On the use of the conjugate gradient method for the solution of the neutron transport equation

On the use of the conjugate gradient method forthe solution of the neutron transport equation

Anurag Guptaa,*, R.S. ModakbaReactor Physics Design Section, Bhabha Atomic Research Centre, 5th Floor, Central Complex,

Mumbai-400085, IndiabTheoretical Physics Division, Bhabha Atomic Research Centre, 5th Floor, Central Complex,

Mumbai-400085, India

Received 10 August 2001; accepted 4 January 2002

Abstract

The possibility of using the standard conjugate gradient (CG) method to directly solve the

Sn-equations based on the diamond difference scheme is studied for mono-energetic neutrontransport problems with isotropic scattering. It is shown that such a direct use is possible forpractical heterogeneous problems with a significant speed-up over the conventional sourceiteration (SI) method except for the problems that are prone to unphysical negative fluxes.

Some recipes are suggested to make use of the CG-method even in those cases which neednegative flux fix-up in the SI-method. The transport synthetic acceleration scheme, recentlydeveloped by Ramone [Nucl. Sci. Eng. 125 (1997) 257] and others, is shown to be useful in

such cases. A symmetrisation scheme for the coefficient matrix has also been presented toenable the use of the CG-method. This scheme is compared with another approach of usingweighted inner products. # 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

Recently, the conjugate gradient-like methods are emerging as efficient techniquesfor the solution of large sparse sets of simultaneous linear algebraic equations (e.g.Axelsson, 1994; Saad, 1996). These methods are known to have good convergenceproperties and are also suitable for parallelisation. A large sparse set of algebraicequations can arise when the differential equations governing various physical phe-nomena are discretised in order to obtain a numerical solution. Here, we considerthe linear neutron transport equation (Bell and Glasstone, 1970; Carlson andLathrop, 1968) which occurs in the field of nuclear reactor physics. Specifically, we

Annals of Nuclear Energy 29 (2002) 1933–1951

www.elsevier.com/locate/anucene

0306-4549/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0306-4549(02 )00017 -8

* Corresponding author.

E-mail address: [email protected] (A. Gupta).

are concerned with the discretised form obtained by the well-known discrete ordi-nates (or Sn) method. This note presents certain investigations on the use of theconjugate gradient (CG) method to solve such discretised equations. The set of lin-ear equations can be written as:

Ax ¼ b ð1Þ

where A is a square matrix, x is an unknown vector and b is a known vector. Thereexist many CG-like methods for solving Eq. (1) depending on the nature of matrix A(Saad, 1996). There is a standard CG method listed in the Appendix (Hestens andSteifel, 1952), applicable for symmetric positive definite (SPD) matrix, which issuperior with respect to computational efficiency and memory requirements. All theCG-like methods are primarily based on repeated calculation of the vector which isa product of matrix A with another vector.We consider the steady state ‘source problem’ in neutron transport theory over a

finite region using single energy group. In this problem, a specified steady ‘externalsource’ of neutrons is supposed to be present in the system and the aim is to evaluatethe resulting steady neutron fluxes. This problem is of a fairly general interestbecause many other problems such as multi-group source problems, K-eigenvalueproblems and time-dependent problems are often solved in terms of repeated solu-tion of such a mono-energetic source problem. The transport equation for the onegroup source problem is:

�r r;�ð Þ þ St rð Þ r;�ð Þ � Ss rð Þ

ðf r;�0 ! �ð Þ r;�0ð Þd�0 ¼ qðr;�Þ ð2Þ

Here (r, �) denotes angular flux at a point r in a direction �. q (r, �) is the spaceand direction dependent known external source. �t and �s are the macroscopic totaland scattering cross-sections while f is a scattering function. The equation is solvedin the presence of appropriate boundary conditions.Eq. (2) is often discretised in direction by the well-known discrete ordinates (Sn)

method. There are several ways to discretise Eq. (2) in space such as the finite dif-ference method, the finite element method, the method of characteristics and thenodal methods. In general, the discretisation leads to a set of linear algebraic equa-tions represented by Eq. (1) where A is a square matrix of order N.M, where N isnumber of spatial degrees of freedom and M is the number of discrete directions.The vector x has N.M elements which are the unknown angular fluxes. The elementsof vector b are external sources in various directions.One of the approaches of using CG-like methods to solve Eq. (2) is based on an

explicit construction of the relevant matrix A occurring in Eq. (1). The matrix Awould be, in general, non-symmetric because the transport operator (in particular,the streaming term) is not self-adjoint. As a consequence of this, the standard CGmethod which is valid for SPD matrices, cannot be used to solve Eq. (1). One has totake recourse to other methods (Oliveira and Deng, 1998) such as conjugate gradientsquared (CGS) method, generalised minimum residual (GMRES) method etc.,

1934 A. Gupta, R.S. Modak /Annals of Nuclear Energy 29 (2002) 1933–1951

which are valid for non-symmetric matrices. These methods, however, usually needmore computational effort and memory than the CG method. The symmetrisa-tion of the transport operator including use of the second order self-adjointforms of transport equation have been discussed by Vladimirov (1963) andrecently by Morel and McGhee (1999). The discretisation of second order formsleads to SPD matrices.The present paper is based on a quite different approach of using the CG-method,

which has the following features:

1. The fact that the ‘‘removal operator’’ made up of the first two terms on theleft-hand side (LHS) of Eq. (2) reduces to a triangular matrix, called T, formany discretisation schemes for certain ordering of the variables is exploited.Then an equation of type T x=y, where x is the unknown vector can besolved easily because x=T�1 y can be evaluated by backward substitution.This is actually the mesh-angle sweep procedure commonly used in the con-ventional source iteration (SI) method (Carlson and Lathrop, 1968).

