Symposium of AER

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 Keff 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 ( Keff ). In this paper, we propose a cyclic iterative method with variable parameters for accelerating the outer iterations in a process used to calculate Keff 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 Keff , 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 Keff and the group fluxes of neutrons  (1 ,.., g ) , where g is the number of groups, can be written in the form

 L  S (1.1) Keff

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, i2 ,.., in ) , i=1,2,…,g . Equation (1.1), which determines the Eigen values, is transformed to the standard form

AU  U, (1.2)

1  where A  SL . Let K 'эфф =1 >2  3  … n  0 be the nonnegative eigenvalues of А n and 1 , 2 ,..., n be the corresponding eigenvectors forming a basis in the space R .

Our problem is to find the eigenpair ( 1 ,1 ). (Then, we set K 'эфф =1 ). We assume that

1  0 and set

n U  Ui p0 (i) , p0 (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 ): 0 0 0 0 Given an initial approximation U  (U1 ,...,U n ), where Ui  0 ,

n 0 0 0 0 0 U  a1 1  a2 2  a3 3   ai i , (1.3) i4

0 and a1  0 , the subsequent approximations

n k k k k k U  a1 1  a2 2  a3 3   ai i (1.4) i4 are constructed by the formulas

k k1 k 1 V  AU   kU , U k  V k /V k , (1.5) where

n k k1 V  (i   k )ai i k =1,2,…,N . (1.6) i1

Neglecting the intermediate normalizations performed for these approximations, we obtain

N 0 0 U  PN (A)U / │ PN (A)U │, (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 2 QN 1(z)  (z 1)  (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 zi ( i=2, N-2).

The roots zi 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

Q (z) E cos((N  3)  ( )) N 1 = N 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

2 P ()  Q (1 ) . (2.4) N N 1 M

M   ,   /  The best convergence of the iteration is attained when 2 2 1 . Given these N N ai equalities, we estimate the decrease in the rations si  N (see (1.4)), corresponding to k a1 0 0 ai =N, compared to si  0 (see (1.3)); here, i  2,3,.., n . If a1

 2 2 2   1 , t1    ( 1)  1 ,      1 1, then [12] 1 

2 s N  s0 . (2.5) i  N 1 i

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 1 is estimated by the quantity    if 2 / 1  1. 30 In our calculations for WWER-type reactors, we set N=30. Then , compared to si , the ratios 1 s0 ( i  2,3..., n ) decrease by a factor that is greater than    1 ( 29 ) . i 2 For instance, if   0.97 , we have   1.419 , and   13179 , for   0.98, we have   1.329 and   1949 . 1 z Let z (i  1,2,....,30 ) be the roots of the polynomial Q (z) (see(2,3), and y  i , i 30 i 2

0  M  1 .

We arrange zi so that y1  y11  y21  0 . To make the calculations stable, we arrange the 3 remaining 27 values yi by the algorithm presented in [2, 6] setting N  3  3 .

Then, the polynomial P30 () (2.4) has the roots

 i  Myi , 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  Myi >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 L0 (U)  Ui p0 (i) , L1 (U )  t(i)Ui p1 (i) , L2 (U)  (i / n  E2 )(i / n  E3 )Ui p2 (i) , i1 i1 i1 (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

0 0 0 L1 (U )  0 , L2 (U )  0 , L2t(i)U  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 , E2 , and E3 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 ,a2 ,a3 , b1 ,b2 ,b3 , c1 ,c2 ,c3 :

3 3 3 m m m ai i  Am , bi i  Bm , ci i  Cm , m  0,1,2,3. (3.3) i1 i1 i1

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  A2b  A1c  A0d  0 , B3  B2b  B1c  B0d  0 , C3  C2b  C1c  C0d  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  3and k3 ai  0 ( i  1,2,3); then , we have

k3 k3 k3 k3 U  a1 1  a2  2  a3 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 k3 (1.5) and (1.6) in terms of the coefficients i ( i = 1, 2, 3). k3 k2 k1 k Calculating the functional L0U , L0U , L0U , L0U , we obtain a system of four equations k3 with respect to the powers of i and the quantities l0i  L0 (i 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). k3 k2 k1 k k3 k2 k1 k Similarly, calculating L1U , L1U , L1U , L1U ( L2U , L2U , L2U , L2U ), k3 we obtain system (3.3) in which bi ( ci ) are linear combinations of l1i = L1 (ai i ) ( k3 l2i L2 (i i ) ) while the scalars Bi (Ci ) are known. Having determined the coefficients b, c and d of cubic equation (3.4) by (3.5), we then find k k k the roots of this equation by the method proposed in [7]: 1  1 , 2  2 , 3  3 . We set

1 equal to the nearest root to

~ k k1 k1 Kэфф = L0 (AU ) /U . (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 k 0  M  3 , then one should expect the regular convergence of i to i (i =1,2,3) k k and a slower convergence of U to 1 . If 3  M  2 , then we expect that only i (i = k 1,2) converge to i ; if 2  M  1 , then only 1 is expected to converge to 1 . For instance, the following algorithm can be used: we set in formula (2.7)

~ k Kэфф  Kэфф . (4.3)

k k k If in our iterative process, the ratios 2 / 1   converge to a certain limit  <1 and  <  < 1, then  in formula (2.7) is replaced by  [12].

5. NUMERICAL RESULTS

To calculate Kэфф for WWER-type reactors, we have incorporated our method in the multigroup software, namely, two-dimensional programs like PERMAK-A (see[11]), three- dimensional 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 Method Cyclic Quantity the variant Lusterniks Iterative points Кэфф  = λ2 / λ1 method

Perm core  k = 0.96 355 89 118669 1.10057

Var 4_7 271 103 118669 1.0090169  k =0.96

Var 5_6 155 47 118669 1.0817735  k =0.96

Var 5_7 355 79 118669 1.0057181  k =0.96

Таb. 2 ПРМАК-2D , g=6

Name of Method Cyclic Quantity the variant Lusterniks Iterative points Кэфф  = λ2 / λ1 method

VAR63_1 239 37 20055 1.0333908  k = 0.96 (g=6 )

Таb.3 Three-dimensional programs ПРМАК-3D, g=4

Name of Method Cyclic Quantity the variant Lusterniks Iterative points Кэфф  = λ2 / λ1 method

00B100TE30 221 47 509111 1.021773  k = 0.98 K

00B100TE30 263 65 2234911 1.021538  k =0.98

Таb. 4 TVSM , g=4 Name of Method Cyclic Quantity the variant Lusterniks Iterative points Кэфф  = λ2 / λ1 method

VAR360_4 88 27 6253 1.253667  k = 0.96

VAR360_41 135 27 6253 1.288449  k =0.96

VAR360_5 209 44 1039 1.300738  k =0.96

VAR360_51 45 19 1039 1.316162  k =0.96

-1.00 -0.50 0.00 0.50 1.00

Fig. 1 - Q30 (z)

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