Spectral Analysis of the Matrix for Three-dimensional
Discrete Ordinates Problems
Marco T. M. B.de Vilhena
Eliete Biasotto Hauser∗
Depto. de Matemática - PUCRS
Depto. de Matemática - UFRGS
[email protected]
[email protected]
Ricardo Ricardo C. Barros
Ruben Panta Pazos
IPRJ - Universidade do Estado do Rio de Janeiro
[email protected]
[email protected]
Based on a spectral analysis of the LTSN matrices, we describe a new algorithm to generate an analytical solution for discrete ordinates problems in
three-dimensional cartesian geometry with isotropic
scattering and one energy group that appear as:
i
h
∂
∂
∂
+ ηm ∂y
+ ξm ∂z
+ σt Ψm (x, y, z) =
µm ∂x
Q(x, y, z) +
Depto. de Matemática - UNISC
M
σs X
wn Ψn (x, y, z) ,
8 n=1
(1)
where m = 1 : M are positive integer numbers,
M = N (N + 2) is the cardinality of the discrete
ordinates set, N represents the order of the angular quadrature, σt is the macroscopic total crosssection, σs is the diferential scattering cross-section,
Q(x, y, z) the source term , Ψm (x, y, z) is the angular flux in the discrete direction Ωm = (µm , ηm , ξm )
and wm is the weight of the angular quadrature set
used to model the transport problem (Lewis [9]).
to generate the systems of linear algebric equations,
that is
(sI − Ax )Ψx (s) = Ψx (0) + S x (s),
(sI − Ay )Ψy (s) = Ψy (0) + S y (s),
(2)
(sI − Az )Ψz (s) = Ψz (0) + S z (s).
The solutions of the linear systems (2) are given
by:
Ψx (s) = (sI − Ax )−1 [Ψx (0) + S x (s)],
Ψy (s) = (sI − Ay )−1 [Ψy (0) + S y (s)],
(3)
Ψz (s) = (sI − Az )−1 [Ψz (0) + S z (s)].
The entries of the matrix Az , with i = 1 : M and
j = 1 : M , have the form

8σt − σsii wi


se i = j

 −
8ξi
az (i, j) =
, (4)

