ACCELERATING OUTER ITERATIONS IN MULTIGROUP PROBLEMS ON K[eff].
Galina Kurchenkova, Viachaslav Lebedev
RRC “ Kurchatov Institute ”, Russia
ABSTRACT
A new cyclic iterative method with variable parameters is proposed for accelerating the outer
iterations in a proposed used to calculate K[eff] in multigroup problems. The method is based
on the use of special extremal polynomials that are distinct from Chebyshev polynomials and
take into account the specific nature of the problem. To accelerate the convergence with
respect to K[eff], the use of three Orthogonal functionals is proposed. Their values
simultaneously determine the three maximal eigenvalues. The proposed method was
Incorporated in the software for neutron-physics calculations for WWER reactors. To
calculate K[eff] for WWER-type reactors, we have incorporated our method in the multigroup
software, namely, two-dimensional programs like PERMAK-A , three-dimensional
programs like PERMAK 3-D , and the TVS-M program . Previously, the iterations in these
programs had been accelerated by the Lyusternik method. Our calculations and a comparison
of about 20 typical versions of the programs have shown the reduction in the Execution time
by a factor ranging from three to seven.
INTRODUCTION
Multigroup problems for determining the multiplication K eff and the corresponding neutron
fields are the basic and most labor-consuming class of problems in neutron-physics reactor
calculations. Mathematically, the problem reduces to solving a partial Eigen value problem,
namely, to finding the maximal Eigen value ( K eff ).
In this paper, we propose a cyclic iterative method with variable parameters for accelerating
the outer iterations in a process used to calculate K eff in multigroup problems. The method is
based on the use of special extremal polynomials that are distinct from Chebyshev
polynomials and take into account the specific nature of the problem.
To accelerate the convergence with respect to K eff , the use of three orthogonal functionals is
proposed. Their values simultaneously determine the three maximal Eigen values.
The proposed method was successfully incorporated in the software for neutron-physics
calculations for WWER reactors.
1. FORMULATION OF THE PROBLEM AND AN INTERATIVE METHOD TO
SOLVE IT
The multigroup system of difference diffusion equations for determining K eff and the group
fluxes of neutrons   ( 1 ,..,  g ) , where g is the number of groups, can be written in the
form
L 

K eff
S
(1.1)
Here L  is the multigroup operator consisting of the difference operators for diffusion,
absorption, and group transitions; U =SФ ,where U =( U1 ,U 2 ,...., U n ) is the fission-source
operator,   (  1 ,  2 ,..,  g ) is the spectrum of the fission-neutrons, n is the number of grid
points in which the solution is sought, and  i  ( i1 ,  i 2 ,..,  in ) , i=1,2,…,g .
Equation (1.1), which determines the Eigen values, is transformed to the standard form
AU  U ,
(1.2)
where A  SL1  . Let K 'эфф = 1 >  2   3  …   n  0 be the nonnegative eigenvalues of А
and 1 ,  2 ,...,  n be the corresponding eigenvectors forming a basis in the space R n .
Our problem is to find the eigenpair ( 1 , 1 ). (Then, we set K 'эфф = 1 ). We assume that
1  0 and set
n
U   U i p 0 ( i ) , p 0 (i )  0 .
i 1
We examine the following cyclic iterative method with the period N and the variable
parametrs (such that  k  N   k ) for determining ( 1 , 1 ):
Given an initial approximation U 0  (U10 ,..., U n0 ), where U i0  0 ,
n
U 0  a101  a 20 2  a30 3   ai0 i ,
(1.3)
i4
and a10  0 , the subsequent approximations
n
U k  a1k 1  a 2k  2  a3k  3   aik  i
(1.4)
i 4
are constructed by the formulas
V k  AU k 1   k U k 1 ,
U k  V k /V k ,
(1.5)
where
n
V k   ( i   k )aik 1 i
k =1,2,…,N .
