Annals of Nuclear Energy 38 (2011) 1033–1038

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Annals of Nuclear Energy

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Application of the hierarchical domain decomposition boundary element methodto the simplified P3 equation

Go Chiba ⇑Japan Atomic Energy Agency, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan

a r t i c l e i n f o

Article history:Received 25 October 2010Received in revised form 9 December 2010Accepted 5 January 2011Available online 4 February 2011

Keywords:Simplified PN methodBoundary element methodHDD-BEM

0306-4549/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.anucene.2011.01.011

⇑ Tel.: +81 29 282 5337; fax: +81 29 282 6122.E-mail address: chiba.go@jaea.go.jp

a b s t r a c t

In this paper, the hierarchical domain decomposition boundary element method (HDD-BEM), which hasbeen developed to solve the diffusion equation, is applied to the simplified P3 (SP3) equation. The HDD-BEM solution for the SP3 equation is provided in the present paper. A computer program, ABEMIE, basedon the HDD-BEM is developed, and a two-dimensional one-group anisotropic-scattering benchmarkproblem is solved with it to verify the present HDD-BEM for the SP3 equation.

Through numerical benchmarking, it is shown that the present method results in good agreement withthe solution obtained using the existing SPN solver based on the finite element method for both eigen-value and neutron flux distribution. This benchmark result suggests that the HDD-BEM is suitable forapplication to the SPN equation.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction computational burden, i.e., computer memory storage and compu-

In recent years, the simplified PN method (Gelbard, 1960; Larsenet al., 1996; Brantley and Larsen, 2000), or the SPN method, hasattracted much attention in the reactor physics field since thismethod makes it possible to partly capture the neutron transporteffect without a severe increase in computational burden. Variouscomputer programs based on the SPN method have been developedfor thermal reactor applications to date (Tatsumi and Yamamoto,2003; Beckert and Grundmann, 2007; Baudron and Lautard,2007; Tada et al., 2008). Generally speaking, the neutron transporteffect in fast reactors is larger than that in thermal reactors. Thus,the SPN method may be considered promising as a fast reactor de-sign calculation tool. Hèbert has developed a finite element-basedSPN method for hexagonal geometries and has incorporated it intothe existing diffusion theory code TRIVAC (Hèbert, 2010a,). Thisdevelopment may be intended to fast reactor applications of theSPN method. These computer programs based on the SPN methodemploy the finite difference method, the nodal method or the finiteelement method for spatial discretization. These spatial discretiza-tion methods have been utilized with great success in diffusiontheory codes for reactor core calculations.

The boundary element method (BEM) is applied to partial dif-ferential equations and has been widely used in the field of engi-neering. A specific feature of the BEM is that the dimensions of aproblem can be reduced by one. For example, a two-dimensionalproblem can be reduced to a one-dimensional problem. Thus the

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tation time, can be greatly reduced. Another advantageous featureof the BEM is that preparation of input data describing problemgeometry is easier than in other methods.

The first application of the BEM to the neutron diffusion equa-tion can be found in Koskinen’s seminal work (Koskinen, 1965),in which the neutron diffusion equation is transformed to a corre-sponding boundary integral equation. Twenty years later, Itagakiapplied the BEM to general neutron diffusion problems; namely,criticality, fixed-source and multi-region problems (Itagaki,1985). This work has been followed by other researchers and sev-eral numerical methods based on the BEM have been proposed(Ozgener and Ozgener, 1993; Purwadi et al., 1998; Maiani andMontagnini, 1999; Chiba et al., 2001a,b; Cavder and Ozgener,2004; Cossa et al., 2010). Among these BEM-based numericalmethods, the hierarchical domain decomposition boundary ele-ment method (HDD-BEM) (Purwadi et al., 1998; Chiba et al.,2001a,b) was very probably the first to be applied to realisticthree-dimensional multi-group problems, since such applicationtook place almost ten years ago (Chiba et al., 2001b). It has alsobeen applied successfully to parallel computers (Tsuji and Chiba,2000). The HDD-BEM is quite unique. The neutron diffusion equa-tion is first transformed to a set of mode flux equations and theseequations are then solved by the BEM. The eigenvalue and innerboundary conditions between neighboring homogeneous regionsare estimated iteratively by Newton’s method.