σsij wj



se i =
6 j
8ξi
T
Ψz (0) = [ Ψ1z (0) Ψ2z (0) · · · ΨM x (0) ] ,
the vector of unknowns is
T
Ψz (s) = Ψ1z (s) Ψ2z (s) · · · ΨM z (s)
(5)
(6)
and the vector Sz (s) has generic components
i
h x
x
µi
−
(0,
s)
−
(a,
s)
S zi (s) = Qzξ(s)
Ψ
Ψ
i
i
aξm
i
Figure 1: Ωm = (µm , ηm , ξm ), m = 1 :
M
8
,N = 4
Furthemore, we first transverse integrate the SN
equations and then we apply the Laplace transform
∗ Partially
supported by FAPERGS and CNPq
−
i
ηi h y
y
Ψi (0, s) − Ψi (b, s) .
bξi
(7)
In this work we describe the diagonalization
Az = Vz Dz Vz(−1) ,
(8)
where, Dz is the diagonal matrix of eigenvalues of
Az and Vz the matrix of the respective eigenvectors.
(−1)
(−1)
Similarly, Ay = Vy Dy Vy
and Ax = Vx Dx Vx .
This is possible from the complete description of the
spectra of these matrices, because any eigenvalues
are multiples(Tabel 1) but the respectives eigenvectors are independet lineary.
Ψx (x) = [Vx eDx x Vx−1 ]Ψx (0) + [Vx eDx x Vx−1 ] ∗ Sx (x) ,
−1
Dy y −1
Ψy (y) = [Vy eD
Vy ] ∗ Sy (y) ,
y yVy ]Ψy (0) + [Vy e
Ψz (z) = [Vz eDz z Vz−1 ]Ψz (0) + [Vz eDz z Vz−1 ] ∗ Sz (z) .
References
N
2
4
6
8
10
12
14
16
σt
ξ1
σt
ξ2
σt
ξ3
σt
ξ4
σt
ξ5
σt
ξ6
σt
ξ7
σt
ξ8
3
7
11
15
19
23
27
31
3
7
11
15
19
23
27
3
7
11
15
19
23
3
7
11
15
19
3
7
11
15
3
7
11
3
7
3
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in Spectral Nodal Methods Applied to Discrete Ordinates Transport Problems, Progress
in Nuclear Energy, 12-33 (1998), 117-154.
[2] R. C. Barros, E. W. Larsen, A spectral Nodal
Method for One-group X,Y-geometry Discrete
Ordinates Problems, Nuclear Science and Engineering, 34-111 (1992), 34-45.
[3] R. C. Barros, E. W. Larsen, A Numerical
Method for One-Group Slab-Geometry Discrete Ordinates Problems, Nuclear Science and
Engineering, 19-104 (1990), 199-208.
Table 1: Multiples eigenvalues of Az for
Ωm = (µm , ηm , ξm ),m = 1 :
M
2
We assume a solution in the exponential form for
the homogeneous equations associate with the onedimensional transverse integrated SN nodal equations. Therefore we obtain the eigenvalue problem
M
σs X
αn wn ,
8 n=1
M
σs X
αm (s)(σt + sηm ) =
αn wn ,
8 n=1
M
σs X
αm (s)(σt + sξm ) =
αn wn .
8 n=1
αm (s)(σt + sµm ) =
(9)
[4] K. Case, E. Zweifel, “ Linear Transport Theory”, Addison-Wesley Publishing Company,
1967.
[5] E.B.Hauser, M. T. B. Vilhena, R. C. Barros, Análise Espectral da Matriz LT SN
para o Problema de Ordenadas Discretas
Bidimensional Cartesiano com Fonte Fixa
e Espalhamento Isotrópico , Tendências em
Matemática Aplicada e Computacional, SBMAC, 3 ,2 (2002)135-140.
[6] E. B. Hauser, R.P.Pazos, M.T.M.B.Vilhena,
An Error Bound Estimate and Convergence of
the Nodal-LTSN Solution in a Rectangle, Annals of Nuclear Energy, 32 (2005) 1146-1156.
In order to determine the angular fluxes we
apply the inverse Laplace transform to (3)
−1
] [7] E. B. Hauser,, Panta Pazos, R., Barros,
£−1 [(sI − Az )−1 ] = £−1 [ sVz Vz−1 − Vz Dz Vz−1
R.C., Vilhena, M.T., Solution and study of
Dz z −1
−1
−1
−1
Nodal Neutron Transport Equation applying
Vz .
= Vz £ [(sI − Dz ) ]Vz = Vz e
the
LTSN DiagExp Method. Proceedings of
In [8] was determined two algorithms whose difthe
18th
International Conference on Transference lies in the representation of the transverse
port
Theory,
pp. 303307, Rio de Janeiro,
leakage terms that appear in the transverse inBrazil,2003.
tegrated SN two-dimensional equations. These
terms in version LT SN 2D−Diag are written as lin[8] E. B. Hauser, ”Estudo e Solução da Equação
ear combinations of the eigenvectors multiplied by
de Transporte de Nêutrons Bidimensional
exponential functions of the corresponding eigenvapelo Método LT SN Para Elevadas Ordens
lues. As with in the LT SN 2D − DiagExp method,
de Quadraturas Angulares: LT SN 2D - Diag
the transverse leakage terms are represented by exe LT SN 2D - DiagExp”, Doctoral Thesis,
ponential functions with decay constant depending
PROMEC, UFRGS, Porto Alegre, RS, 2002.
on the characteristics of the material associated to
[9] E. Lewis, W. Miller, “Computational Methods
the medium.
of Neutron Transport”, John Wiley-Sons, New
We proceed in a similar form with threeYork, 1984.
dimensional problem and representing the convolution operation by ∗, we determine the analyti- [10] R. P. Panta, M. T. B. Vilhena, Convergence in
cal matrix form for the edge-average angular flux
Transport Theory , Applied Numerical Matheas function of the average angular flux at the bonmatics, 9 (1999),79-92.
daries.