(1.6)
i 1
Neglecting the intermediate normalizations performed for these approximations, we obtain
U N  PN ( A)U 0 / │ PN ( A)U 0 │,
(1.7)
where
N
PN ( )   (   i ) .
(1.8)
j 1
This process is called the outer iteration. We assume that the operator L1 (including the
thermalization groups) is determined sufficiently accurately using some iterative method
(which we call the inner iteration).
2. THE CHOUCE OF THE POLINOMIAL PN ( )
The Chebyshev polynomials of the first kind TN (z ) (see [1, 2]) provide an effective tool for
accelerating the convergence of methods for partial eigenvalue problems. However, the
problem under discussion has a number of special features:
1) The eigenspace N(0) associated with the zero eigenvalue has a large dimension, and the
_
zero eigenvalue itself is a limit point of the spectrum. The dimension of N(0) is at least g  m ,
_
where m is the number of grid points without fission sources.
2) The operator SL1 is an approximation of the corresponding compact operator in the
differential problem. Therefore, significant portion of summands in expansions (1.3) and (1.4)
are associated with small
eigenvalues.
3) The operator L1 is not known exactly but is formed using the inner iteration. In practical
calculations, this may cause the transition operator to have complex eigenvalues with small
moduli.
4) The round-off errors in iterative approximations produce new components in the subspace
N(0) .
5) The coefficients of Eq.(1.1) may depend on Ф which changes ( i ,  i ) and the eigenspace
N(0) .
To effectively take into account these features, it is reasonable to supplement the iterative
method by a simple iteration performed once in a while (when  k =0) .Its aim is to suppress
all the errors in the subspace N(0) . The other parameters of PN ( ) should be chosen so that
PN 1 ( ) be a polynomial with the least deviation from zero on the interval [0, M], where
0  M  1 and PN ( ) = PN 1 ( ) .
We change the variable according to z  1  2 / M to transform [0, M] onto [-1, 1]; then , the
point   0 is mapped to z  1 , and М is mapped to z  1 . Define   M / 1 and
ti  1  2i / M ; then   1 and t1  1  21 / M  1 . Let
N 2
Q N 1 ( z )  ( z  1) 2  ( z  zi )
(2.1)
i 2
Be the polynomial of degree N  1 with the least deviation from zero on the interval [-1, 1]
having the double root z = 1; the other roots are z i ( i=2, N-2).
The roots z i can be calculated using the program KLM-10, which implements the method
developed in [ 3 - 5].
Setting z  cos , 0  Re    , we can write QN 1 ( z ) in the form
QN 1 ( z ) = E N cos(( N  3)   N ( )) ,
(2.2)
where  N ( ) is a phase function.
Then , as z  1 , the maximal modulus of the values of the polynomial
QN ( z )  (1  z )QN 1 ( z )
(2.3)
converges to zero witch a linear rate (see Fig.1). We set
PN (  )  QN 1 (1 
2
).
M
(2.4)
The best convergence of the iteration is attained when M   2 ,   2 / 1 . Given these
equalities, we estimate the decrease in the rations siN 
aiN
(see (1.4)), corresponding to
a1N
ai0
k =N, compared to s  0 (see (1.3)); here, i  2,3,.., n . If
a1
0
i

2
1 ,
1
t1    (
siN 
2

2
 N 1
 1)  1 ,      2  1  1 , then [12]
si0 .
(2.5)
Recall that, in the power method, the errors decrease in accordance with the geometric
progression with the ratio  . In the method that we propose, the average rate of convergence
is estimated by the quantity  1  
if 2 / 1  1.
In our calculations for WWER-type reactors, we set N=30. Then , compared to s i30 , the ratios
si0 ( i  2,3..., n ) decrease by a factor that is greater than   1  1 ( 29 ) .
2
For instance, if   0.97 , we have   1.419 , and   13179 , for   0.98 , we have
  1.329 and   1949 .
