High order backward discretization of the neutron diffusion equation

Pergamon Ann. Nucl. Energy, Vol, 25, No. 1-3, pp. 47-64, 1998

© 1997 Elsevier Science Ltd. All fights reserved PII: S0306-4549(97)00046-7 Printed in Great Britain

0306-4549/97 $17.00 + 0.00

HIGH ORDER BACKWARD DISCRETIZATION OF THE NEUTRON DIFFUSION EQUATION

D. GINESTAR, t G. VERDI~I, 2 V. VIDAL, 3 R. BRU, 1 J. M A R i N 1 and J. L. MU/qOZ-COBO 2

1Departamento de Matemfitica Aplicada, 2Departamento de Ingenieria Quimica y Nuclear, 3Departamento de Sistemas Informfiticos y Computacirn.

Universidad Politrcnica de Valencia, Camino de Vera S/N, 46071 Valencia, Spain

(Received 8 March 1997)

Abstract--Fast codes capable of dealing with three-dimensional geometries, are needed to be able to simulate spatially complicated transients in a nuclear reactor. We propose a new discretization technique for the time integration of the neutron diffusion equation, based on the backward difference formulas for systems of stiff ordinary differential equations. This method needs to solve a system of linear equations for each integration step, and for this purpose, we have developed an iterative block algorithm combined with a variational acceleration technique. We tested the algorithm with two benchmark problems, and compared the results with those provided by other codes, conclud- ing that the performance and overall agreement are very good. © 1997 Elsevier Science Ltd

1. INTRODUCTION

Complicated space and time phenomena related to neutron flux oscillations have been observed in several boiling water reactors (Takeuchi et al., 1994; Verd6 et al., 1995). Furthermore, rapid transients of reactor power caused by a reactivity insertion, due to a postulated drop or abnormal withdrawal of control rods from the core, have strong space, dependent feedback associated with them. Therefore, a fast three-dimensional transient analysis code is needed to simulate these phenomena.

Recently, some authors have tried to reduce codes CPU time in several ways, for example, in Koclas et al. (1996) the authors solve the kinetics equations using a quasistatic method and new schemes to compute the precursors equations to reduce the CPU time. Another strategy is found in the paper by Zimin and Ninokata (1996) where the authors develop new techniques for the acceleration of outer iterations for the space-dependent neutron kinetics based on Chebyshev semi-analytical and semi-iterative methods, but as noted in Section 4, the big handicap of this proposal is the spatial discretization used, which is based on the mesh centered finite differences method. This discretization requires

47

48 D. Ginestar et al.

finer mesh size than the usual nodal methods for a comparable level of accuracy and this implies that the CPU times necessary to solve the problem become prohibitive. Another recent work (Chen et al., 1997) is focused on determining the best solution for the resulting linear systems, but the results presented are incomplete because the performance of the solution depends strongly on the spatial discretization used.

The time counterpart of the neutron diffusion equation can be treated with either a direct or an indirect approach. One example of the latter is the improved quasistatic method (Ott, 1969; Verdfi et al., 1995; Koclas et al., 1996). Direct methods such as those based on difference equations discretizations are more appropriate if high accuracy is required (Verdfi et al., 1995; Ginestar, 1995).

In the same way as nodal methods have improved the efficiency for solving the spatial problem, algorithms to deal with sparse matrices have improved the efficiency of solving time-dependent problems. Direct methods should, therefore, be the standard approach to improve the accuracy of the solutions with a reasonable computational cost (Diamond, 1996).

According to the ideas expressed above, we present a new scheme for the time integration of the neutron diffusion equation, based on the backward difference formula (BDF) methods for stiff differential equations. Moreover, we take advantage of the algebraic matrix structure, a result of the spatial discretization, to develop a variational strategy to accelerate the convergence of the solution of the resulting linear systems.

The paper is structured as follows. In Section 2, we present the nodal collocation method used for the spatial discretization of the neutron diffusion equation, and the time discretizations, based on BDF methods. Section 3 is devoted to the solution of the resulting linear systems. The application of the methodology to two neutron kinetics benchmark problems is given in Section 4, and the main conclusions obtained are presented in Section 5.