The present study is the first application of the HDD-BEM to theSP3 equation. In the present paper, the application of the HDD-BEMto the SP3 equation will be described, and then it will be verifiedthrough a benchmark calculation using a two-dimensional one-group anisotropic-scattering problem.

1034 G. Chiba / Annals of Nuclear Energy 38 (2011) 1033–1038

2. Hierarchical domain decomposition boundary elementmethod for the SP3 equation

Region

rS

r

rkS

kr Nodal point

Fig. 1. Definition of region and boundary elements.

2.1. Mode flux representation for the zero-th and second angularmoments of neutron flux

In the conventional HDD-BEM for the multi-region diffusionequation, the original diffusion equation is transformed to a setof mode flux equations for each homogeneous region, as proposedby Koskinen (1965). Derivation of mode flux equations for the mul-ti-region SP3 equation is straightforward. Here, we consider a one-group equation for simplicity.

A one-group SP3 equation can be written as follows:

�D0r2ð/0 þ 2/2Þ þ ðr� r0Þð/0 þ 2/2Þ

¼ 1kmrf /0 þ 2ðr� r0Þ/2; ð1Þ

�2735

D2r2/2 þ ðr� r2Þ/2 ¼25ðr� r0Þ/0 �

1kmrf /0

� �; ð2Þ

where /0 and /2 are the zero-th and second angular moments ofneutron flux, r is the total cross section, rn is the nth Legendre com-ponent of scattering cross section, mrf is the production cross sec-tion, k is the eigenvalue, and

D0 ¼1

3ðr� r1Þ; ð3Þ

D2 ¼1

3ðr� r2Þ: ð4Þ

This one-group SP3 equation can be written in a matrix form asfollows:

½$2I þ AðkÞ�w ¼ 0; ð5Þ

where

AðkÞ¼�r�r0

D0þ 1

D0kmrf � 2D0kmrf þ 2

D0ðr�r0Þ

� 3527D2

�25ðr�r0Þþ 2

5kmrf

� �� 35

27D2ðr�r2Þþ 4

5ðr�r0Þ� 45kmrf

� � !

;

ð6Þ

and wt ¼ ð/̂0;/2Þ ¼ ð/0 þ 2/2;/2Þ.The procedure to derive mode flux equations is the same as that

for the diffusion equation.For a homogeneous region r, angular flux moments /̂r

0 and /r2

can be expressed with the ‘‘mode flux’’ ur as follows:

/̂r0 ¼

X2

j¼1

Cr0ju

rj ; ð7Þ

/r2 ¼

X2

j¼1

Cr2ju

rj : ð8Þ

The coupling coefficients Cr0j and Cr

2j are determined from the groupconstants of the region r. The mode flux satisfies the followingequation:

r2 þ Br2j

� �ur

j ¼ 0; ð9Þ

where the values Br2j are the roots in the following algebraic equa-

tion for r2:

det $2I þ ArðkÞh i

¼ 0: ð10Þ

As described above, the multi-region SP3 equation is trans-formed to a set of mode flux equations for each homogeneous re-gion. If the eigenvalue and boundary conditions are given to each

homogeneous region, the mode flux equations can be solved andsolutions for /̂0 and /2 for the homogeneous region can beobtained.