2. In the case of isotropic scattering, it is possible to ‘‘formally’’ reduce Eq. (2)to the form Ax=b as in Eq. (1), where vector x contains ‘‘total fluxes’’ ratherthan angular fluxes. The matrix A then has a much smaller size than in theearlier approach.

3. The matrix A is not explicitly constructed. As will be seen in Sections 2 and 3,A can be expressed in terms of T�1 and hence the effect of matrix A on anyvector can be found easily by employing the mesh-angle sweeps. This is suf-ficient to implement the CG-method to solve Ax=b.

The present note is concerned with some investigations on the use of the CG-method implemented in this manner with the diamond difference scheme used forthe discretisation. The plan of this paper is as follows. In Section 2, the commonlyused SI method to solve the transport equation is described. In Section 3, the earlierwork on the use of the CG-method is quoted and the investigations presented in thispaper are described. The actual studies are presented in Sections 4–8. Section 9 givesthe overall conclusions.

2. Source Iteration (SI) method

Although the method is fairly general, we restrict our attention to isotropic scat-tering, which is of concern in the present studies. In the case of isotropic scattering,the scattering function f in Eq. (2) is unity. Then, by transferring the scattering termto the right hand side (RHS), Eq. (2) can be schematically written as:

T ¼ �L þ q ð3Þ

where the operators T, � and L have appropriate meaning. L equals the total flux�. T is streaming plus collision operator. � denotes the effect of multiplying by

A. Gupta, R.S. Modak /Annals of Nuclear Energy 29 (2002) 1933–1951 1935

scattering cross-section Ss Pre-multiplying both sides of above equation by theoperator LT�1 we get:

L ¼ LT�1�L þ LT�1q ð4Þ

Replacing L by �, we get:

� ¼ LT�1��þ LT�1q ð5Þ

Let Eq. (2) be discretised in space by finite differencing and in direction by the discreteordinates (Sn) method, in a manner similar to the well-known DTF-IV code (Lathrop,1965). In particular, let us consider use of the popular ‘‘diamond difference scheme’’.Let us assume that Eqs. (3)–(5) represent the corresponding discretised forms also.Then, the continuous functions are replaced by vectors. The operators are matrices.The SI-method (Carlson and Lathrop, 1968; Lewis and Miller, 1984) to solve the

discrete form given by Eq. (5) is as follows. Denoting LT�1 by K and iterationnumber by n, the method can be described as:

�nþ1 ¼ K��n þ Kq ð6Þ

A guess value (say zero) is assumed for the vector �1. Using this, scattering sourcedensity vector ��1 is evaluated. The effect of operator K on (��1+q) has to be foundout. This needs evaluation of the effect of T�1 as K is equal to LT�1. This is done bythe well-known sweeping procedure based on principle of directional evaluation(Carlson and Lathrop, 1968). This gives all the angular fluxes. From this total fluxesare computed which gives �2. Then, using this value of flux, RHS is re-evaluated.This gives �3. The iterations are continued till scattering source converges. Theseiterations converge if and only if (Varga, 1962):

� K�ð Þ<1:0 ð7Þ

where � denotes the spectral radius.Menon and Sahni (1985) have shown the convergence of this iteration scheme by

explicit construction of iteration matrix for the finite differenced Sn-equations. Inparticular, they have shown that, for a slab geometry with isotropic scattering andvacuum boundary conditions, the eigenvalues of K� are all real. Their results arevalid for any positive weighted difference scheme for any material heterogeneity andnon-uniform mesh. This includes the diamond differencing scheme also if the case issuch that negative fluxes do not occur.It is known from theoretical and computational analysis that the source iterations

converge very slowly in thick highly scattering regions where loss of neutrons byleakage or absorption is very small. A very large number of iterations are neededand the computational effort is then very large. The most common methodsemployed to improve the convergence have been coarse mesh rebalancing and dif-fusion synthetic acceleration (DSA) (Lewis and Miller, 1984).


3. Use of the CG-method

The CG-method has been used to develop schemes which are more efficient thanthe SI-method. The basic idea is to write Eq. (5) in the form:

ðI - K�Þ� ¼ Kq ð8Þ

so that it is of the form Ax=b. The effect of coefficient matrix (I–K�) on any guessvector can be found using the mesh-angle sweeps mentioned in Section 2. Hence theCG method can be applied. The earlier work on the use of the CG-method in thisway is as follows.