2. DISCRETIZATION OF THE NEUTRON DIFFUSION EQUATION

The starting point is the standard time-dependent two groups neutron diffusion equation, written in matrix form as (Weston and Stacey, 1970)

K

Iv-l] q~ + £~b = (1 - fl).M4) + E k k C k X , (1) k = l

Ck = ~ka~¢~ -- ~.kCk; k = 1 . . . . . K

where K is the number of delayed neutron groups,

and

E l Eo o] -X12 -V(D:X7) + Xa2

(2)

High order backward discretization

The boundary conditions for the problem are

4>lr = o,

where F is the reactor boundary.

49

2.1. Spatial discretization

The first step in the analysis of a transient by means of the neutron diffusion equation is to choose a spatial discretization scheme for these equations. To do this, the reactor is divided into cells or nodes where the nuclear cross-sections are assumed to be constant, and a nodal collocation method is applied (H6bert, 1987; Verdfi et al., 1994, 1995). This nodal collocation method is based on a Legendre polynomial expansion of the solution in each node, and the imposition of adequate continuity conditions for the fluxes and cur- rents.

After a proper ordering of the indices has been chosen, we are able to approximate the partial differential equations (1) and (2) by the systems of ordinary differential equations

K

k=l (3)

Ck = ~k[MIIMI2]~ - XkCk. (4)

where ~p and Ck are vectors whose components are the Legendre coefficients of the expansions of ~ and Ck in the different nodes, and L, M, and [v-q are matrices with the following block structure (D6ring and Kalluhl, 1993; Verdfi et al., 1994, 1995)

r ,_.,,, o ], r,+, L = [_L21 ] v-I = = [o]

It is important to note that the nodal collocation method leads to symmetric and diagonal dominant blocks Lll and L22, and to diagonal blocks Mll and M12.

2.2. Temporal discretization

The next step consists of integrating the above ordinary differential equations over a series of time intervals [tn tn÷ 1]- To integrate equation (4) from tn to t~+ 1, we will suppose that the term [MI 1 MI2] ~P varies linearly between these instants. Then equation (4) in the interval [tn, tn+ 1] can be approximated by

R,. = --XkCk + flk[MllM12]ndl n + h ( t - tn)([MllM12]n+l~l n+l - [Ml lMl2]n~f l ) (5)

where h = t n ÷ l - t n and [MI1 Mi2]nVs ~ is [MI1 Ml2]~P evaluated at time tn. Integrating equation (5) the solution Ck in tn + 1 can be expressed as


G + I = Ge-~.kh + flk(ak[MllM12]nl~n _~_ bk[MllMl2]n+l ~/n+l) (6)

where the coefficients ak and bk are given by

ak m (1 + ,~kh)(1 - e -xkh) 1 Zkh -- 1 + e "x*h

~.~h Xk' b k - X2 h

To integrate equation (3), we must take into account that it constitutes a system of stiff differential equations, mainly due to the elements of diagonal matrix [v-l]. This makes it necessary to use implicit BDF methods for the integration (Grainville, 1988; Ginestar, 1995).

Given a first order differential equation

U(t) = f i t , U(t)),

an m-step general backward method consists of the difference equation

U(tn+l) + Oll U(tn) q- ot2U(tn_l) +. . . -F Otm(tn+l-m) = hflQf(tn+l, U(tn+l)) (7)

where h = tn+ 1--tn is the integration step, 30 > 0, and al , -.., Otm are chosen to maximize the order of h of the truncation error. Table 1 shows the values of the backward difference coefficients for several values of m, together with the order of the truncation error in each case.

Selecting these parameters, BDF methods are absolutely stable (Grainville, 1988) and moreover, these methods are implicit. Therefore, it is necessary to solve a system of linear equations for each time step but, unfortunately, it is not possible to construct an explicit method that does not perform poorly for stiff problems.

Hence, we are going to build explicitly several backward difference methods to integrate equation (3).