2.2. Formulation of boundary integral equation

Here we define the boundary of a homogeneous region r as Sr, asshown in Fig. 1. Using the weighted residual formulation (Brebbia,1978), Eq. (9) can be converted to a boundary integral equation as

~ciurj ðriÞ ¼ �

ZSr

urj ðrÞu�r0j ðr; riÞdsþ

ZSr

ur0j ðrÞu�rj ðr; riÞds; ð11Þ

where ur0j is a normal derivative of ur

j , and u�rj and u�r0j are Green’sfunction and its normal derivative, respectively. Green’s functionu�rj is given as the fundamental solution of the following equationwith a point source d(r, ri):

r2u�rj ðr; riÞ þ Br2j u�rj ðr; riÞ þ dðr; riÞ ¼ 0: ð12Þ

The constant ~ci in Eq. (11) is unity if ri lies inside of the region and 1/2 if ri lies on a smooth boundary.

In order to discretize the boundary integral Eq. (11), the bound-ary is divided into K boundary elements. These boundary elementsare denoted as Sr

k. On each boundary element, we assign nodalpoints on which values of mode flux and its derivative are deter-mined. The spatial distributions of the mode flux and its derivativeon the boundary element are expanded by polynomials by usingthe values on the nodal points.

Here, we assume that one nodal point is assigned to all theboundary elements. In this case, the mode flux and its derivativeare assumed to be constant on each boundary element. Thus theboundary integral Eq. (11) can be written as

~ciurj ðriÞ ¼ �

XK

k¼1

urj ðrkÞ

ZSr

k

u�r0j ðr; riÞdsþXK

k¼1

ur0j ðrkÞ

�Z

Srk

u�rj ðr; riÞds; ð13Þ

where rk corresponds to the location of the nodal point on theboundary element Sr

k. If we set ri as rk in Eq. (13), we obtain the fol-lowing K simultaneous equations:

~ckurj ðrkÞ ¼ �

XK

k0¼1

urj ðrk0 Þ

ZSr

k0

u�r0j ðr; rkÞdsþXK

k0¼1

ur0j ðrk0 ÞZ

Srk0

u�rj ðr; rkÞds; k ¼ 1;2; . . . ;K: ð14Þ

A set of the boundary integral Eq. (14) can be written in the follow-ing matrix equation:

Gur0j ¼ Hur

j : ð15Þ

As shown in Eq. (14), we have K simultaneous linear equations.Thus if K of ur

j ðrkÞ and ur0j ðrkÞ are known, others can be obtained

G. Chiba / Annals of Nuclear Energy 38 (2011) 1033–1038 1035

by solving matrix Eq. (15). Generally the mode fluxes or its deriva-tives on the outer boundaries are known and those on inner bound-aries between neighboring two homogeneous regions are unknown.In the HDD-BEM, the inner boundary conditions, i.e., values of modefluxes, and eigenvalue are initially assumed. Thus the boundaryintegral equation can be solved for each homogeneous region.

For finer spatial discretization, the mode flux and its derivativecan be expanded by the polynomials if two or more nodal pointsare assigned on a boundary element. On these boundary elements,two of the nodal points are placed on the extremes of the boundaryelement. If the extreme of the boundary element coincides with a‘‘corner point’’ where the derivative of the boundary curvaturealong the boundary changes discontinuously, the concept of thenon-conforming elements has to be introduced. On non-conform-ing elements, nodal points are not placed on the extremes but ona position slightly inside of the extremes. Details of the non-con-forming element can be found in Chiba et al. (2001a).

2.3. Modification of eigenvalue and inner boundary conditions byNewton’s method

In the HDD-BEM, the eigenvalue and the inner boundary condi-tion, i.e., mode fluxes on the inner boundaries, are initially as-sumed, and these are modified through iterations. This procedurefor the SP3 equation is the same as that for the diffusion equation.

Consider an inner boundary element Ip between two homoge-neous neighboring regions. On this boundary element, two setsof mode fluxes and their derivatives are defined, i.e., those includedin one region denoted as r(Ip+) and those included in another re-gion denoted as r(Ip�). In the HDD-BEM, mode fluxes on the innerboundary in the region r(Ip+) are initially assumed. By using thecontinuity condition for /̂0 and /2, mode fluxes on this innerboundary in the other region r(Ip�) are automatically determined.

