SECOND ORDER THREE STAGE STOCHASTIC RUNGE-KUTTA
METHOD FOR NUCLEAR REACTORS
Daniel Suescún-Díaz
1, Nathaly Roa-Motta
1and Freddy Humberto Escobar
21
Departamento de Ciencias Naturales, Universidad Surcolombiana, Avenida Pastrana, Neiva, Huila, Colombia 2Universidad Surcolombiana/CENIGAA, Avenida Pastrana, Neiva, Huila, Colombia
E-Mail: [email protected]
ABSTRACT
This paper describes and implements the numerical Runge-Kutta method for solving stochastic point kinetic equations. The mean and variance of neutron and delayed neutron precursor populations are calculated for different time steps for constant reactivities. The advantage of the proposed method is that it requires the development of few derivatives and has low computational cost, as it does not require the square root calculation of a matrix. The numerical results obtained indicate that the method is efficient and precise for the study of stochastic point kinetics with constant reactivity.
Keywords: stochastic point kinetic equations, concentration of delayed neutron precursors density, Runge-Kutta stochastic method.
INTRODUCTION
A nuclear reactor is a device from which nuclear energy is produced, and obtained by the nuclear fission process. This process occurs as a result of the high probability of interaction of thermal neutrons with fissionable material. The nuclear reactor allows the initiation, maintenance and control of fission reactions. The energy extracted is that required by existing demand, and the reactor is designed for that purpose. The fluctuation in the operation phase can be adjusted by controlling the density of the neutron population. Having a clear knowledge of the dynamics of the nuclear reactor permits careful control of the neutron distribution to guarantee energy production [1].
Point kinetic equations are a system of nonlinear deterministic differential equations, however, the operability of the real dynamic process of a nuclear reactor is stochastic in nature [2-3]. The density of neutrons and delayed neutron concentrations vary over time. In critical and subcritical reactors, random behavior in neutron density and in the concentration of delayed neutrons is representative, but in supercritical reactors, random behavior is negligible [2]. Point kinetic equations can be used to estimate the first and second order moments of random functions [4].
Previous studies have given solutions to stochastic point kinetic equations. The following have been reported: The Partial Constant Approximation method (PCA) and Monte Carlo [5], Euler-Maruyama (EM) [6], Taylor 1.5 [6-7], the simplified stochastic kinetic method (SSPK) [8], the analytical exponential model (AEM) [9], in the efficient stochastic model (ESM) [10], the double diagonalization and decomposition method (DDDM) [11] and the implicit Euler-Maruyama method [12] (implicit EM).
This work is organized as follows: the first section presents the stochastic model describing the kinetics of a nuclear reactor. In the next section we develop the proposed explicit second order three stage
THEORETICAL CONSIDERATIONS STOCHASTIC POINT KINETIC EQUATIONS
Nuclear reactor kinetics study the dependence over time of the processes involved in the temporal behavior of the neutron flux in the reactor; these processes are stochastic by nature [2-3]. The stochastic equation of point kinetics models the evolution of neutron density and the concentration of delayed neutron precursors, these equations are given by:
1
( ) 2 ( )
( ) ( )
ˆ ˆ
ˆ
t t
t t
dx dW
Ax Q B
dt dt (1)
where
( ) 1( ) ( ) 2( )
( )
ˆ
t
t
t t
m t
n
c
x c
c
(2)
( )
1 2
1
1 2
2
0 0
0 0
0
0 0
t
m
m
m
A
( ) ( ) 0 ˆ 0 0 t t q Q (4) 1( ) 2( ) 3( ) ( ) 1( ) ˆ t t t t t W W W W W (5)
1 2 1
1 1,1 1,2 1, 2 2,1 2,2 2,
,1 ,2 ,
m
m
m m m m m
a a a a b b b a b b b B
a b b b
(6) where
2 ( ) ( ) ( ) 11 2 1 m
t
t i i t t
i
n c q
1
1 ( ) ( )i
i t i i t
a n c
, ( ) , ( )
i j
i j t i j i j t
b n c
( )t
n is the neutron density, ci t( )is the density of precursors of the i-th group of precursors,
i is the fraction ofdelayed neutrons of the i-th group of precursors, is the total fraction of delayed neutrons,
i is the decay constantof the i-th group of precursors, is the velocity of fission neutrons q( )t is the source of neutrons, and i j, is the Kronecker delta. Bis the covariance matrix in equation (8) and Wˆ( )t is the Weiner process given in equation (5).