2.3. One-step backward method

If we discretize equation (3) by means of a backward method of one step (m = 1), we obtain

K --1 1 +1 ..{_ Ln+l~b~+l ~)Mn+l~n+l E ~ . k G + I x [v ]~(q/' -~b ~1 = ( 1 - + (81

k=l

Table 1. Backward methods coefficients

m /30 cq ~2 ~3 ~4 Truncation error

1 1 - 1 o (h) 2 2/3 -4/3 1/3 O(h 2) 3 6/11 - 18/11 9/11 -2/11 O(h 3) 4 12/25 -48/25 36/25 - 16/25 3/25 O(h 4)

High order backward discretization 51

Taking into account equation (6) and the structure of matrices L and M, we re-express equation (8) as the system of linear equations

Tll T1217/,+1 IRR2~2]I/f n ~ K 0 k=l

where

1 _ , L7~1 " T~ = ~.~ + - (1 - , ) ~ - ' - ~ X k , ~ b k ~ -~ k=l

K T12 = -(1 - fl)M~z~ 1 - Z ~'k~kbkMT+l

k=l

721 = -L~.~ -1

1_1 LF1 T22 = ~ v2 +

1 -1 K RI| = ~ v 1 + ZXkflkakMTll

k=l

K

RI2 = ~ Xk~kak~2 k--1

1 -1 R22 = h I)2

The above backward method has an associated truncation error proportional to the integration step, h; this implies the necessity of using very small time steps to obtain accurate solutions. Therefore, it would be more suitable to dispose backward methods with a high number of steps and so to be able to increase the time step while keeping the accuracy of the solution. For the one-step backward method, the integration step can be variable, however, for higher order BDF methods this is not possible, and it is necessary to develop elaborate strategies for this purpose.

We now give the explicit formula for the two-step backward method and the four-step backward method. These methods can be combined to achieve larger integration steps, as shown below.

2.4. Two-step backward method

Discretizing equation (3) by means of a two-step backward method, we obtain

[v-l] ( h 4 ~ ) 2 ( ~ K X) ~b.n+l _ ~ff/, + gin-1 = 3 _L,,+l ~,n+l + (1 - fl)M ~+17/'+1 + ~ XkC~k +1


and, taking into account equation (6), we re-express it as the system

where

TI2 ]~n+l = n 1 T21 T22J R12 ¢ "F R22 ~tn-l--F

-F- -- ~,k e - k k h

_ 2 K +1 1 2--rn+l -- ~ (1-- /~)M~l +1 - TII = v-~ 1 ~ + 3 ~H 3 k~=i )'k~kbkM~ll

(lo)

2 K

T 2 1 = - - - 2 L~_ 1 3

1 .a 21. n+l T22 = v21 ~ . 3 ~22

41 -1 2 x

2 /~ 4 1 - x R]2 = -~ Y~ ~.kakflkM~12, R~2 = -~-~ v2

11 -1 11 - l R211 = - -~1)1 , R222 = - - - ~ v 2

In order to use the two-step backward method, it is necessary to know the solution in two consecutive time steps, ¢0 and ¢1.

2.5. Four-step backward method

Using a four-step backward discretization for equation (3), we obtain

[l~-l]h \ (1/In+l _ ~_~¢n + ~ _ ~ . - 4 8 36 . I - ~V/'16 --2 + ~ . 3 --3\)

_-- __12 _ L n + l ~ +1 + ( 1 _ ¢i)/1,10+1~+1 + E ~ . k ~ + l 25 k=l


which can be grouped as the system

I °1 ~,, 5 , ~ I ¢ + , i 12 i = ~ + ~-~ T21 122J R12 R22

0 R~2 ¢ - 2 + R12 ¢ -3 12.---, K ~ ~1-

- - ~ . k e - kn

where

1 12 n+l 12 12 ,~K +l Tll = V1-1 ~ + ~ L ll - ~ (1 - fl)M~ +1 - 2---5 ~ ~'kflkbkM~ll

(11)