3.1. Earlier work

One of the earliest papers that considered the use of the CG method to solve thetransport equation was Lewis (1977). He considered finite differenced Sn-equationsfor 1-D slab cases. In general, the coefficient matrix (I–K�) in Eq. (8) need not besymmetric. Then the standard CG method (see Appendix) by Hestens and Steifel(1955) valid for SPD matrices cannot be applied to Eq. (8). The coefficient matrixwas symmetrised by pre-multiplying Eq. (8) by (I–K�)T. This gives:

ðI-K�ÞTðI-K�Þ� ¼ ðI-K�ÞTKq ð9Þ

The coefficient matrix in Eq. (9) is always SPD and hence the CG-method wasused to solve it. Lewis has, however, mentioned that the convergence behaviour isnot satisfactory. It is not like that of a normal iterative method. This might bebecause the condition number of the coefficient matrix in Eq. (9) is larger than thatin Eq. (8) and this has an adverse effect on convergence. It may also be noted that,with Eq. (9), for each CG-iteration the sweeping procedure implicit in the operatorK has to be performed twice.Faber and Manteuffel (1989) have used the finite element method for spatial dis-

cretisation and the Sn method for directions. A 1-D homogeneous slab with uniformmesh containing uniform source was considered. With these simplifications, thecoefficient matrix (I–K�) in Eq. (8) is SPD and no symmetrisation is required. TheCG-method was used to solve it. As is known, each CG iteration involves pre-mul-tiplying a guess vector by coefficient matrix (I–K�). This was done by using thesweeping procedure mentioned earlier. The vector on the RHS in Eq. (8) has to beevaluated once in the beginning by finding the effect of K on the known externalsource density vector q, by using the sweeping procedure. Faber and Manteuffel(1989) have shown that the CG-method applied to Eq. (8) can reduce the number ofiterations substantially. They have also given a theoretical explanation for this.Recently, some studies have been reported (Zika and Adams, 2000a,b and refer-

ences therein) on the use of the CG-method to implement the so-called transportsynthetic acceleration (TSA) scheme. The scheme is analogous to the DSA scheme.In the TSA scheme, one considers a ‘‘low order equation’’ based on transport theory


rather than the diffusion equation as in the DSA scheme. The actual problem caninvolve isotropic or anisotropic scattering and is primarily solved by the SI itera-tions. However, between two SI iterations, the low order equation, which is a kindof corrective equation, is solved and this solution is used to accelerate the con-vergence of SI iterations. The low order equation is always taken as an isotropicscattering problem so that the CG method can be used (Ramone et al., 1997) tosolve this equation efficiently. This, in turn, makes the overall scheme more efficientthan simply using the SI method alone.Since the low order equation is based on isotropic scattering, it would be of the

form of Eq. (8) and the coefficient matrix is not symmetric if scattering cross-sectionand mesh size vary with space. Then the standard CG algorithm cannot be directlyused. Ramone et al. (1997) have tackled this problem as follows. The vectors ri, pi etcoccurring in the standard CG algorithm (see Appendix), when applied to Eq. (8),contain a number of elements equal to the number of space variables or meshes. Theinner products of these vectors occurring in the algorithm are re-defined as weightedinner products (Saad, 1996), the weighting being done by respective mesh volumeand scattering cross-section of the material in the mesh. Then the standard CGalgorithm can be used even if A is non-symmetric. This procedure is based on theobservation that the operator in Eq. (8) is self-adjoint under an inner product whichis a volume integral weighted by the scattering cross-section �s(r). This procedure ofapplying the CG-method is referred to as CG(IPR) in this paper, IPR signifying theinner-product redefinition.Ramone et al. (1997) have used the above procedure with the step characteristic

and discontinuous finite element discretisations. They have solved external sourceproblems with isotropic scattering. Subsequently, Zika and Adams (2000a) used thisscheme with the long characteristic method for discretisation. They considered cri-ticality problem for a single ‘‘fuel assembly’’ in a nuclear reactor in which oppositereflective boundary conditions are used.

3.2. Present work

In the context of the earlier work described above, the main features of the workpresented here are as follows:

1. The use of the CG-method is analysed for the classical finite difference dis-cretisation of Sn-equations based on the well-known ‘‘diamond difference’’scheme. This widely used scheme is attractive because it is very easy toimplement and is also second order accurate (Lewis and Miller, 1984).Moreover, it has asymptotic diffusion limit (Larsen et al., 1983).

2. A simple scheme is suggested to symmetrise the coefficient matrix A in Eq. (8)so that the standard CG-method can be used. This scheme is calledCG(SYM) and is different than the one mentioned by Lewis (1977).

3. The symmetrisation procedure in CG(SYM) is compared with the alter-native equivalent procedure called CG(IPR) used by Ramone et al. (1997)where instead of symmetrising the coefficient matrix, the inner products


are re-defined. The later method was found to have better convergenceproperties. This observation may be of interest to users of the CG-method.

4. Primarily, the use of the CG-method to directly solve the discretised equa-tions for practical heterogeneous problems is considered in which there areno SI-iterations. This is different than using the CG-method for solving theTSA equations which in turn accelerates the SI-iterations.

5. The diamond differencing scheme occasionally leads to the problem ofoccurrence of unphysical negative fluxes. This is more likely to happen inregions with high absorption having negligible scattering or other types ofsources. It is found that the CG-method cannot be directly used in such cases.Some recipes to make use of the CG-method in such case are described.

6. In the case of global criticality calculation for a nuclear reactor (or any sub-critical system) modeled by homogenised fuel assemblies, there exists a fairlywell distributed neutron source arising from fissions or group-to-group neu-tron scattering. As a result, the occurrence of negative flux is less likely. It isshown that the CG-method can then be directly used in place of SI-methodfor an efficient solution.

In Section 4, the symmetrisation scheme CG(SYM) for the coefficient matrix isdescribed. In Section 5, the scheme CG(SYM) is applied to a realistic 1-D sourceproblem and is shown to be much more efficient than the SI method. In Section 6,the two apparently similar schemes CG(SYM) and CG(IPR) are compared. In Sec-tion 7, use of the CG-method for a full reactor K-eigenvalue test problem is descri-bed. In Section 8, the problems arising from negative flux are discussed with possiblerecipes to handle them.