When the covariance matrix B0, the discrete version of the point kinetic equations is obtained.
The stochastic point kinetic equation given in equation (1) involves a random variable, in which its increments are given by a deterministic part, plus a part with uncertainty; this type of equation corresponds to the Itô differential equation, from the form
ˆ( )ˆ ˆ ˆ
, , t
t
t t
dW dX
a t X b t X
dt dt t0 t T (7)
To give a numerical solution to a differential equation of Itô, as that given in equation (7), the stochastic Runge-Kutta method is developed in the next section.
STOCHASTIC RUNGE-KUTTA METHOD
The explicit stochastic Runge-Kutta scheme of stages[13], is given by
1
1 1 1
ˆ
, ,
s m s
k k k
n n j n j j n j n j j
j k j
X X a t W b t R
(8)where 1 0
1 Xn
1 1
1 1 1
ˆ
, ,
j m j
k k k
j n ji n i i n ji n i i
i k i
X a t W b t
2,...,
j s
R is an appropriate term, , , k,
j j j ji
and k
ji
are numerical constants and Wˆ( )t represents Brownian
motions.
The parameters and term R must be chosen so that the approximation of equation (8) would be
-equivalent to the simplified order of Taylor’s scheme
. A considerable disadvantage of Taylor’s simplified schemes is that the determination of many derivatives is required, and to avoid this disadvantage in the Runge-Kutta scheme, the free adjustment term will be chosen so that for each family of parameters the number of derivatives in R must be notably smaller than in Taylor’s scheme. The functions a t
n j ,j
and
n j , j
b t are second order truncated expansions, these being given by
(2) 2, 1 1
2 3 3 4 2
2
, 1 , , 1 , , , 1
2 3
,
2
1 1
2 4 2
1 2
i j
d d
t i j i
i j i
ij
d d d d
ij kl ij kl
i j l i j k i j k l
i j i j k l i j k l
jk
i i j
f f X X f
f t X X f X
t x x x
f f c f f
c c c c
t t x x x x x x x x x x
f f
c
t x x x x
1 , 1
d d
i k
i j k
Where ( 2)
refers to the approximation of a second order function. For the scaling case dm1 and considering, in equation (9), that 2
cb
4 2 (2)
2 3
10 01 20 12 01 03 04 2
2
11 03 02 ,
4 2
2 2
t
b
f t X X f f f X f b f b b f f
X b
f f X f
(10) Notation i j
ij i j
f g t x
is implemented for
simplicity. The Brownian motions Wˆn in the scaling case take the form of
2
3
4
52 3
ˆn ˆn ˆn ˆn 3 ˆn
EW E W E W E W E W K
(11) With K0 . The combination of products of the
form i
Wˆn j with i j 1, 2,...,n for theapproximation
2 corresponds to:
1 2
2 (2)
(2)
(2)
1 2 1 2
if
ˆ ˆ
0 if
ˆ
3 if
ˆ ˆ ˆ
ˆ if j
5
ˆ ˆ ˆ 0 if
2 2
m
i j
n n
i
i j k n
n n n i
n
j j j
i m m
n n n
i j
W W
i j
W i j k
W W W
W k i
j j j
W W W i
(12)
SECOND ORDER STOCHASTIC RUNGE-KUTTA SCHEME
To obtain the general case of the second order third stage stochastic Runge-Kutta scheme we consider in equation (8) s2 for the scaling case dm1
1 1 2 , 1 2 , 3 , ˆ
n n n n n n
X X aa b b b W R (13) where
ˆ ˆ ˆ , ; , n n n n n n n nn n n n
t
X a b W X a b W
X a b W a a t X b b t X
(14)