12 1 fl)M]'l +1 12 K +1

12 n+l T21 = - ~-~L21

1 12 n+l T22 = v21 ~ + ~-~ L22

48 1 -1 12 K RI, = ~ g v , + ~ X ~ a ~ & ~ ,

k=l

12 x R]2 = 25 E ~'kakflk M72

k=l

48 1 -1 R;~ = 5g~ v2 ,

36 1 v~_l, R~2 - 25 h

161 R~2 = yg ~ v~ -1,

36 1 R211 - 25 h Vl l

161 - l

3 1 R~I -- 25hVi-1

3 1 -1 R4~ = - ~3 ~ ':~


To use the four-step backward method it is necessary to solve equation (11) for each integration step, starting with the value of the solution at four consecutive points, ¢0, ~.l, ~2 and ~V 3.

2.6. Combined algorithms

2.6.1 Fixed step algorithm As mentioned above, to make use of the four step backward algorithm, it is necessary

to know the solution at four consecutive time steps, ,/i0, ~pl, ~fl and ap 3. In order to obtain these solutions, it is possible to combine the one-, two- and the four-step backward algorithms in the following way (Fig. 1).

We start with the solution at the initial point, ~0, and calculate the solutions for the instant tl using the one-step backward method, with h = t l-to as the time step. Next, using the two-step backward algorithm and the solutions at to and tl we perform two iterations, obtaining the solutions at t2 and t 3. Then, with the four-step backward algorithm and the solutions at to tl, t2 and t3 we calculate the solution in the following step, t4. The integration continues using, in each step, the solutions obtained for the four previous instants.

2.6.2 Variable step algorithm The four-step backward method possesses a truncation error O(h4), while the trunca-

tion error for the one-step algorithm is O(h) and for the two-step algorithm is O(h2). This implies that it is possible to use a bigger integration step for the first method than for the others, keeping the same level of accuracy in the solution.

We now present an algorithm which makes use of the one-, two- and four-step backward methods and which allows increase of the integration step until a maximum, hmax, imposed by the user. This maximum step will depend on the transient to be analyzed.

The next scheme explains the developed algorithm (Fig. 2). We begin with the solution for the initial time, ~p0, and calculate the solutions for the

instants tl and tz using h = h- to as the integration step. Using the solution at instants to and t2 we perform five iterations with the two-step backward method using 2h as the integration step. With the solutions at to, t4, t8 and t12 we perform three iterations with the four-step backward method using 4h as the integration step. Once this process is finished, it is possible to use the four-step backward method with the solutions at to, t8, t16 and t24 doubling the integration step (8h). It is then possible to repeat the blocks of three iterations with the four-step backward method to double the integration step until hmax is achieved. Once this maximum is reached, the integration follows with the four-step backward method until the final integration time is achieved using hmax as the integration step.

to h I step i

t l (I) (2)

2 step i q to tl t2

4 step ~ i i to tl t2

I

t3 (1) .-.

I ! I I I 13 t4

Fig . 1. F i x e d s tep c o m b i n e d a l g o r i t h m .


1 step

2 step

4 step

4 step

h I I I

to tl t2 (1) (2) (3) (4) (5)

! ~ ~ t I I I

to t2 t4 t6 ts tlo t12 (2) (3)

I I I I I I I

to t4 t8 t12 t16 t20 t24 (1)

I I I I I . . .

to t8 t16 t24 t32

Fig. 2. Variable step combined algorithm.

3. SOLUTION OF THE LINEAR SYSTEMS

To use the backward discretization methods shown above it is necessary, for each integration step, to solve a system of linear equations large and sparse, of the form

where the solution is

and the independent term is function of the solution at previous time steps, depending on the backward method utilized.

Due to the large size of the system [equation (12)] an iterative method is recommended for its resolution. The nodal collocation method used for the spatial discretization of the neutron diffusion equation leads to blocks Tll, T12, T21 and Tz2, with special properties. In particular, blocks Tll and T22 are symmetric and diagonal dominant matrices, and blocks T12 and T21 are diagonal matrices. The whole matrix of the system of linear equations, T, has ,none of these properties, and the usual iterative methods present convergence problems if they are directly applied to this matrix. In order to tackle this problem, we propose a block iterative algorithm (Verdfi et al., 1995).