4. Proposed CG(SYM) method

It will be shown that the coefficient matrix (I–K�) in Eq. (8) is non-symmetricbecause of two reasons: (i) the scattering cross-section changes in space due tomaterial heterogeneity and (ii) the mesh is non-uniform. The symmetrisation proce-dures for these two situations are described separately. The discussion is specific tothe diamond differenced Sn discretisation scheme.

4.1. Heterogeneous system with uniform mesh

Let us first consider a system which is spatially heterogeneous so that �s canchange spatially but all the meshes are of same size.It can be seen that K is symmetric as follows. As can be seen from Eq. (8), K

operates on the isotropic scattering source density vector (��) and gives the uncol-lided total fluxes in different meshes. It is known from reciprocity relations (Bell andGlasstone, 1970) that the total flux at point r1 due to a unit isotropic source atanother point r2, is equal to the total flux at r2 due to a unit isotropic source at r1. Asa discrete analog of this, one expects that the average flux at the ith mesh due to unit


isotropic source in the jth mesh is equal to the average flux at the jth mesh due tounit isotropic source in the ith mesh. It may be mentioned that the Sn-method,which is used here for angular discretisation, obeys this reciprocity relation for thequadrature coefficients, which are invariant under geometric transformations (Carl-son and Lathrop, 1968, p. 197). For uniform meshes, the discrete reciprocity rela-tions clearly imply that K is symmetric.K�, however, need not be symmetric. The reason is as follows. The operator � is a

diagonal matrix of order equal to the number of spatial variables. Its elements arethe scattering cross-sections in various meshes. For a spatially homogeneous pro-blem, all the diagonal elements of � are equal and it is a scalar matrix. But for aheterogeneous problem, different materials in general can have different scatteringcross-sections. As a result, all the elements of diagonal matrix � are not identicalwhich can lead to non-symmetricK�. This is the reason why (I–K�) is non-symmetric.Suppose there are no regions having zero scattering cross sections in the problemdomain. Then � would be a diagonal positive definite non-singular matrix and ��1

exists. Eq. (8) is written in the form:

ð��1�KÞð��Þ ¼ Kq ð10Þ

Let us consider the unknown vector to be (��) rather than �. Then, the coefficientmatrix in Eq. (10) is (��1–K). Since both ��1 and K are symmetric, (��1–K) is alsosymmetric. Its positive definiteness can be shown as follows.Since K is symmetric and � is SPD, K� has real eigenvalues. Moreover from Eq.

(7), all eigenvalues of K� are less than unity in magnitude. As mentioned earlier, thishas also been shown by Menon and Sahni (1985) for the finite difference scheme.This implies that (I–K�) has all real positive eigenvalues.(��1–K) is related to [��1/2 (��1–K) �1/2] by similarity transformation. Hence both

have the same eigenvalues. The later matrix is congruent to (I–K�) as follows.

ðI � K�Þ ¼ �1=2½��1=2ð��1�KÞ�1=2�1=2 ð11Þ

Hence the number of positive eigenvalues of [��1/2 (��1�K) �1/2] are equal to that of(I�K�) (Axelsson, 1994). But we have seen that (I�K�) has all positive eigenvalues.As a result, [��1/2 (��1�K) �1/2] and hence (��1�K) has all real positive eigenvalues.Thus coefficient matrix (��1�K) in Eq. (10) is SPD and hence the CG-method can

then be used to solve it to obtain the unknown vector (��), from which the requiredsolution � can be recovered easily.

4.2. Effect of non-uniform mesh

If the mesh size is not uniform, the operator K, which relates source density in amesh with flux in any other mesh, is not symmetric in the light of discrete form ofreciprocity relations mentioned earlier, which relate total source (rather than sourcedensity) in a mesh with flux in some other mesh. Hence the coefficient matrix in Eq.(10) is non-symmetric. However it can be symmetrised as follows. Let V be a diagonal


matrix with elements v1, v2,. . ., vN which are the mesh volumes of all the N meshes.Then, if both sides of Eq. (10) are pre-multiplied by V, we get

V�ð��1 � KÞ ð��Þ ¼ V�Kq ð12Þ

The coefficient matrix is now symmetric. Further, since V is obviously SPD,V(��1�K) has all real positive eigenvalues like (��1�K). Hence the CG-method canbe used to solve the equations.

5. Utility of CG(SYM) over the SI-method

Computer programs were developed to solve the mono-energetic external sourceproblems using the SI method and the CG(SYM) method. All the computationswere done on an Intel 667 MHz Pentium-III processor. In both the CG and SI-methods, the following relative point convergence criterion is used:

" ¼ maxi

�nþ1i � �ni

�� ni�� 4 "0 ð13Þ

where i denotes mesh number and n is iteration numberWe compare the CG(SYM) method with the SI-method for a heterogeneous 1-D

slab test case close to a realistic problem. There are four regions with iron in thethird region and water elsewhere, as shown in Fig. 1. A uniform isotropic source ofneutrons of total strength unity is present only in the first region which extends up to12 cm. The resultant fluxes are to be found out. The mesh sizes in water and iron aredifferent. There is a vacuum boundary on both sides. This case was chosen becauseit involves substantial heterogeneity in cross sections and mesh size and the fluxvaries significantly; in fact, it falls by a factor of 105. A similar test case has beenused by McCoy and Larsen (1982) for analysis.We consider a total of 88meshes, the region-wise numbers being 40, 10, 8 and 30. The