Using equation (10) and bearing in mind the equivalences of equation (12) the following is obtained,
n,
a , b
n,
Wˆn and b
n,
Wˆn:
(2)
2 2 2 2
10 01 02
1
ˆ
,
2
n n
a a a aa W a b (15)
(2) 2
01 10 01
2 2 2 2
02 11 03 02
ˆ ˆ ˆ ˆ ˆ
,
3 ˆ 1
2 2
n n n n n n
n
b W b W bb W b W ab W
b b W b b b b abb
(16)
(2) 201 10 01
2 2 2 2
02 11 03 02
ˆ ˆ ˆ ˆ ˆ
,
3 ˆ 1
2 2
n n n n n n
n
b W b W bb W b W ab W
b b W b b b b abb
(17)
by replacing equations (15) to (17) in equation (13), the following is obtained
2
1 1 2 1 2 3 2 3 01
2 2 2 2
2 10 2 01 2 3 11 03 2 02 2 3 02
2 2 2
2 01 2 3 10 2 3 02 2 3 01
ˆ ˆ
1 1
2 2
3 ˆ
2
n n n n
n
X X a b W bb W
a aa b b b b a b abb
a b b b b ab W R
(18)
Now, Taylor’s second order is simplified scheme is given by [14]
22 2 2
1 01 10 01 02
2 10 01 02 01
1 1
ˆ ˆ
2 2
1 1 ˆ
2 2
n n n n
n
X X a b W bb W a aa a b
b ab b b ba W
(19)
2 2
1 2 2 3 2
1 2 3 2 3 2
2
2 3 2 3
2 2
01
2 3 2
1 1 1 6 2 1 1 1 2 2 1 1 1 2 2 2 1
1 1 ˆ
2
2 2 R bb Wn
(20)
With the equalities given in equation (20), it can be seen that the system has only one solution [13] when
1
and 12 12. Substituting these two values in equation (20), gives:1 2 2 2 3 2 1 ; 1 2 1 1 ; 3 2 6 3 ; 0 2 6 (21)
Replacing the equivalences of equation (21) in equation (13) we find the second order third stage stochastic Runge-Kutta (RK2-3st):
1 01
2 2
1 1 ˆ 1 ˆ 1 ˆ
,
2 2 2 2
1 ˆ ˆ 3 1 ˆ ˆ
, ,
3
2 6 2 6
n n n n n n n
n n n n n n n n
X X a b W a t X a b W bb W
b t X a b W W b t X a b W W
(22)
The scheme given by equation (22) is 2-equivalent to the scheme of equation (19) of Taylor´ second order.
The explicit RK2-3st method given by equation (22), can be applied to solve the equations of stochastic point kinetics given by equation (1), in the following way
2
1 1 1
2 2 2
1 2 2
2
1 1 3 ˆ
, , ,
2 1 3 1 3
,
1 ˆ ˆ 1 ˆ
2 n n n n 4
n n n n n n n n n
n n
t n t n n
n
x x B t x B B W
dB t x
A x Q A Q W
dx (23) where
1 2 1 2ˆ , ˆ
1
ˆ , ˆ
3
n n
n n
n n
n n t n t n n n
n n t n t n n n
t
x A x Q B t x W
x A x Q B t x W
(24)
In the following section, the RK2-3st method is implemented to solve the stochastic point kinetic equations, applying equation (23). Several numerical experiments with different reactivity values are developed for one and six groups of delayed neutron precursors.
RESULTS
In this section, the explicit RK2-3st method with
1 3
in equation (23) is implemented, using Matlab computational software. Several numerical experiments were developed, where one and six groups of delayed neutron precursors with constant reactivity are considered for various initial conditions and experimental data. Owing to the stochastic nature of the dynamic process explained by the point kinetic equation, numerical simulations with 5000 Brownian trajectories were considered for the computational development. In the simulation the pseudo-random numbers with normaldistribution were generated with seed 200 implementing the Matlab 'state’ command.
