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 WWERreactors. 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 acceleratedby the Lyusternik method. Our calculations and a comparison of about20 typical versions of the programs have shown the reduction in theExecution time by a factor ranging from three to seven.
INTRODUCTION
Multigroup problems for determining the multiplication 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 ().
In this paper, we propose a cyclic iterative method with variable parameters for accelerating the outer iterations in a process used to calculate 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 , 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.
- FORMULATION OF THE PROBLEM AND AN INTERATIVE METHOD TO SOLVE IT
The multigroup system of difference diffusion equations for determining and the group fluxes of neutrons, where g is the number of groups, can be written in the form
(1.1)
Here is the multigroup operator consisting of the difference operators for diffusion, absorption, and group transitions; =SФ ,where =( is the fission-source operator, is the spectrum of the fission-neutrons, is the number of grid points in which the solution is sought, and , i=1,2,…,g.
Equation (1.1), which determines the Eigen values, is transformed to the standard form
(1.2)
where . Let =…0 be the nonnegative eigenvalues ofА and be the corresponding eigenvectors forming a basis in the space
Our problem is to find the eigenpair (). (Then, we set =). We assume that and set
, .
We examine the following cyclic iterative method with the period N and the variable parametrs(such that ) for determining ():
Given an initial approximation where ,
(1.3)
and , the subsequent approximations
(1.4)
are constructed by the formulas
,
, (1.5)
where
k =1,2,…,N . (1.6)
Neglecting the intermediate normalizations performed for these approximations, we obtain
/ ││, (1.7)
where
. (1.8)
This process is called the outer iteration. We assume that the operator (including the thermalization groups) is determined sufficiently accurately using some iterative method (which we call the inner iteration).
2. THE CHOUCE OF THE POLINOMIAL
The Chebyshev polynomials of the first kind (see [1, 2]) provide an effective tool foraccelerating 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 , where is the number of grid points without fission sources.
2) The operator 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 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 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 =0) .Its aim is to suppress all the errors in the subspace N(0) . The other parameters of should be chosen so that be a polynomial with the least deviation from zero on the interval [0, M], where and =.
We change the variable according to to transform [0, M] onto [-1, 1]; then , the point is mapped to , and М is mapped to . Define and; then and . Let
(2.1)
Be the polynomial of degree with the least deviation from zero on the interval [-1, 1] having the double root z = 1; the other roots are ( i=2, N-2).
The roots can be calculated using the program KLM-10, which implements the method developed in [ 3 - 5].
Setting , , we can write in the form
=, (2.2)
where is a phase function.
Then , as z 1 , the maximal modulus of the values of the polynomial
(2.3)
converges to zero witch a linear rate (see Fig.1).We set
. (2.4)
The best convergence of the iteration is attained when . Given these equalities, we estimate the decrease in the rations (see (1.4)), corresponding to =N, compared to (see (1.3)); here, . If
, , , then [12]
. (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 if 1.
In our calculations for WWER-type reactors, we set N=30. Then , compared to, the ratios ( ) decrease by a factor that is greater than .
For instance, if , we have , and , for , we have and .
Let () be the roots of the polynomial (see(2,3), and , .
We arrange so that . To make the calculations stable, we arrange the remaining 27 values by the algorithm presented in [2, 6] setting .
Then, the polynomial (2.4) has the roots
, i=1,…,30, (2.6)
where 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 (=== 0 ). The remaining obey the relations >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 would be the best choice; however, is unknown and is calculated in the iterative process.
The number M is determined by the formula
, (2.7)
Where is assigned at the start of iteration (for instance, =0.97). Both 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 , , we use three linearly independent (even mutually orthogonal) functional of the form
, , ,
(3.1)
Here, are prescribed; ,,, and , , are chosen so that
, ,. (3.2)
In this derivation, we assume that the determinant formed of the rows where , is nonzero.
Given the values of functional (3.1) at the members of iterative sequence (1.5), the three largest eigenvalues 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 >0, =1,2,3 and =0 ( +1,1,2,3); then, () would be biorthogonal.
The quantities and are updated after the current cycle of 30 steps has been completed. The formulas given above are used for this update with replaced by the current approximation .
Suppose that we have the following moment system of 12 equations with the unknowns , , , :
, , , . (3.3)
Here , , are prescribed scalars. It is required to find from this system only the quantities b=, c= , d=-. Then can be determined as the roots of the cubic equation
. (3.4)
We obtain the following system of linear equations with respect to the coefficients b, c, d (see (3.4)):
, , (3.5)
4. DETERMINING THE COEFFICIENTS OF SYSTEM (3.5) AND=,.
We drop the summands corresponding to i =4,5,…,n in the formulas for . Let and ( ); then , we have
. (4.1)
The right-hand sides of system (3.3) are obtained by using (4.1) and the vectors , and . In the vectors, the coefficients of the three Eigen functions are expressed by formulas (1.5) and (1.6) in terms of the coefficients ( = 1, 2, 3).
Calculating the functional , we obtain a system of four equations with respect to the powers of and the quantities (=1,2,3). This system is then transformed to a system of form (3.3), where the scalars (=1, 2, 3, 4) are known and are well-defined linear combinations of (=1, 2, 3, 4).
Similarly, calculating(),
we obtain system (3.3) in which ()are linear combinations of =() while the scalars ()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]: , , . We set equal to the nearest root to
= . (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 , then one should expect the regular convergence of to (=1,2,3)
and a slower convergence of to . If , then we expect that only (= 1,2) converge to ; if , then only is expected to converge to . For instance, the following algorithm can be used: we set in formula (2.7)
. (4.3)
If in our iterative process, the ratios converge to a certain limit <1 and < 1,then in formula (2.7) is replaced by [12].
5. NUMERICAL RESULTS
To calculate 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. 1Calculation different method of some variants
two-dimensional programsПЕРМАКА- 2D, g=4
Name of
the variant /Method
Lusterniks / CyclicIterative
method / Quantity
points / Кэфф / = λ2 / λ1
Perm core
/ 355 / 89 / 118669 / 1.10057 / = 0.96Var 4_7
/ 271 / 103 / 118669 / 1.0090169 / =0.96Var 5_6
/ 155 / 47 / 118669 / 1.0817735 / =0.96Var 5_7
/ 355 / 79 / 118669 / 1.0057181 / =0.96Таb. 2ПРМАК-2D,g=6
Name of
the variant /Method
Lusterniks / CyclicIterative
method / Quantity
points / Кэфф / = λ2 / λ1
VAR63_1
(g=6 ) / 239 / 37 / 20055 / 1.0333908 / = 0.96
Таb.3 Three-dimensional programs ПРМАК-3D, g=4
Name of
the variant /Method
Lusterniks / CyclicIterative
method / Quantity
points / Кэфф / = λ2 / λ1
00B100TE30
K / 221 / 47 / 509111 / 1.021773 / = 0.98
00B100TE30 / 263 / 65 / 2234911 / 1.021538 / =0.98
Таb.4 TVSM , g=4
Name of
the variant /Method
Lusterniks / CyclicIterative
method / Quantity
points / Кэфф / = λ2 / λ1
VAR360_4 / 88 / 27 / 6253 / 1.253667 / = 0.96
VAR360_41 / 135 / 27 / 6253 / 1.288449 / =0.96
VAR360_5
/ 209 / 44 / 1039 / 1.300738 / =0.96VAR360_51
/ 45 / 19 / 1039 / 1.316162 / =0.96
Fig. 1 -
LITERATURE
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2. V.I. Lebedev, Functional Analysis and Computational Mathematics (Fizmatlit, Moscow,2005)[in Russian]
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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
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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