S4 gauss quadrature set is used. "0 gives the relative point-wise convergence in succes-sive iterations. The same value of "0 in the CG and SI-methods does not imply that theabsolute error in computed fluxes is the same. To make a proper comparison of the SI

Fig. 1. 1-D heterogenous iron–water test problem with cross sections in cm�1.


and CG methods, the following procedure was used. The CG(SYM) calculation wasdone with a convergence criterion of "0=10�7. Then many calculations by the SI-method were done with gradually tightened convergence criteria varying from "0=10�6

to "0=10�10 and the fluxes at these convergence levels are noted. It is found that thefluxes given by the SI-method gradually approach those by the CG-method. This canbe clearly seen from Table 1 in which the fluxes at some selected meshes are listed. Thenumber of iterations needed are also given in the table. The convergence level of the SI-method of "0=10�10 gives fluxes very close to those by the CG(SYM) method with"0=10�7. Hence, the number of iterations needed by these two cases are compared. It isseen that, with the CG-method. the number of iterations are reduced by about 24 times.In the above case, the ratio (�s/�t), denoted by c, was about 0.994 and 0.87 for

water and iron regions. The above computations were repeated by modifying thetwo c-values to 0.999. Then the reduction in number of iterations by CG(SYM) isabout 100 times. This is because the SI-method converges slowly when c tends tounity and hence the speed-up obtained by CG(SYM) method is higher. On the otherhand, in highly absorbing media, with small c, the SI-method converges rapidly andbenefits of the CG-method would be less.The above problem was solved with higher quadratures up to S12 also. Similar

reduction in the number of iterations was obtained. All the above results have beenverified against the standard code DTF-IV (Lathrop, 1965).

6. Comparison of CG(SYM) and CG(IPR)

The scheme CG(SYM) discussed in Sections 4 and 5, converts the coefficientmatrix to an SPD form. Subsequent implementation of CG-iterations then producesresidual vectors ri (see Appendix) which are mutually orthogonal. The CG(IPR)method, described in the earlier work in Section 3, does not symmetrise the coeffi-cient matrix but considers the inner products in the CG-algorithm, to be weighted by

Table 1

Comparison of the CG(SYM) and SI-methods for a 1-D heterogeneous test case

Mesh number at

each zonal boundary

Computed fluxes (�) at

selected meshes

CG(SYM)

"0=10�7

SI-method

"0=10�6 "0=10�8 "0=10�10

1 0.5743411 0.5742614 0.5743403 0.5743411

41 1.8562882 1.8555772 1.8562810 1.8562881

51 0.1251067 0.1250346 0.1251060 0.1251067

59 1.0660161�10�3 1.0651511�10�3 1.0660075�10�3 1.0660163�10�3

88 6.0712178�10�6 6.0191655�10�6 6.0706105�10�6 6.0712217�10�6

No. of iterations 105 1140 1784 2430


mesh volume and scattering cross sections. The implementation of the CG-iterationsthen generates residual vectors ri which are orthogonal only with respect to therevised weighted inner product. Thus, in general, the two approaches produce dif-ferent iterates. Hence it is of interest to compare their numerical convergence beha-viour. A computer program was developed to implement the CG(IPR) method forfinite-differenced Sn equations, with matrices V and �, defined in Section 4, used asweighting factors for inner product.

6.1. 1-D test cases

The CG(SYM) and CG(IPR) schemes are compared for the 1-D test case (Fig. 1)for which CG(SYM) was compared with the SI-method in Section 5. A total of 88meshes and S4 Gauss quadrature was considered. The convergence level " reachedby the two methods is plotted against the number of iterations in Fig. 2. TheCG(IPR) scheme reaches almost complete convergence ("=10�15) in 88 iterations.This is consistent with the theoretically known fact that the CG method gives exactanswer in at the most N (=order of matrix, here 88) iterations. It is also known thatthis may not be always true in practice due to propagation of rounding errorsresulting from finite precision. This is what is found to happen in case of theCG(SYM) scheme which requires more than 88 iterations to reach convergence levelof "=10�15. The fluxes computed by the CG(SYM) scheme are found to graduallyapproach those given by the CG(IPR) scheme in 88 iterations.Apart from the case just described, some additional 1-D test cases containing

three types of materials were designed with different values of �t, �s and mesh sizes.In all cases, the CG(IPR) scheme was found to have a better convergence thanCG(SYM).

Fig. 2. Convergence of CG (IPR) and CG (SYM) with iteration number for the 1-D iron–water test case.


6.2. 2-D test case

There is a well-known iron–water benchmark problem (Waering et al., 1994)studied by many researchers. Fig. 3 gives a full description of the benchmark. Thisbenchmark, when analysed by the SI-method, needs negative flux fix-up. In such acase, the CG-method cannot be used in a straightforward manner and this aspectwill be fully discussed in Section 8. Here, we want to compare the two CG schemesfor a 2-D case without negative flux complications. For this purpose, we modifyslightly the benchmark problem. A uniform external neutron source is assumed tobe present in all regions instead of the localised source present in the actual bench-mark, as shown in Fig. 3. With this modification, no negative flux fix-up is requiredin the SI-method and the CG-methods also work well. For this particular case, thetwo schemes CG(SYM) and CG(IPR) were tested. The convergence level " reachedis observed against number of iterations. The CG(IPR) scheme reaches "=10�15 inless than 200 iterations while CG(SYM) needs about 690 iterations.From all the above numerical experiments, it is seen that the CG(IPR) scheme

exhibits better convergence than the symmetrisation scheme. As a result, theCG(IPR) method will be used in further analysis in this paper. Section 7 gives directuse of CG(IPR) for the K-eigenvalue problem and Section 8 presents studies on theoccurrence of negative fluxes.