3.1. Block iterative algorithm

This algorithm can be scheduled in the following steps: 1. Solve the system

= e l - h 2 C

In this first step we chose ~o.


2. Solve the system r22~ = e2 - :r21~

Once this first iteration is finished, an extrapolation process is used, passing to step 3. 3. Solve the system

Tll~Jl +l = E l - - Tl2(ta~//2 + (1 - - 09)1///2-1).

4. Solve the system T22~/t~ +l = E2 - T21(co~ +1 + (I - w)~) .

where ¢o is the extrapolation factor which is typically chosen as 1.5. 5. The stop criterion is checked

II ~1 +1 - ~1 II< tol and II ~2 +1 - ~2 Il< tol

If these conditions are not fulfilled, we set l = 1+ 1 and the process is repeated from step 3. If these conditions are satisfied then we set ~/t7+1 = ~+1 and ~r~2+1 = ~//fl. In the developed code tol has been chosen as 5x 10 -5.

Using this process, only the linear systems associated with matrices Tli and T22 have to be solved. These matrices are sparse, symmetric and diagonal dominant, therefore these linear systems can effectively be solved by means of a standard iterative method such as the conjugate gradient method.

Usually, to solve equation (12), it is necessary to perform many external iterations of the process, and so it would be convenient to accelerate the convergence. To accomplish this we alternated these external iterations with iterations of the following variational acceleration technique.

3.2. Variational acceleration

For a generic system of linear equations

T¢= E

the variational acceleration technique relies on iterations of the form

(13)

where the gradient vectors are

gn : E - - Tff/n, Gn = ~'n - ~ n-1

and the acceleration factors, ot and fl, are calculated minimizing the quadratic error

E2= ( E - T~)T(E - T~)

The solution of the normal equations

leads to the values


~2 &~2

(g~r Trg~)( Gnr Tr TGn) - ( G~r Trg~)(g~r TT TG ~) (g~rTrTg~)(G~rTrTG~) - (GnrTrTg~)(g~rTrTG n)

57

(gnTTrg~)(GnrTTg n) -- (G.rTTTg~)(g"rTTg n )

fl = (gnTTT Tgn)( Cr,,TTT TGn) - ( G"rTT Tgn)(gnrT'r TGn)

For the problems studied in the next section we observed that alternating five external iterations of the block iterative method with one iteration of the variational acceleration method leads to an efficient algorithm to obtain the solution of the linear system (12).

4. NUMERICAL RESULTS

The developed algorithm has been written in FORTRAN and called NOBACK, and all the studied cases have been run on a HP Exemplar S Class. We studied two transients, the bi-dimensional Seed-Blanket reactor and the three-dimensional Langenbuch reactor.

4.1. Seed-Blanket reactor

To analyze the capabilities of the backward method, we studied a simplified reactor kinetics problem which is shown in Fig. 3, where the spatial units are measured in cm. The reactor has been discretized in square nodes 8 cm wide.

This problem is similar to the TWIGL problem (Hageman and Yasinski, 1969; Lan- genbuch et al., 1977; Jae-Woong Song and Jong-Kyung Kim, 1993). It is modeled with two energy groups, one delayed neutron precursor group, and a quarter core symmetry. The kinetic parameters, t3 and ;~, have been modified to accelerate the transient (Verdfi et al., 1995). The two group constants for the problem are given in Table 2.

The transient consists of decreasing the thermal absorption cross-section in region 1 as the following ramp function

0.0035 Ea2(t)=0.15 - - t , 0 < t < 0 . 2

0.2

where the time is measured in seconds. The relative neutronic power evolution for the transient is shown in Fig. 4. These results

have been obtained using the fourth order backward algorithm with fixed integration steps h = 1.25×10 -3, h=5 .0x l0 -3 and h = 1.0x 10-2s.

As we can see, the high order backward algorithm gives very accurate results for long integration time steps.

58

0 0

24

Y

56

80

D. Ginestar et al.

37 24 56 80

3 2

2 1

3

Fig. 3. Quadrant of the Seed-Blanket reactor.