1  zi
Let zi ( i  1,2,....,30 ) be the roots of the polynomial Q30 ( z ) (see(2,3), and yi 
,
2
0  M  1 .
We arrange zi so that y1  y11  y21  0 . To make the calculations stable, we arrange the
remaining 27 values yi by the algorithm presented in [2, 6] setting N  3  33 .
Then, the polynomial P30 ( ) (2.4) has the roots
 i  My i ,
i=1,…,30 ,
(2.6)
where yi are the following numbers:
0.,0.5522435,0.1153057,0.9415190,0.7087638,
0.8432142,0.2391296,0.3900389,0.0314653,0.993384,
0.,0.6058870,0.1526971,0.9643438,0.7567910,
0.8805971,0.2871593,0.4436831,0.0543441,0.9817008,
0.,0.4979634,0.0823947,0.9134939,0.6582651,
0.8017836,0.1941332,0.3376597,0.0139536,0.999267.
A step of the simple iteration occurs after each ten iteration steps (  1 = 11 =  21 = 0 ). The
remaining  i obey the relations  i  My i >0 and are chosen so that the polynomial
constructed from the first 10 roots and the one constructed from the first 20 roots are nearly
optimal.
Recall again that M  2 would be the best choice; however,  2 is unknown and is calculated
in the iterative process.
The number M is determined by the formula
M  K 'эфф  ,
(2.7)
Where  is assigned at the start of iteration (for instance,  =0.97). Both K 'эфф and  are
refined in the iterative process (see Section 4).
If the required accuracy is not attained after 30 iteration steps, the process is continued
cyclically with the period 30 using a corrected value of M.
3. CHOOSING THREE LINEAR FUNCTIONALS AND SOLVING MOMENT SYSTEMS
OF EQUATIONS
To obtain approximate values of 1 ,  2 ,  3 we use three linearly independent (even
mutually orthogonal) functional of the form
n
n
n
i 1
i 1
i 1
L0 (U )   U i p0 (i ) , L1 (U )   t (i )U i p1 (i ) , L2 (U )   (i / n  E 2 )(i / n  E3 )U i p2 (i ) ,
(3.1)
Here, p j (i )  0 are prescribed; j  1,2,3 , t(i)  sign(i  0.5(n  0.1))  E1 , i  1,2,3 ,
L0 (1 )  0 and E1 , E2 , E3 are chosen so that
L1 (U 0 )  0 , L2 (U 0 )  0 ,
L2 t (i )U 0  0 .
(3.2)
In this derivation, we assume that the determinant formed of the rows ( Li (1 ), Li ( 2 ), Li ( 3 ))
where i  0,1,2 , is nonzero.
Given the values of functional (3.1) at the members of iterative sequence (1.5), the three
largest eigenvalues 1 , 2 , 3 can be approximately determined by solving moment systems of
equations.
The orthogonality of the functional makes it possible to improve the condition of these
systems. The ideal choice would be linear functional for which Li 1 ( i ) >0, i =1,2,3 and
Li ( k ) =0 ( k  i +1, k  1,2,3); then, ( Li ,  k ) would be biorthogonal.
The quantities E1 , E 2 , and E 3 are updated after the current cycle of 30 steps has been
completed. The formulas given above are used for this update with U 0 replaced by the
current approximation U 30 .
Suppose that we have the following moment system of 12 equations with the unknowns
1 , 2 , 3 , a1 , a 2 , a3 , b1 , b2 , b3 , c1 , c2 , c3 :
3
 ai im  Am ,
i 1
3
 bi im  Bm ,
i 1
3
c 
i 1
i
m
i
 Cm ,
m  0,1,2,3 .
(3.3)
Here A0 , A1 , A2 , A3 , B0 , B1 , B2 , B3 , C0 , C1 , C2 , C3 are prescribed scalars. It is required to
find from this system only the quantities b=  (1  2  3 ) , c= 1 * 2  1 * 3  2 * 3 ,
d=- 1 * 2 * 3 . Then 1 , 2 , 3 can be determined as the roots of the cubic equation
3  b2  c  d  0 .