Since all the mode fluxes on inner boundaries are known,boundary integral Eq. (15) can be solved and the derivatives ofthe mode fluxes can be obtained. If the assumed eigenvalue and in-ner boundary conditions are correct, the following equation shouldbe preserved since the derivatives of /̂0 and /2 on the inner bound-aries should be continuous:

F I ¼F I

0

F I2

!¼

F I10

F I20

..

.

F IP0

F I12

F I22

..

.

F IP2

0BBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCA

¼ 0; ð16Þ

where

F Ip0 ¼

FIp ;00

FIp ;10

..

.

FIp ;Lp0

0BBBBB@

1CCCCCA; F Ip

2 ¼

FIp ;02

FIp ;12

..

.

FIp ;Lp2

0BBBBB@

1CCCCCA; ð17Þ

and

FIp ;l0 ¼ DrðIpþÞ

0

X2

j¼1

CrðIpþÞ0j u0rðIpþÞ

j;l þ DrðIp�Þ0

X2

j¼1

CrðIp�Þ0j u0rðIp�Þ

j;l ; ð18Þ

FIp ;l2 ¼ DrðIpþÞ

2

X2

j¼1

CrðIpþÞ2j u0rðIpþÞ

j;l þ DrðIp�Þ2

X2

j¼1

CrðIp�Þ2j u0rðIp�Þ

j;l ; ð19Þ

in which P corresponds to the total number of inner boundary ele-ments in the system, Lp corresponds to the number of nodal pointson the boundary element Ip, and FIp ;l

0 and FIp ;l2 correspond to the con-

tinuity conditions of the derivatives of /̂0 and /2 on the l-th nodalpoint of the boundary element Ip. The problem of determining thecorrect eigenvalue and inner boundary conditions is then reducedto the problem of determining the roots of Eq. (16). While the eigen-value and the 2� L ¼

PPp¼1Lp

� �inner boundary conditions (i.e.,

urðIpþÞj ) are unknown, the number of the continuity conditions, ex-

pressed by Eq. (15), is 2L. Thus it is required to add another condi-tion. In the conventional HDD-BEM, the following scaling equationis added:

Fc ¼ 1�XP

p¼1

XLp

l¼1

X2

j¼1

urðIpþÞ¼0j;l : ð20Þ

The roots of Eqs. (16) and (20) are searched iteratively by New-ton’s method as follows:

JðnÞduI

dk

� ¼ � F IðuIðnÞ; kðnÞÞ

FcðuIðnÞ; kðnÞÞ

" #; ð21Þ

uIðnþ1Þ

kðnþ1Þ

" #¼

uIðnÞ

kðnÞ

" #þ duI

dk

� ; ð22Þ

where n denotes the number of iterations and the vector uI is com-posed of urðIpþÞ

j;l . The Jacobian matrix J(n) is defined as

JðnÞ ¼J Iu J I

k

Jcu 0

" #; ð23Þ

where

J Iu ¼

@F I

@uI

!; J I

k ¼@F I

@k

!; Jc

u ¼@Fc

@uI

�: ð24Þ

The sizes of these Jacobian submatrices J Iu; J I

k and Jcu are (2L � 2L),

(2L � 1) and (1 � 2L), respectively.When the total number of nodal points on the inner boundaries

is large, it is difficult to solve matrix Eq. (21) directly. Thus in theHDD-BEM, the block Jacobi method is applied to obtain du andNewton’s method is applied to obtain dk as follows.

At the mth iteration of Newton’s method which modifies the va-lue of dk to dkm, the values of d uI on the inner boundaries are mod-ified iteratively by the following procedure using the results of theprevious iteration:

duIpðm;lþ1Þ ¼ J Ip

uIp

� ��1�XP

q¼1;q–p

J IpuIq

duIqðm;lÞ � F Ip � J Ip

k dkðmÞ !