~2

2.15

. . . . . . . . . . . . . . . . . . i~'1 a !

p p

!

h P

j#

i /

/

J J

/ / /

j /

/"

/ f"

. / 0.1 0.2

Time (seconds)

~25e-3 .e-3

1 .e-2

Fig. 4. Neutronic power evolution for the Seed-Blanket transient.

Table 2. Group constants for the Seed-Blanket problem

Region Group Dg(cm) ]~ag(Cm- 1) vEfg E12

1 1 1.4 0.01 0.007 0.01 2 0.4 0.15 0.2

2 1 1,4 0.01 0.007 0.01 2 0.4 0.15 0.2

3 1 1.3 0.008 0.003 0.01 2 0.5 0.05 0.06

~1 Xi l/vl l/v2 0.0064 0.08 10 -s 10 -7


Table 3. CPU time comparison between the one-step and the four-step backward methods

h Time steps CPU time (s) CPU time (s) order 1 order 4

6.25× 10 -4 320 10.2 10.6 1.25x 10 -3 160 5.3 5.5 2.50x 10 -3 80 2.9 3.0 5.00X 10 -3 40 2.8 1.7 6.25x 10 -3 32 2.6 1.7 1.00× 10 -2 20 2.2 1.7

In Table 3, we present a comparison of the CPU times spent for the Seed-Blanket problem, using backward methods of order one and four and fixed integration step. We observe that the CPU time reduces when the integration step increases, reaching a limit due to the fact that the benefits of reducing the number of time steps is balanced by the bigger CPU time necessary to achieve the convergence of the linear system solutions, which are calculated by means of an iterative method. Also, it is interesting to note that the fourth order backward method spends less CPU time than the backward method of order one, when we choose a big integration step (greater than 5x 10-3). We believe that this is because this method produces matrices with a better behaviour for the convergence of the iterative process used to solve the linear systems (Table 4).

As noted above, the iterative block algorithm we proposed for the resolution of the linear systems for each integration step has a slow convergence. Therefore, to accelerate this convergence, we proposed using a variational strategy. To clarify the convenience of using this technique we compare the CPU times to solve the linear systems with and without the acceleration strategy, for both variable and fixed step algorithms.

As observed from the results, the accelerated resolution is approximately twice as fast for this problem.

4.2. Langenbuch reactor

The second problem analyzed was introduced by Langenbuch et al. (1977) and repre- sents an operational transient in a small LWR core with 77 fuel assemblies and two types of fuel. Figure 5 shows the geometry of the reactor model. The numerical solution has been obtained by many authors but, as far as we know, this problem does not have a reference solution.

The macroscopic cross-sections and other data defining the transient are shown in Table 5.

Table 4. Comparison between the accelerated and non accelerated solutions

h hmax time steps CPU time (s) CPU time (s) accelerated non-accelerated

1.25x10 -3 5.0x10 -3 44 1.8 3.7 1.25x10 -3 1.0xl0 -2 27 1.8 3.2 5>( 10 -3 5.0x 10 -3 40 1.7 3.6


All the calculations for the reactor have been performed with the quarter-core symmetry, using 330 nodes. The steady-state was calculated using MOD3D code (Verdfi et al., 1994) and the value obtained for the fundamental eigenvalue was 0.9992. The transient was initiated by the withdrawal of a bank of four partially inserted control rods at a rate of 3 cm/s -1 over 0<t<26.7 s. A second bank of control rods is inserted at the same rate over 7.5<t<47.5 s. The transient is followed for 60 s.

The CPU times measured in seconds for the Langenbuch transient spent by the fourth order backward algorithm are shown in Table 6.

The relative average power for several fixed integration steps h = 0.125, h = 0.25, h = 0.5 and h = 1 s is shown in Fig. 6. We observe that the results are practically coincident except for the power peak which shows a rod cusping effect when integration steps greater than 0.5 s are used. The tendency of our code to give wrong results for the peak of the core relative power can be attributed to the treatment of the control rod cross-sections for a partially rodded node. The methodology requires all the cross-sections to be spatially

~7~ Absorber. rod group 1

Absorber. rod group 2

Y,

111 Fheli2 / /

®

i Z

Transversal section of Langenbuch Reactor

5

4

3

2 1

initial positron of absorber-rods

8

20c 6

8 i

1 I Final position of absorber-rods

cm

Axial sections of Langenbuch Reac tor

Fig. 5. Transversal and axial sections of Langenbuch Reactor.