(3.4)
We obtain the following system of linear equations with respect to the coefficients b, c, d
(see (3.4)):
A3  A2 b  A1c  A0 d  0 , B3  B2 b  B1c  B0 d  0 , C3  C2 b  C1c  C0 d  0
(3.5)
4. DETERMINING THE COEFFICIENTS OF SYSTEM (3.5) AND K 'эфф = 1 , 2 , 3 .
We drop the summands corresponding to i = 4,5,…,n in the formulas for U k . Let k  3 and
aik 3  0 ( i  1,2,3 ); then , we have
U k 3  a1k 31  a 2k 3 2  a3k 3 3 .
(4.1)
The right-hand sides of system (3.3) are obtained by using (4.1) and the vectors U k 2 ,
U k 1 and U k . In the vectors, the coefficients of the three Eigen functions are expressed by
formulas (1.5) and (1.6) in terms of the coefficients  ik 3 ( i = 1, 2, 3).
Calculating the functional L0U k 3 , L0U k 2 , L0U k 1 , L0U k , we obtain a system of four equations
with respect to the powers of i and the quantities l0i  L0 ( ik 3 i ) ( i =1,2,3). This system is
then transformed to a system of form (3.3), where the scalars Ai ( i =1, 2, 3, 4) are known and
ai are well-defined linear combinations of l0i ( i =1, 2, 3, 4).
Similarly, calculating L1U k 3 , L1U k 2 , L1U k 1 , L1U k ( L2U k 3 , L2U k 2 , L2U k 1 , L2U k ),
we obtain system (3.3) in which bi ( c i ) are linear combinations of l1i = L1 ( aik 3 i )
( l 2i  L 2 ( ik 3 i ) ) while the scalars Bi ( C i ) are known.
Having determined the coefficients b, c and d of cubic equation (3.4) by (3.5), we then find
the roots of this equation by the method proposed in [7]: 1  1k , 2  k2 , 3  k3 . We set
1 equal to the nearest root to
~k
= L0 ( AU k 1 ) / U k 1 .
K эфф
(4.2)
An inevitable issue is the one of choosing a strategy for solving problems of this kind in the
absence of information about the exact value of the quantity M in formulas (2.6) and (2.7). If
0  M  3 , then one should expect the regular convergence of ik to i ( i =1,2,3)
and a slower convergence of U k to  1 . If
3  M  2 , then we expect that only ik ( i =
1,2) converge to i ; if 2  M  1 , then only 1k is expected to converge to 1 . For
instance, the following algorithm can be used: we set in formula (2.7)
~k
K эфф  K эфф
.
(4.3)
If in our iterative process, the ratios k2 / 1k   k converge to a certain limit  <1 and  <  <
1, then  in formula (2.7) is replaced by  [12].
5. NUMERICAL RESULTS
To calculate K эфф for WWER-type reactors, we have incorporated our method in the
multigroup software, namely, two-dimensional programs like PERMAK-A (see[11]), threedimensional programs like PERMAK 3-D (see[10]) and the TVS-M program (see[8, 9]).
Previously, the iterations in these programs had been accelerated by the Lyusternik method.
Our calculation and a comparison of about 20 typical versions of the programs have shown
the reduction in the execution time by a factor ranging from three to seven.
We are grateful to M.P. Lizorkin, V.D. Sidorenko, S.S. Aleshin, P.A. Bolobov and
A.Yu. Kurchenkov who provided us with their programs and their numerous variants for our
comparison calculations and helped us with the incorporation of our method.