7. CG-method for K-eigenvalue problems

The direct use of the CG(SYM) or CG(IPR) method is possible in those caseswhich are not prone to the occurrence of negative fluxes in the SI-method. The

Fig. 3. The 2-D iron–water test case.


K-eigenvalue problem for full nuclear reactor (or any sub-critical system) containinghomogenised fuel assemblies may offer such a possibility. This problem is usuallysolved in terms of repeated solution of many external source problems. This is the well-known inner–outer iteration technique. The inner iterations are required to solve theexternal source problems, where the external sources contain the fission source in addi-tion to the scattering source. Because of the presence of a widely distributed fissionsource, the negative fluxes are less likely to occur than for the source problems in non-multiplicative media. The use of the CG-method will hasten each source calculation andhence the K-eigenvalue problem will be solved faster. One such test case is studied here.A one-group K-eigenvalue problem was solved for a 2-D X–Y geometry global

model of a nuclear reactor (Barros et al., 1999). There are three types of fuel regionssurrounded by reflector, as shown in Fig. 4. The problem has quarter core sym-metry. In both the directions, 105 uniform meshes of 1.0 cm length were taken. TheS4 fully symmetric quadrature set was used. The K-eigenvalue was computed byusing the SI-method as well as the CG(IPR) method. It was found to be 0.9619. Thiscompares well with the reference value 0.9622 obtained by Barros et al. (1999) withmuch finer mesh structure. The CG-method takes about 16 s of CPU time which ismuch less than that about 480 s taken by the SI-method.It may be mentioned that a 2-group K-eigenvalue problem for a PHWR core was

also analysed. It simulates the initial state of benchmark problem-1 given in McDon-nell et al. (1977). Again the CG-method was found to give a similar speed-up.

Fig. 4. The 2-D test case for the K-eigenvalue problem with three fuel zones and reflector.


8. The negative flux problem

As mentioned in Section 3, the diamond difference scheme can occasionally pro-duce an unphysical negative value of angular flux in the SI-method. In this method,the negative flux problem is treated as soon as it occurs during the mesh-anglesweeps (Lewis and Miller, 1984). For this purpose, either the negative flux fix-up(set-to-zero) scheme is used or one resorts to the step scheme for that mesh. Finally anon-negative solution is obtained. An important observation is that, in case of thoseproblems which need negative flux fix-up in the SI-method, the direct use of the CG-method as described in Sections 5–7, leads to the occurrence of some negative valuesof total flux in the final solution. For instance, this happens in the case of the 2-Diron–water benchmark described in Fig. 3. Such negative fluxes are clearly notacceptable. Further difficulty is that, the procedure of negative flux fix-up followedin the SI-method, cannot be used in the CG-method for the reasons explainedbelow.When the CG-method is used to solve the equation Ax=b, the vector Ap (see

Appendix) has to be repeatedly evaluated. The vector p, which is found from resi-dual vector r, can in general, have both positive and negative elements. In thetransport problem, finding Ap involves finding T�1 p. As explained in Section 2, thisinvolves carrying out mesh-angle sweep which gives the uncollided fluxes resultingfrom the presence of source vectors p containing both positive and negative ele-ments. Thus, these intermediate fluxes obtained by mesh-angle sweep can be justifi-ably positive as well as negative because the source vector p can contain positive aswell as negative elements. Hence, it is not possible to impose the negative flux fix-upprocedures during the sweeps as in the SI-method.Thus, a question arises as to how to make use of the good convergence properties

of the CG-method for those problems which need negative flux fix-up in the SI-method. One simple solution is to refine the mesh sufficiently so that negative fluxesdo not occur at all. However, this can be too costly. Another remedy was tried asfollows. The vector p is expressed as the difference of two vectors: p=p1�p2. Vectorp1 contains all positive elements of p and zero values elsewhere. p2 Contains absolutevalues of all negative elements of p and zero values elsewhere. Thus, both p1 and p2contain non-negative elements. Then Ap1 and Ap2 are evaluated by using the con-ventional negative flux fix-up procedure during the sweeping. Finally Ap=Ap1-Ap2,required in CG-iterations is found. However, such a scheme was found to beunstable.In what follows, two recipes are suggested which were found useful to make use of

the CG-method even in cases prone to the occurrence of negative fluxes.

8.1. Use of the TSA scheme

The TSA scheme proposed by Ramone et al. (1997) can be used to handle theproblems arising from negative flux. The problem is primarily solved by the SI-method; however, after few SI-iterations, a ‘‘corrective TSA equation’’ is formulatedand this equation is solved by the CG method. The SI-iterations do involve the


conventional negative flux fix-up procedure while CG-iterations are carried out asusual. The CG-iterations are useful to accelerate the overall convergence.The above procedure was applied to the 2-D iron–water benchmark problem

(Fig. 3) analysed by many investigators (Waering et al., 1994). The cross-sections ofiron and water are the same as in the 1-D problem described in Fig. 1. There is aunit-localised source of neutrons and the resulting flux distribution is to be foundout. A total of 88 meshes are used in the X and Y direction as shown in Fig. 3. TheS4 fully symmetric quadrature set is used. This problem, when solved by the SI-method, needs negative flux fix-up. The direct use of the CG(IPR) method leads tosome unphysical negative total flux values in the final solution. The TSA scheme wasapplied to handle this problem. After every 5 SI-iterations, the corrective TSAequation was solved by the CG(IPR) method. The TSA equation was solved with alower order S2 quadrature set. The number of CG-iterations for solving the TSAequation was restricted to 5. With this procedure, the problem was solved in a CPUtime of about 11 s for "0=10�6. No negative fluxes were found in the solution. If theSI-method alone with negative flux fix-up is used, a larger CPU time of about 64 s isrequired. The results are presented in Table 2.The region-wise fluxes are also shownin Table 2 and compare well with the reference values taken from Waering et al.(1994).

