ ( δψ L δψ R ) δψ L ] [( ) δψ L ( ) δψ R ] = s h β l β l [( ) δφl ( ) δφr ] 4 / / / t h β / β / µ [ δψl ( δψl δψr )] [( ) δψl ( ) δψr ] = s h β l β l [( ) δφl ( ) δφr ] 4 (3.3) (3.4) where the error of the flux s gven as: δψ = ψ ψ (3.5) / / L L L δψ = ψ ψ (3.6) / / R R R δφ = φ φ (3.7) l l L L L δφ = φ φ (3.8) l l R R R and where the paraeter β s the ass lupng paraeter wth β = / 3 correspondng to LDG and β = correspondng to LLDG. We assue an nfnte hoogeneous unfor esh ( h = h). The spatal dependence of the dscrete flux error s defned by a sngle Fourer ode.e. δψ = δψ exp( jλ x ) (3.9) / / L L / δψ = δψ exp( jλ x ) (3.0) / / R R / l l δφl = δφlexp( jλ x /) (3.) l l δφr = δ ΦRexp( jλ x /) (3.)

47 37 where j = and < Λ <. We set a paraeter θ = Λ h where h s the cell wdth. Substtutng Eqs. (3.9)-(3.) nto the exact SI equatons for the error.e. Eqs. (3.)- (3.4) and dvdng by exp( jλ x /) we obtan the follows: For µ > 0 h β µ [ ( δψ e δψ ) δψ ] [( ) δψ β jθ / s h β l β jθ l ( ) e δψ R ] = [( ) δφ L ( ) e δφr] 4 / jθ / / t / L R R L h β µ [ e δψ ( δψ e δψ )] [( ) δψ β jθ / s h β l β jθ l ( ) e δψ R ] = [( ) δφ L ( ) e δφr]. 4 jθ / / jθ / t / R L R L (3.3) (3.4) For µ < 0 h β µ [ ( δψ e δψ ) δψ ] [( ) δψ β jθ / s h β l β jθ l ( ) e δψ R ] = [( ) δφ L ( ) e δφr] 4 / jθ / / t / L R L L jθ / / jθ / t h β / µ [ e δψ L ( δψ L e δψ R )] [( ) δψ L β jθ / s h β l β jθ l ( ) e δψ R ] = [( ) δφ L ( ) e δφr] 4 (3.5) (3.6) Fro Eqs. (3.3) and (3.4) for µ > 0 and fro Eqs. (3.5) and (3.6) for µ < 0 respectvely we obtan the followng equaton for any gven drecton µ : / l δψl δφ L M / = l δψr δφ R (3.7)

48 38 where M s the coeffcent atrx solved fro Eqs. (3.3) and (3.4) for µ > 0 and fro Eqs. (3.5) and (3.6) for µ < 0. We wrte M as: M. = (3.8) Multplyng Eq. (3.7) by w and sung over =... N we obtan the equaton we desred.e. the scalar flux error at step l / n ters of the scalar flux error at stepl : δφ δφ δφ / l L L = H / SI l R δφr (3.9) where N / / L δ L w = δφ = Ψ (3.0) N / / R δ R w = δφ = Ψ (3.) and where H SI represents the source teraton egenfuncton atrx: H SI h h = h h (3.) N h = w for = and j =. (3.3) j j = Let α ( ) H SI ax θ be the larger of the two egenvalues of H SI for a gven value of θ. Then the global spectral radus that relates the scalar flux error at step l / to the scalar flux error at step l after suffcently any teratons s gven by: ρ = ax α ( θ) (3.4) SI H ax all θ SI