Таb. 1 Calculation different method of some variants
two-dimensional programs ПЕРМАКА- 2D, g=4
Name of
the variant
Method
Lusterniks
Cyclic
Iterative
method
Quantity
points
Кэфф
Perm core
Var 4_7
355
271
89
103
118669
118669
1.10057
1.0090169
Var 5_6
155
47
118669
1.0817735
Var 5_7
355
79
118669
1.0057181
 = λ2 / λ1
 k = 0.96
 k =0.96
 k =0.96
 k =0.96
Таb. 2 ПРМАК-2D , g=6
Name of
the variant
VAR63_1
(g=6 )
Method
Lusterniks
239
Cyclic
Iterative
method
37
Quantity
points
20055
Кэфф
 = λ2 / λ1
1.0333908
 k = 0.96
Кэфф
 = λ2 / λ1
1.021773
 k = 0.98
Таb.3 Three-dimensional programs ПРМАК-3D, g=4
Name of
the variant
Method
Lusterniks
Cyclic
Iterative
method
Quantity
points
00B100TE30
K
221
47
509111
00B100TE30
263
65
2234911
1.021538
 k =0.98
Таb. 4 TVSM , g=4
Name of
the variant
Method
Lusterniks
Cyclic
Iterative
method
VAR360_4
88
27
VAR360_41
135
VAR360_5
VAR360_51
Quantity
points
Кэфф
 = λ2 / λ1
6253
1.253667
 k = 0.96
27
6253
1.288449
 k =0.96
209
44
1039
1.300738
 k =0.96
45
19
1039
1.316162
 k =0.96
1.00
0.50
0.00
-0.50
-1.00
-1.00
-0.50
0.00
Fig. 1 - Q30 ( z )
0.50
1.00
LITERATURE
1. V.I. Lebedev and G.I.Marchuk, Numerical Methods in the Theory of Neutron Transport
(Atomizdat,Moscow,1981)[in Russian].
2. V.I. Lebedev, Functional Analysis and Computational Mathematics (Fizmatlit,
Moscow,2005)[in Russian]
3. V.I. Lebedev, “A new method for determining the roofs of polynomials ofleast devicetion
on a segment with weight and subject to additional conditions”. Part I. // Russ. J. of Numer.
Anal. and Mathem. Modelling.1993, v. 8, N 3, p. 195--222.
4. V.I. Lebedev, “A new method for determining the roofs of polynomials ofleast devicetion
on a segment with weight and subject to additional conditions”. Part II.
// Russ. J. of Numer. Anal. and Mathem.Modelling., 1993, v. 8, N 5, p. 397--426.
5. V.I. Lebedev, “Extremal Polynomials and Optimization Techniques for Computational
Algorithms”,
6. Lebedev V.I.,Finogenov S.A.,” Some Algorithms for Computing of Chebyshev normalized
first Kind polynomials by roots”.
// Russ. J. Numer. Anal.Modeling., 2005, v. 20, N 4.
7. V.I. Lebedev , “ On formulae for roots of cubic equations”. // Sov.J.Num.An.Math.Mod.,
8. A.Yu. Kurchenkov and V.D.Sidorenko, “Estimate of the Doppler Effect Change Taking
into Account the Thermal Motion of Nuclei and the Resonance Behavior of the Scattering
Cross Section in the Scattering Indicatrix”.(Atomizdat, Moscow,1997), Vol.82,issue 4, pp.
321-327.
9. Sidorenko V.D., Bolschagin S.N., Lazarenko A.P e.a. Spectral code TVS-M for calculation
of characteristics of cells, supercells and fuel assemblies of VVER-type reactors. – In:
Material of 5 AER Symp.,1995
10. Aborina I., Bolobov P., Krainov Yu. Calculation and experimental studies of power
distribution in the normal and modernized CR of the VVER-440 reactor. Institut of Nuclear
Reactors, RRC “Kurchatov Institute”, Moscow, sept.2000, Simposium of AER.
11. Novikov A..N., Pshenin V.V., Lizorkin M.P. et al., Code package for
WWER cores analysis and some aspects of fuel cycles improving,
VANT, Series “Nuclear Reactor Physics”, 1992, v. 1 (in Russian).
12. G.I.Kurchenkova, V.I.Lebedev. Solving Reactor Problems to Determine the
Multiplication: A New Method of Accelerating Outer Iterations. Zhurn. Calculative. Matem.
end Matem.Physics.2007,Vol.47, No.6, pp.1007-1014