The approximations obtained with the explicit RK2-3st method are compared with stochastic methods reported in literature such as PCA, Monte Carlo, Taylor 1.5, AEM, ESM, DDDM and implicit EM. In the numerical results presented in the figures, three densities will be taken at random and the mean density of the 5000 Brownian trajectories.
The first numerical experiment considers a group of delayed neutron precursors in a nuclear reactor, and the following parameters were used:decay constant
1 1 0.1 s
, fraction of delayed neutrons, fission neutrons
2.5
, external source of neutrons q200 s1, neutron generation time 1 s
3
; a constant reactivity was used 1
3
, N 40 steps in a time interval
0, 2 s and the initial condition assumes equilibrium values(0) 400
n and c1(0) 300.
mean values obtained by the proposed stochastic RK2-3st method, and the results with the deterministic method of
fourth order Runge-Kutta, are also shown in Table-1.
Table-1. Comparison of methods for 1 3
with one group of precursors.
Method E n ( 2) n( 2) E c 1(2) c1(2) Montecarlo 400.0300 27.3110 300.0000 7.8073 PCA 395.3200 29.4110 300.6700 8.3564
EM 412.2300 34.3910 315.9600 8.2656
AEM 396.2800 31.2120 300.4200 7.9576 ESM 396.6200 0.91990 300.3000 0.0016 Double DDM 402.3500 28.6100 305.8400 7.9240 Implicit EM 399.9874 0.5439 299.8730 6.8405 RK2-3st 399.7225 31.7264 299.7662 8.0793
RK O(ℎ4) 400.0000 - 300.0000 -
Figure-1 shows the density of neutrons as a function of time, and Figure-2 shows the sum of precursor density obtained with the RK2-3st method. It shows that the proposed method can be applied to this problem of rigidity which is a characteristic of point kinetics.
0,0 0,5 1,0 1,5 2,0
385 390 395 400 405 410 415 420
N
eut
ro
n
D
ensity
(
n
/cm
3 )
Time (s)
Sample Neutron 1 Sample Neutron 2 Sample Neutron 3 Neutron Mean
Figure-1. Variation of neutron density for the first numerical experiment with constant reactivity
1 3
.
0,0 0,5 1,0 1,5 2,0
297,0 298,5 300,0 301,5 303,0 304,5
S
u
m o
f
Pr
ec
u
rso
r D
ensity
(
n
/cm
3 )
Time (s)
Sample Precursor 1 Sample Precursor 2 Sample Precursor 3
PrecursorMean
Figure-2. Variation of the sum of the density of precursors for the first numerical experiment with constant
reactivity 1 3 .
In the second and third numerical experiments, six groups of delayed neutron precursors were considered, using the parameters for the combustible element of 235
92U. Decay constant: 1 0.0127 s 1
; 2 0.0317 s 1
;
1
3 0.115 s
; 4 0.311 s 1
;5 1.4 s1
and
1
6 3.87 s
, the fraction of delayed neutrons
1 0.000266
;
20.001491;
30.001316;4 0.002849
;
50.000896 and
6 0.000182, the0.007
, time interval
0, 0.001 s The initial conditions
for both experiments are n(0)100 y i(0) 100 ii c . For the second experiment, the results obtained with the RK2-3st method for neutron density are presented in Figure-3 and for the sum of the density of precursors Figure-4. For the third experiment, the neutron density is
shown Figure-5 and the sum of the density of precursors Figure-6. Tables 2-3 present the mean values and the variance of the neutron density and the sum of density of delayed neutron precursors for the second numerical experiment in the timet0.1 s, and for the third experiment int0.001 .s
Table-2. Comparison of methods for 0.003 with six groups of precursors.