8.2. Use of weighted diamond difference (WDD)

If the step-difference scheme is used, which is known to be positive (Carlson andLathrop, 1968), the negative fluxes do not arise in the SI-method and then the CG-method can be used in a straightforward manner. However, the step scheme is notused because it leads to poor accuracy. It is found that if the diamond differencingrelations are slightly modified, the CG method works well and a positive solution,with reasonable accuracy, is obtained.To be specific, the WDD relation between angular fluxes at the two mesh edges in

x-direction and at the mesh center for the ith mesh can be written as:

i;m ¼ w i-1=2;m þ ð1� wÞ iþ1=2;m ð15Þ

Table 2

Comparison of WDD and TSA recipes with the SI-methodfor the 2-D iron–water test case

Average flux by regions (cm�2 s�1) ("0=10�6)

Zone no. Waering (1994) CG(IPR) with WDD TSA SI-method with flux fix-up

1 40.93 40.75 40.99 40.99

2 9.31 9.96 9.92 9.91

3 0.229 0.226 0.210 0.210

4 1.91�10�3 1.91�10�3 1.55�10�3 1.55�10�3

CPU times (s) – 7.31 11.37 64.60


The standard diamond difference scheme is equivalent to using w=0.5. This ismodified so that more weightage is given to the downstream edge. Thus, forinstance, when m is positive, w is chosen to be 0.49. For negative m, w is taken as0.51. Similar weights are taken for the y-direction also. Using such weighted dia-mond relations, the 2-D iron–water test problem (Fig. 3) mentioned above wassolved by the CG(IPR) method. Then all the computed fluxes are found to be posi-tive. As shown in Table 2, the problem is solved in a CPU time of about 7 s which ismuch smaller than the usual SI-method with negative flux fix-up. The computedaverage zonal fluxes are also satisfactory.It may be mentioned that the method is not rigorous and the choice of w is to

some extent arbitrary. To start with, values of w between 0.5 and 0.495 were tried.This gives negative fluxes in the final solution. Hence, smaller values of w namely0.495, 0.490 and 0.485 were used. The results are presented in Table 3. It is seen thatthe results are not too sensitive to slight variations in w. It can be expected that as wdeviates more and more from 0.5, the truncation errors would be larger. With avalue of w close to 0.5, reasonably good results seem to be obtained.

9. Conclusion and discussion

Some studies have been presented on the use of conjugate gradient methods inneutron transport problems with isotropic scattering. The discretisation based onthe diamond difference scheme was considered because it is easy to implement and isaccurate. Basically the idea is to use the CG-method directly in place of the con-ventional SI-method. A simple procedure is suggested for the symmetrisation of therelevant coefficient matrix. It is shown that the CG-method provides a good speed-up. It should be noted that the coefficient matrix is not explicitly formed. Rather, itseffect is considered through the well-known mesh-angle sweeps.For the above type of problems, there exists an alternative mathematically

equivalent approach (Saad, 1996) in which one re-defines the inner products whilethe coefficient matrix is unchanged. Ramone et al. (1997) have also used thisapproach. Numerical analysis was done using some test cases to compare these twoapparently similar approaches. It was found that numerical convergence of the laterapproach is better. This result may be of interest to users of the CG-method.

Table 3

Average zonal fluxes with different weight factors w for the 2-D iron–water test case

Average flux by regions (cm�2 s�1)

Zone No. w=0.48 w=0.485 w=0.49 w=0.495

1 40.50 40.62 40.75 40.88

2 9.98 9.97 9.96 9.95

3 0.242 0.234 0.226 0.218

4 2.37�10�3 2.13�10�3 1.91�10�3 1.70�10�3


The diamond difference scheme suffers from the problem of occasional occurrenceof negative flux in the conventional SI-method. It was found that in such cases theuse of the CG-method leads to unphysical negative fluxes. Further, it is seen that theflux fix-up procedures used in the SI-method cannot be used in the the CG-method.Thus, it is necessary to look for some recipes to make use of CG-method in suchcases. It is shown here that the technique of using the CG-method through thetransport synthetic acceleration (TSA), developed by Ramone et al. (1997), can beused to handle such cases. This is an additional application of the TSA scheme. Theuse of weighted diamond differencing is also shown to be useful in such cases. It isplanned to further study both these recipes for a wide variety of problems.In the case of solution of the K-eigenvalue problem for a system such as a nuclear

reactor containing homogenised fuel assemblies, negative fluxes may not occur dueto the presence of widely distributed strong fission source in addition to the scatter-ing source. The direct use of the CG-method is shown to be very efficient in suchcases.The case of anisotropic scattering is not covered in this paper. It is felt that the