49 39 We refer to α ( ) ax θ as the local spectral radus snce t s the effectve spectral H HSI radus for the sngle Fourer ode assocated wth a gven value of θ. Now let s ove on to the low order S equatons. Our next task s to obtan the S scalar flux error estate n ters of the scalar flux error at step l. The dscrete for of the low order S schee for the LDG ethod s gven n Eqs. (.65)-(.68). The spatal dependence of the S flux error s defned by a sngle Fourer ode.e. f = F exp( jλ x ) (3.5) / / ± L ± L / f = F exp( jλ x ) (3.6) / / ± R ± R / Substtutng Eqs. (3.)-(3.) Eqs. (3.0)-(3.) and Eqs. (3.5)-(3.6) nto the loworder S schee n Eqs. (.65)-(.68) and dvdng by exp( jλ x /) we get: For the postve drecton / jθ / / t h β / β jθ / [ ( F L e F R ) F R ] [( ) F L ( ) e F R ] = 3 s h β / / β jθ / / [( )( F L F L ) ( ) e ( F R F R )] 4 s h β / l β jθ / l [( )( ΦL Φ L) ( ) e ( ΦR Φ R)] 4 (3.7) jθ / / jθ / t h β / β jθ / [ e F R ( F L e F R )] [( ) F L ( ) e F R ] = 3 s h β / / β jθ / / [( )( F L F L ) ( ) e ( F R F R )] (3.8) 4 s h β / l β jθ / [( )( ΦL Φ L) ( ) e ( Φ l R Φ R )]. 4 For the negatve drecton

50 40 / jθ / / t h β / β jθ / [ ( F L e F R ) F L ] [( ) F L ( ) e F R ] = 3 s h β / / β jθ / / [( )( F L F L ) ( ) e ( F R F R )] 4 s h β / l β jθ / l [( )( ΦL Φ L) ( ) e ( ΦR Φ R )] 4 (3.9) jθ / / jθ / t h β / β jθ / [ e F L ( F L e F R )] [( ) F L ( ) e F R ] = 3 s h β / / β jθ / / [( )( F L F L ) ( ) e ( F R F R )] (3.30) 4 s h β / l β jθ [( )( ΦL Φ L) ( ) e ( Φ / l R Φ R )]. 4 Substtutng Eq. (3.9) nto Eqs. (3.7)-(3.30) followed by consderable anpulaton we obtan the equaton we desred.e. the S scalar flux error estate n ters of the scalar flux error at step l : F δφ F / l L L = H / S l R δφ R (3.3) where the S scalar flux error: F = F F (3.3) / / / L L L F = F F (3.33) / / / R R R. and where H S represents the S error egenfuncton atrx. Untl now we have expressed the scalar flux error at step l / n ters of the scalar flux error at step l n Eq. (3.9) and the egenfuncton atrx H SI. We have also expressed the S scalar flux error estate n ters of the scalar flux error at stepl n Eq. (3.3) and the egenfuncton atrx H S. Our fnal task s to fnd out the egenfuncton

51 4 atrx H SSAthat relates the scalar flux error at step l to the scalar flux error at stepl. Based on Eq.(.5) the scalar flux error at step l s equal to the scalar flux error at step l / nus the scalar flux error estate fro the S equatons. Thus we subtract Eq. (3.3) fro Eq. (3.9) to obtan: / / l δφl FL δφl ( H / / SI HS ) = l δφr FR δφr (3.34) Thus δφ δφ δφ l L L = H SSA l R δφr (3.35) where H SSArepresents the SSA egenfuncton atrx: H = H H (3.36). SSA SI S Letα ( ) H S SA ax θ be the larger of the two egenvalues of H SSAfor a gven value of θ. Then the global spectral radus that relates the scalar flux error at stepl to the scalar flux error at step l for suffcently any teratons s gven by: ρ = ax α ( θ) (3.37) S SA H S SA ax all θ We refer to α ( ) H HS SA ax θ as the local spectral radus snce t s the effectve spectral radus for the sngle Fourer ode assocated wth a gven value of θ. At ths pont we have derved an expresson for the spectral radus of the SI schee ρ SI. We have also obtaned an expresson for the desred spectral radus SSA ρ that relates the scalar flux error at step l to the scalar flux error at step l. Thus we