Table 5. Group constants for the Langenbuch problem

61

Region Group Dg(cm) I2ag(Cm- 1) v~,fg ~12

Fuel 1 1 1.423913 0.01040206 0.006477691 0.01755550 2 0.3563060 0.08766217 0.1127328

Fuel 2 1 1.425611 0.01099263 0.007503284 0~01717768 2 0.3505740 0.09925634 0.1378004

Absorber 1 1.423913 0.01095206 0.006477691 0.01755550 2 0.3563060 0.09146217 0.11273228

Reflector 1 1.634227 0.002660573 0.0 0.02759693 2 0.2640020 0.04936351 0.0

Del. Prec Group 1 Group 2 Group 3 Group 4 Group 5 Group 6

/3i 0.000247 0.0013845 0.001222 0.0026455 0.000832 0.000169 ~i (s - l ) 0.0127 0.0317 0.115 0.311 1.4 3.87

1 .73

o

e~

0.398

I I

! I

/

/ , \

,/ \ \

2 • , , - . , , , .

0 I 0 20 30 40 50 Time (seconds)

o.125

6.25 hi5 l

60

Fig. 6. Relative power evolution for the Langenbuch transient.

Table 6. CPU times for Langenbuch transient

h hmax Time steps CPU time (s)

0.0625 0.0625 960 256.30 0.125 0.125 480 172.50 0.125 0.250 241 100.56 0.125 0.5 124 66.18 0.125 1.0 67 44.88 0.125 2.0 40 30.66 0.125 4.0 28 23.05


constant within a node, In the actual status of the code, volume weighted average cross- sections are used for partially rodded nodes, but in order to simulate adequately the rod motions, we plan to correct this error. We will update the rodded cross-sections using a reconstruction of the neutronic flux inside the node, ~b(r-], from the provisional Legendre coefficients obtained in the iterative process. The new rodded cross-sections will be averaged as

Z(z)~(7)dV

S ¢~(-~)dV vj

where the integration is extended over the rodded node, and ~(z) is the corresponding cross-section for the node depending on the height of the control rod.

Table 7 compares the mean neutronic power density (W cm -3) obtained with the backward algorithm (NOBACK) fixing h =hmax=0.125, with the results given by the codes CSA (Zimin and Ninokata, 1996), ARROTA (Huang, 1994), CUBBOX (Langenbuch et

al., 1977) and QUABOX (Langenbuch et al., 1977) with a time step of 0.125 s. The reference solution considered was the CUBBOX solution with an integration step

of 125 ms. We can see that the overall agreement is very good, and the maximum relative error is less than 1.5%.

Table 8 shows the results for several integration steps for the codes CUBBOX, ARROTA and NOBACK, together with the relative deviations obtained by subtracting the calculation with the integration step of 0.125 s from the value obtained with the corresponding integration step and normalizing to this value. The results show that the temporal convergence rate of the developed code, NOBACK, is good compared with the results given by CUBBOX and ARROTA.

Furthermore, if we take into account the time results shown in Table 6, we are able to conclude that, within a reasonable degree of accuracy, our code is very fast compared with, for example, the code CSA (Zimin and Ninokata, 1996) based on an improved finite difference algorithm. This is because the finite difference methods require a finer discretization than the usual nodal methods for a comparable level of accuracy, hence the size of the problem increases with the corresponding penalization in CPU time terms.