; ð25Þ

where the superscript l denotes the iteration number of the blockJacobi method. Note that the inversion of the Jacobian sub-matrixJ IpuIp

can be done once before the Newton’s method iteration for dksince this matrix does not depend on dk(m).

When the iteration process with the block Jacobi method is car-ried out, the value of dk(m) is modified by Newton’s method as

dkðmþ1Þ ¼ dkðmÞ � dkðmÞ � dkðm�1Þ

JcuduIðmÞ � Jc

uduIðm�1Þ ðJcuduIðmÞ þ FcÞ; ð26Þ

where duI(m) is a converged solution of the block Jacobi method atthe mth Newton’s method iteration.

The above mentioned procedure based on Newton’s method canfind all the eigenvalues and eigenvectors of the system as dis-cussed in Purwadi et al. (1996). In order to achieve convergenceto the fundamental mode solution, the proper setting of initial esti-mates for the eigenvalue and eigenvector is necessary.

Mixture 1

Mixture 2

Mixture 3

VacuumB.C.

19cm

ReflectiveB.C.

ReflectiveB.C.

Fig. 2. Description of the hexagonal-geometry 2D one-group benchmark.

1036 G. Chiba / Annals of Nuclear Energy 38 (2011) 1033–1038

2.4. Boundary condition treatment

In the SP3 equation, the following Marshak-like boundary con-ditions are usually used for vacuum boundaries:

12

/̂0 þ D0r/̂0 ¼38

/2; ð27Þ

2140

/2 þ2735

D2r/2 ¼3

40/̂0: ð28Þ

These equations are described by the mode fluxes as

12

X2

j¼1

C0juj þ D0

X2

j¼1

C0ju0j ¼38

X2

j¼1

C2juj; ð29Þ

2140

X2

j¼1

C2juj þ2735

D2

X2

j¼1

C2ju0j ¼3

40

X2

j¼1

C0juj: ð30Þ

As shown in Eqs. (29) and (30), fluxes and their derivatives of differ-ent modes couple with each other. It is the same as with the vac-uum boundary conditions for the diffusion equation. In theprevious studies of the HDD-BEM for diffusion equations, the valuesof boundary fluxes and their derivatives of different modes were as-sumed to be those at the previous iteration step. In the presentstudy, the same technique is utilized.

2.5. Volume-integrated neutron flux calculation

After convergence, mode fluxes and their derivatives on all theinner boundaries are determined. Thus mode flux at an arbitraryposition inside each homogeneous region can be calculated bysolving Eq. (13).

On the other hand, volume-integrated neutron flux is easily cal-culated. For example, volume-integrated /̂0 is calculated asZ

Vr

/̂0 dr ¼Z

Vr

X2

j¼1

C0juj dr ¼X2

j¼1

C0j

ZVr

uj dr

¼ �X2

j¼1

C0j

ZVr

r2uj

Br2j

dr ¼ �X2

j¼1

C0j

Br2j

ZSr

u0j ds: ð31Þ

In this derivation, the relation r2uj þ Br2j uj ¼ 0 is used.

3. Verification

3.1. Calculation procedure and benchmark system

In order to verify the HDD-BEM for the SP3 equation, we devel-op a computer code, ABEMIE, as one of the solvers in a reactorphysics simulation code system CBG (Chiba, 2008). The ABEMIEis written using the C++ computer language and can solve one-group SP3 equations and one- or two-group diffusion equations.This code can treat an arbitrary two-dimensional geometry whichis composed of lines. The maximum number of nodal points on aboundary element is four. Thus mode fluxes and their derivativescan be expanded at most by third-order polynomials. When twoor more nodal points are assigned to a boundary element, theboundary element is treated as a non-conforming element evenif it does not include any corner points. The locations of the nodalpoints are denoted with a non-dimensional quantity n. The posi-tions of the extremes of boundary elements correspond ton = ±1.0 and the position of the mid-point corresponds to n = 0.The locations of nodal points are n = 0 for constant elements,n = ±0.65 for linear elements, n = ±0.85 and n = 0 for quadratic ele-ments, and n = ±0.85 and n = ±0.3 for cubic elements.