52 4 have copleted the descrpton of the Fourer analyss for the LDG (and LLDG) ethod. We can now copute the egenvalues for the nfnte hoogenous edu and unfor esh proble. The quadrature set s chosen to be S 8 Gauss quadrature. The scatterng rato s chosen to be ether c =.0 (pure scatterng) or c = The total cross sectons s chosen to be t =.0. Results for the LDG ethod ( β = / 3) are gven n Fgs It can be observed fro Fgs. 3. and 3. whch show the global spectral radus of SI h = t 0.0 s equal to the scatterng rato c and the spectral radus of SSA s at roughly 0.c. We can also observe that the local spectral radus has a perodc dependence upon θ ( θ =Λ h ). Fro Fgs. 3.3 and 3.4 whch show the global spectral radus as a functon of h t t can be seen that the SSA schee reans effectve for all optcal cell thcknesses.e. ρ < 0.c. The SSA n the probles wth absorpton gves a S SA spectral radus of 0 n the lt as scatterng ( c = ) proble. t h whle t converges to 0. n a pure

53 SI SSA Spectral radus θ Fg. 3.. Local spectral rad fro LDG wth c =.0 and = 0.0. t h 0.9 SI SSA Spectral radus θ Fg. 3.. Local spectral rad fro LDG wth c = 0.98 and = 0.0. t h

54 SI SSA Spectral radus t h Fg Global spectral rad fro LDG wth c = SI SSA Spectral radus t h Fg Global spectral rad fro LDG wth c = 0.98.

55 45 Results for the LLDG ethod ( β = ) are gven n Fgs and they are qute slar to the results for the LDG ethod. It can be observed fro Fgs. 3.5 and 3.6 that for = 0.0 the global spectral radus of SI s equal to the scatterng rato c t h and the global spectral radus of SSA s at roughly 0.c. We can also observe that the local spectral radus has a perodc dependence upon θ ( θ = Λ h ). Fro Fgs. 3.7 and 3.8 whch show the global spectral radus as a functon of h t t can be seen that the SSA schee reans effectve for all optcal cell thcknesses.e. ρ < 0.c. The SSA n the probles wth absorpton gves a S SA spectral radus of 0 n the lt as t h whle t converges to 0. n a pure scatterng ( c = ) proble whch gves the sae results as the LDG ethod. Dervaton of Fourer analyss for the LDLS ethod In ths secton we perfor a Fourer analyss for the LDLS ethod to deterne the teratve convergence rate of both the SI and SSA process. The analyss process s slar to that of the LDG ethod. Our frst task s to obtan the scalar flux error at step l / n ters of the scalar flux error at step l. The exact LDLS soluton s gven by Eqs. (.46)-(.5) and the SI schee s gven by Eqs. (.73)-(.80). Based on these equatons elnatng the Lagrange ultpler λ and after soe anpulaton usng Maple we obtan the exact soluton for the error:

56 SI SSA Spectral radus θ Fg Local spectral rad fro LLDG wth c =.0 and = 0.0. t h 0.9 SI SSA Spectral radus θ Fg Local spectral rad fro LLDG wth c = 0.98 and = 0.0. t h

57 SI SSA Spectral radus t h Fg Global spectral rad fro LLDG wth c = SI SSA Spectral radus t h Fg Global spectral rad fro LLDG wth c = 0.98.