Table 7. Comparisons of the neutronic power with different codes

Time CSA ARROTA C U B B O X Q U A B O X NOBACK

1 152.5 152.4 152.4 152.7 153.7 2 155.6 155.5 155.5 155.7 157.0 5 168.6 168.3 168.8 168.7 170.5 10 200.2 199.6 201.1 200.2 201.6 15 234.6 233.6 239.6 238.3 238.1 20 254.3 252.8 260.0 260.5 257.2 25 242.4 240.8 248.8 250.3 246.0 30 205.3 203.4 211.2 213.6 209.5 40 122.7 120.8 125.5 127.5 124.7 50 76.8 75.2 77.1 78.6 77.8 60 59.2 57.7 58.9 60.3 59.8


Table 8. Mean power density (w cm -3) and relative deviations (%) for several integration steps

Time (s) h m a x ( m s ) CUBBOX ARROTA NOBACK

5 125 168.79 168.30 170.49 250 168.16 (-0.37) 168.05 (-0.15) 170.52 (+ 0.02) 500 167.18 (-0.95) 167.35 (-0.56) 170.60 (+0.06)

1000 166.23 ( - 1.52) 165.71 ( - 1.54) 170.70 (+ 0.21)

10 125 201.11 199.59 201.59 250 200.18 (-0.46) 198.83 (-0.38) 200.69 (-0.44) 500 198.18 ( - 1.46) 197.28 ( - 1.16) 200.63 (-0.47)

1000 195.08 (-3.00) 194.20 (-2.70) 197.37 (-2.09)

20 125 260.03 252.80 257.19 250 259.53 (-0.19) 251.46 (-0.53) 254.68 (-0.97) 500 257.81 (-0.85) 248.70 (-1.62) 200.63 (-1.87)

1000 253.28 (-2.60) 243.08 (-3.84) 197.37 (-5.47)

30 125 211.26 203.40 209.49 250 212.61 (+0.64) 203.10 (-0.15) 207.13 (-1.13) 500 214.23 (+ 1.41) 202.35 (-0.52) 204.80 (-2.20)

1000 215.73 (+ 2.12) 200.36 ( - 1.49) 199.64 (-4.70)

40 125 125.46 120.79 124.69 250 127.08 ( + 1.29) 121.10 (+0.26) 123.75 (-0.76) 500 129.80 (+ 3.46) 121.59 (+ 0.66) 122.41 ( - 1.83)

1000 133.60 (+ 6.49) 122.33 (+ 1.27) 119.98 (-3.77)

50 125 77.07 75.19 77.82 250 77.66 (+ 0.77) 75.44 (+ 0.33) 77.51 (-0.38) 500 78.84 (+ 2.30) 75.90 ( + 0.94) 77.66 (-0.20)

1000 91.16 (+ 5.31) 76.74 (+ 2.06) 77.59 (-0.29)

60 125 58.94 57.72 59.84 250 59.21 (+ 0.51) 57.89 (+ 0.29) 59.68 (-0.26) 500 59.77 (+ 1.41) 58.20 (+ 0.83) 60.09 ( + 0.40)

1000 60.76 ( + 3.09) 58.76 ( + 1.80) 60.18 ( + 0.56)

5. CONCLUSIONS

In this paper, we developed a nodal neutronic code, called NOBACK, with a new scheme for the time integration of the neutron diffusion equation based on BDF methods. As these are implicit methods, it is necessary to solve large and sparse linear systems for the integration. We made use of the algebraic structure of the resulting matrix to develop a block iterative algorithm combined with a variational strategy to solve these systems efficiently.

We solved two benchmark problems first, the 2-D Seed-Blanket reactor, where we observed that high order backward methods present better performance than the one-step backward method. We also noticed that the use of the variational strategy is very convenient for resolution of the linear systems.

The other problem analyzed was the 3-D Langenbuch transient. Comparing our results with the ones provided by other codes, such as CUBBOX or ARROTA, we conclude that our method is fast and provides results that have a reasonable accuracy. Nevertheless, the high order backward methods present a rod cusping effect predicting the peak of the core relative power that we attribute to our treatment of the rodded cross-sections.


We can conclude that NOBACK is a versatile code which can be coupled as neutronic module to a standard thermal-hydraulic code such as TRAC. This task is now being carried out.

Acknowledgements--This work was partially supported by the projects CICYT TIC96-0718-C02- 01, and CICYT PB94-0537.

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High order backward discretization of the neutron diffusion equation

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