The initial estimate for the eigenvalue is 1.0, and a constant va-lue is given to the inner boundary conditions as an initial estimate.In order to accelerate the iteration for estimation of the eigenvalue

and inner boundary conditions, the Aitken acceleration method isapplied.

As a benchmark system, we choose the three-region hexagonalproblem with anisotropic-scattering described in Hèbert (2010b).This benchmark is not representative of real-life problems but isdesigned to magnify transport and anisotropic effects both insideand on the vacuum boundary of the domain. This benchmark sys-tem is depicted in Fig. 2 and the cross section data are shown in Ta-ble 1. Please note that calculations using ABEMIE are performed forthis one-sixth core model with reflective boundary conditions.

Reference values are the SP3 solutions obtained by the TRIVACcode (Hèbert, 2010b). For the spatial discretization of the TRIVACcalculation, the finest option, 12 lozenges per hexagon and thequadratic flux expansions, is employed. In the TRIVAC calculation,the symmetric property of the benchmark system is not used andthe whole core is treated.

Comparisons are carried out in eigenvalue and assembly-wisescalar neutron flux. Maximum and average errors in the assem-bly-wise neutron flux are derived as

�max ¼maxi

j/i � /refi j

/refi

( ); ð32Þ

and

�� ¼ 1Vcore

Xi

V i

/i � /refi

��� ���/ref

i

: ð33Þ

3.2. Calculation results

Table 2 summarizes results of the diffusion and SP3 calculationswith ABEMIE. For the spatial discretization, constant, linear, qua-dratic and cubic boundary elements are employed. Ntot corre-sponds to the total number of unknowns in the calculations. Inorder to compare with Ntot of the TRIVAC whole-core calculations,Ntot of the ABEMIE calculations are multiplied by six since ABEMIEcalculates a one-sixth core. The Ntot value of the TRIVAC calculationis cited from Hèbert (2010b).

In the diffusion calculation results, large errors are observed inboth the eigenvalue and the assembly-wise neutron fluxes even ifcubic boundary elements are employed. This is due to errors in thediffusion theory, and these results are consistent with the formerresults obtained by Hèbert (2010b). Regarding the SP3 calculationresults, it is found that the eigenvalue and the assembly-wise neu-tron fluxes converge to the reference solutions as finer boundaryelements are employed. The quadratic boundary elements are suf-ficiently accurate to spatially discretize the mode fluxes and their

Table 1Cross section data of benchmark system.

Mixture r (cm�1) r0 (cm�1) r1 (cm�1) mrf (cm�1)

1 0.025 0.013 0.0 0.01552 0.025 0.024 0.006 0.03 0.075 0.0 0.0 0.0

Table 2Summary of benchmark calculation results with ABEMIE.

Boundary element Ntot keff Dkeff (pcm) �max (%) ��ð%Þ

DiffusionConstant 762 0.95167 �4866 39.1 10.9Linear 1524 0.97114 �2919 22.4 7.3Quadratic 2286 0.97234 �2799 19.3 7.0Cubic 3048 0.97240 �2793 19.4 7.0

SP3

Constant 1524 0.98253 �1780 17.3 4.2Linear 3048 0.99941 �92 3.3 0.6Quadratic 4572 1.00034 +1 0.34 0.07Cubic 6096 1.00034 +1 0.21 0.05

Reference 79,118 1.00033

-2.35-2.26-2.26

-2.20-2.23-2.23 -2.21

-2.10-2.11

-1.08-0.86-0.87

-1.93-1.77-1.78

+2.04+2.14+2.14 +3.53

+3.73+3.72

+4.79+2.64+2.80

+5.89+6.15+6.10 +7.36

+7.90+7.78

+5.65+6.75+6.73

+18.9+19.3+19.4 +12.2

+11.8+11.6

+16.7+17.0+17.1

+22.4+17.9+17.9

LinearQuadratic

Cubic

RelativeError (%)

Fig. 3. Errors in assembly-wise neutron flux of diffusion calculations.