Method E n (0.1) n(0.1) E c i(0.1) ci(0.1) Montecarlo 183.0400 168.7900 447800 1495.7000
PCA 186.3100 164.1600 449100 1917.2000 EM 208.6000 255.9500 449800 1233.3800 Taylor 1.5 199.4080 168.5470 449700 1218.820
AEM 186.3000 164.1400 449000 1911.9100
ESM 179.9300 10.5550 448900 94.7500
Double DDM 187.0500 167.8300 448800 1475.6000 Implicit EM 179.9461 0.2178 448880 60.4267
RK2-3st 179.3368 197.3299 448857 2043.8392
RK O(ℎ4) 179.9528 - 448877 -
Figure-3. Variation of neutron density for the second numerical experiment with constant reactivity
0.003 .
0,00 0,02 0,04 0,06 0,08 0,10
446260 446790 447320 447850 448380 448910
S
u
m o
f
Pr
ec
u
rso
r D
ensity
(
n
/cm
3 )
Time (s) Sample Precursor 1 Sample Precursor 2 Sample Precursor 3 Precursor Mean
Figure-4. Variation of the sum of the density of precursors for the second numerical experiment with constant
Table-3. Comparison of methods for 0.007 with six groups of precursors.
Method
E n
(0.001)
n(0.001) E c i(0.001) ci(0.001) Montecarlo 135.6500 93.3760 446400 16.2260PCA 134.5500 91.2420 446400 19.4440
EM 139.5680 92.0420 446300 6.0710
Taylor 1.5 139.5690 92.0470 446300 18.3370
AEM 134.5400 91.2340 446400 19.2350
ESM 134.9600 6.8527 446400 2.5290
Double DDM 135.8600 93.2100 446300 17.8450 Implicit EM 134.9218 5.9661 446360 6.0686
RK2-3st 134.8104 93.2386 446360 19.2959
RK O(ℎ4) 135.0009 - 446360 -
0,0000 0,0002 0,0004 0,0006 0,0008 0,0010
90 105 120 135 150 165
N
eut
ro
n
D
ensity
(
n
/cm
3 )
Time (s)
Sample Neutron 1 Sample Neutron 2 Sample Neutron 3 Neutron Mean
Figure-5. Variation of neutron density for the third numerical experiment with constant reactivity
0.007 .
0,0000 0,0002 0,0004 0,0006 0,0008 0,0010
446352,5 446355,0 446357,5 446360,0 446362,5 446365,0
S
u
m o
f
Pr
ec
u
rso
r D
ensity
(
n
/cm
3 )
Time (s)
Sample Precursor 1 Sample Precursor 2 Sample Precursor 3 Precursor Mean
Figure-6. Variation of the sum of the density of precursors for the third numerical experiment with constant
reactivity 0.007.
Table-2 shows that in the RK2-3st method, good approximations are obtained in mean values of neutron density and the concentration of delayed neutron precursors. These values are close to the approximations of the implicit Euler-Maruyama method; the RK2-3st method presents a value in the variance of neutron density lower than that reported with the Euler-Maruyama method.
The results obtained for the third numerical experiment are shown in Table-3. It can be seen that the mean values found for the neutron density and the sum of the density of precursors with the proposed method are values which are close to the methods reported in literature. The variance of the sum of precursor density is very similar to the PCA, EM, Taylor 1.5, AEM and Double DDM methods. Comparing the average values obtained with the proposed RK2-3st stochastic scheme, and with the deterministic method of fourth order Runge-Kutta it can be observed in Tables 2-3 that the proposed method has satisfactory approximations for calculating mean values. In addition, the standard deviation of the density of neutrons and of precursors with constant reactivity and six groups of delayed neutron precursors has good approximations.
CONCLUSIONS
In this paper the explicit Stochastic RK2-3st method is implemented. It has a great advantage over other methods of numerical derivation, because it requires the calculation of only one derivative of the covariance matrix function. The proposed method provides good approximations for the study of point kinetics with constant reactivity for one and six groups of delayed neutron precursors, compared with other stochastic methods reported in literature.
Physics FIASUR, and the academic and financial support of the Universidad Surcolombiana.
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