58 48 For µ > 0 A B C D = (3.38) / / l l δψ L δψ R δφl δφr 0 For µ < 0 / / l l / / δψ L δψ R s h δφl δφr µ ( δψ R δψ R ) t h =. (3.39) A B C D = (3.40) / / l l δψ L δψ R δφl δφr 0 / / l l / / δψ L δψ R s h δφl δφr µ ( δψ L δψ L ) t h = (3.4) where the coeffcents A B C D = are polynoal coeffcents. They are all functons of µ h t and s. Substtutng the sngle Fourer ode of the flux error n Eqs. (3.9)-(3.) nto Eqs. (3.38)-(3.4) and dvdng by exp( jλ x /) we obtan: For µ > 0 A Ψ Be Ψ C Φ De Φ = (3.4) / jθ / l jθ l δ L δ R δ L δ R 0 For µ < 0 / jθ / l jθ l jθ / / δψ L e δψ R s h δφ L e δφr µ ( e δψ R δψ R ) t h =. (3.43) A Ψ B e Ψ C Φ D e Φ = (3.44) / jθ / l jθ l δ L δ R δ L δ R 0 / jθ / l jθ l jθ / / δψ L e δψ R s h δφ L e δφr µ ( e δψ L δψ L ) t h =. (3.45)

59 49 Slarly to the LDG ethod we adopt the sae procedure as fro Eq. (3.7) to Eq. (3.4) to obtan the local and global spectral rad of SI schee for the LDLS ethod. Our next task s to obtan the S scalar flux error estate n ters of the scalar flux error at stepl and to obtan the desred spectral radus that relates the scalar flux error at stepl to the scalar flux error at stepl. The dscrete for of the low order S schee for the LDLS ethod s gven n Eqs. (.89)-(.94). The spatal dependence of the S flux error s defned by the sngle Fourer ode gven n Eqs. (3.5)-(3.6). Substtutng Eqs. (3.)-(3.) and Eqs. (3.5)-(3.6) nto the low-order S schee n Eqs. (.89)-(.94) and dvdng by exp( jλ x /) we get: For the postve drecton AF Be F CF De F / jθ / / jθ / 3 L 3 R 3 L 3 R E ( Φ Φ ) Fe ( Φ Φ ) = 0 / l jθ / l 3 L L 3 R R (3.46) / jθ / ( jθ / / F / / ) L e F R s h ( jθ e F R F R t h = FL e FR ) 3 4 s h / l jθ / l [( ΦL Φ L) e ( ΦR ΦR)]. 4 (3.47) For the negatve drecton AF Be F CF De F / jθ / / jθ / 4 L 4 R 4 L 4 R E ( Φ Φ ) Fe ( Φ Φ ) = 0 / l jθ / l 4 L L 4 R R (3.48) / jθ / ( jθ / / F / / ) L e F R s h ( jθ e F L F L t h = FL e FR ) 3 4 s h / l jθ / l [( ΦL Φ L) e ( ΦR ΦR)]. 4 (3.49)

60 50 where the coeffcents A B C D = 34 are polynoal coeffcents. They are all functons of µ h t and s. Slarly to the LDG ethod we adopt the sae procedure as fro Eq. (3.3) to Eq. (3.37) to obtan the local and global spectral rad of the SSA schee for the LDLS ethod. At ths pont we have derved an expresson for the spectral radus of the SI schee ρ SI. We have also obtaned an expresson for the desred spectral radus SSA ρ that relates the scalar flux error at step l to the scalar flux error at step l. Thus we have copleted the descrpton of the Fourer analyss for the LDLS ethod. We can now copute the egenvalues for the nfnte hoogenous edu and unfor esh proble. The quadrature set s chosen to be S 8 Gauss quadrature. The scatterng rato s chosen to be ether c =.0 (pure scatterng) or c = The total cross sectons s chosen to be t =.0. Results for the LDLS ethod are gven n Fgs It can be observed fro Fgs. 3.9 and 3.0 that for = 0.0 the global spectral radus of SI s equal to the t h scatterng rato c and the spectral radus of SSA s at roughly 0.c. We can also observe that the local spectral radus has a perodc dependence upon θ ( θ =Λ h ). Fro Fgs. 3. and 3. whch show the global spectral radus as a functon of h t t can be observed that the SSA schee reans effectve for all cell thcknesses. For sall ean-free-paths ( h < t 0. ) the spectral radus reans at SSA= 0.c whle there s a peak at roughly = 0.8. The SSA schee n the probles wth t h