-0.11-0.03-0.03

+0.03+0.00-0.00 -0.08

+0.03+0.02

-0.22+0.00-0.00

-0.15+0.01-0.00

-0.08+0.00+0.00 -0.14

+0.00-0.00

+1.96-0.10+0.05

-0.19+0.04-0.00 -0.37

+0.08-0.02

-1.02-0.04-0.03

-0.19-0.03+0.09 +0.57

+0.34+0.21

-0.04+0.09+0.14

+3.32+0.22+0.19

LinearQuadratic

Cubic

RelativeError (%)

Fig. 4. Errors in assembly-wise neutron flux of SP3 calculations.

G. Chiba / Annals of Nuclear Energy 38 (2011) 1033–1038 1037

derivatives. It should be noted that the ABEMIE calculations attainhigh accuracy with a much smaller number of unknowns incomparison with the reference finite element-based TRIVAC calcu-lation. It is shown in Hèbert (2010b) that the coarser spatial dis-cretization, 3 lozenges per hexagon and the quadratic flux

expansions, yields results that are nearly consistent with the refer-ence. By using this coarser option, Ntot is reduced to 20,078, whichis yet larger than the Ntot of ABEMIE. This is one of advantages ofthe BEM-based method over the other spatial discretization meth-ods. Note that the matrix structure also significantly affects theconvergence property and the effectiveness of the numerical meth-od. A more detailed comparison between the present HDD-BEMand FEM-based methods is necessary in future.

Figs. 3 and 4 show errors in the assembly-wise neutron fluxes inthe diffusion and SP3 calculations, respectively. While the diffusioncalculations result in large errors around the core peripheral re-gion, these large errors disappear in the SP3 results.

4. Concluding remarks

In the present paper, the HDD-BEM, which has been developedto solve diffusion equations, is applied to the SP3 equation. TheHDD-BEM solution for the SP3 equation is provided in the presentpaper. A computer program, ABEMIE, based on the HDD-BEM hasbeen developed, and a two-dimensional one-group anisotropic-scattering benchmark problem is solved with it to verify the pres-ent HDD-BEM for the SP3 equation.

Through numerical benchmarking, it has been shown that thepresent method results in good agreement with the solution ob-tained by using the existing SPN solver based on the finite elementmethod for both eigenvalue and neutron flux distribution with asmaller number of unknowns. This benchmark result suggests thatthe HDD-BEM can be an efficient solver for the SPN equation.Extension of the present method to three-dimensional and multi-group problems is the next subject. The extension to three-dimen-sional problems can be carried out as performed for diffusion prob-lems (Chiba et al., 2001b). The extension to multi-group problemsmay not be straightforward since the number of modes becomeslarger than 2 and some of the buckling values B2

j can take imagi-nary values if we apply the present method to multi-group SP3

equations. A remedy for such a problem, however, has alreadybeen reported in Cossa et al. (2010). The extension to multi-groupproblems may be possible using this technique.

Please note that an application of one of the BEM-based meth-ods, the BERM method, to the SP3 equation has just recently beenpublished (Giusti et al., 2010). Comparison of the present methodwith the BERM method also can be a future subject.

Acknowledgements

The author is grateful to Dr. Masashi Tsuji of Hokkaido Univer-sity, who instructed the author in the HDD-BEM, and Mr. KazuteruSugino of the Japan Atomic Energy Agency for their comments onthis manuscript.

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