technique of using the CG-method in the corrective TSA equation, which hasalready been used here to handle the negative flux problems, can also be used forsolving problems with anisotropic scattering. In fact, Ramone et al. (1997) haddeveloped the technique precisely to solve anisotropic scattering problems. Ofcourse, they used step characteristic and discontinuous finite element schemes, whichare not prone to negative flux problems. It is planned to study anisotropic scatteringcases in a subsequent work.It is well known that an isotropic scattering problem can be formulated in terms of

total flux alone as in the present paper. A scheme was suggested here to symmetrisethe relevant coefficient matrix. An extension to anisotropic scattering problemswould involve not only the total flux but also some higher order moments dependingon the degree of the anisotropy. It is not known how to symmetrise the relevantcoefficient matrix. One may have to use CG-like methods for non-symmetric matri-ces.In the present work, we do not intend to make any comparison with the diffusion

synthetic acceleration (DSA) technique. Ramone et al (1997) have shown that theuse of the CG-method through TSA is somewhat slower than the DSA technique insome cases. But, the direct use of the CG-method as well as its use through TSA ismuch more easy to implement compared to the DSA. Moreover as mentioned byRamone et al., one need not worry about the consistency of discretised transportand diffusion equations required in the DSA.In the approach used in this paper, the coefficient matrix is not explicitly con-

structed. Hence, it is not straightforward to construct effective pre-conditioners(Saad, 1996) such as incomplete LU decomposition or Gauss–Seidel type, which areobtained from the coefficient matrix.Curvilinear geometries are not discussed here. In the case of spherical geometry,

the reciprocity relation between isotropic source and total flux is not exactly repro-duced by the Sn method (Modak et al., 1995; Carlson and Lathrop, 1968, p. 253).This wouldmake the operatorK slightly non-symmetric and symmetrisation procedure


is not known. Perhaps, the use of the CG-method through the TSA scheme may beuseful in such cases.

Acknowledgements

The authors are grateful to the referee for many useful suggestions and for bring-ing to our notice related work by Zika and Adams. Thanks are due to Drs. D.C.Sahni, S.R. Dwivedi and R. Srivenkatesan for their encouragement. We are alsothankful to Drs. S.V.G. Menon, H.P. Gupta and Vinod Kumar for useful comments.

Appendix. The standard CG algorithm to solve Ax=b

1. Compute r0:=b- Ax0, p0 :=r0.2. For j=0,1,. . ., until convergence, Do:3. �j :=(rj, rj)/(Apj, pj)4. xj+1:=xj+�jpj5. rj+1 :=rj-�jApj6. �j:=(rj+1, rj+1)/(rj, rj)7. pj+1 :=rj+1+�jpj8. End Do

References

Axelsson, O., 1994. Iterative Solution Methods. Cambridge University Press, New York.

Barros, R.C., et al., 1999. Prog. Nucl. Energy. 35, 293.

Bell, G.I., Glasstone, S., 1970. Nuclear Reactor Theory, Van Nostrand Reinhold, New York.

Carlson, B.G., Lathrop, K.D., 1968. Transport theory—the method of discrete ordinates. In: Greenspan,

H., Kelber, C.N., Okrent, D. (Eds.), Computing Methods in Reactor Physics. Gordon and Breach,

New York.

Faber, V., Manteuffel, T.A., 1989. In: Nelson, P. et al., (Eds.). Transport Theory, Invariant Imbedding,

and Integral Equations, Marcel Dekker Inc., New York and Basel.

Hestens, M.R., Steifel, E., 1952. J. Res. Nat. Bur. Standards Sect., B 49, 409.

Larsen, E.W., Morel, J.E., Miller, W.F., 1983. J. Comp. Phys. 69, 281.

Lathrop, K.D., 1965. DTF-IV, A Program for Solving the Multigroup Transport Equation with Aniso-

tropic Scattering, LA-3373.

Lewis, E.E., 1977. Trans. Am. Nucl. Soc. 26, 225.

Lewis, E.E., Miller Jr., W.F., 1984. Computational Methods of Neutron Transport. John Wiley & Sons,

New York.

McCoy, D.R., Larsen, E.W., 1982. Nucl. Sci. Eng. 82, 64.

McDonell, F.N., et al., 1977. Nucl. Sci. Eng. 64, 95.

Menon, S.V.G., Sahni, D.C., 1985. Transport Theory Stat. Phys. 14 (3), 353.

Modak, R.S., Sahni, D.C., Paranjape, S.D., 1995. Ann. Nucl. Energy 22 (6), 359.

Morel, J.E., McGhee, J.M., 1999. Nucl. Sci. Eng. 132, 312.

Oliveira, S., Deng, Y., 1998. Prog. Nucl. Energy 33 (1/2), 155.

Ramone, G.L., Adams, M.L., Nowak, P.F., 1997. Nucl. Sci. Eng. 125, 257.

Saad, Y., 1996. Iterative Methods for Sparse Linear System. PWS, Boston, MA.


Varga, R.S., 1962. Matrix Iterative Analysis. Prentice-Hall Inc, New Jersey.

Vladimirov, V.S., 1963. Report AECL-1661.

Waering, T.A., et.al., 1994. Proc. 8th Intl Conf on Radiation Shielding, 24–28 April, Arlington, TX, USA,

p. 201.

Zika, M.R., Adams, M.L., 2000a. Nucl. Sci. Eng. 134, 135.

Zika, M.R., Adams, M.L, 2000b. Nucl. Sci. Eng. 134, 159.


Documents

On the use of the conjugate gradient method for the solution of the neutron transport equation