Transcript discrete scheme
1 14th International Conference on Hyperbolic Problems:Theory, Numerics, Applications, June 25–29, 2012, Padova, Italy
Time and space discrete scheme to suppress numerical solution oscillations for the neutron transport equations Zhenying Hong
2012.06.28
Cooperated with Pro. Guangwei Yuan and Pro. Xuedong Fu
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Outline
1 Introduction 2 Time discrete scheme
2.1 Typical time discrete scheme 2.2 Second-order time evolution scheme
3 Space linear discontinuous finite element method 4 Numerical results 5 Conclusions
1 Introduction
With the development of nuclear energy, the new fission-type reactor has complex structure:
strong non-uniform medium
strong anisotropic Furthermore, the nuclear device has more complicated characteristic, for example:
Width energy region
Complicated dynamic state
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The time-dependent neutron , photon , transport equation is studied to comprehend the time behavior for charged particle .
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neutron transport equation The time-dependent group form: neutron transport equation may be written as follows in multi Space variable
：
x
，
y
，
z Angular variable
：
μ
，
η Energy variable
：
E time variable
：
t There are seven variables to demonstrate the angular flux.
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Ψ Σ v Q g Tr g g is the angular flux of g- th group neutron; is the velocity of g-th group neutron; is the total macroscopic cross section of g-th group neutron; g is the sum of scattering source(Q s g ), fission source(Q f g )and external source(S g ).
The determine methods for transport equation are: SN (simple) PN (complex) Solve neutrality particle transport equations Nuclear pile Medicine region astrophysics
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We consider t he spherical neutron transport equations which the coordinate is as follows:
e
2
e
r
(
e
x
) x
r
(
e
y
) y o (
e
z
) z
e
1
Spherical system
If each space directions are the same of the spherical device, the equation can be changed to one-dimensional spherical transport equation.
time variable (1)
Space variable(1)
Energy variable(1)
angular variable(1)
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We give the following definition for describe the physical progress.
J
0
g
(
r J w
,
This physical quantity J gives the information about outflux at outermost boundary,which denotes the outflux current of system particle.
N
4
E V
2
r dr
v
This physical quantity N gives the information about flux at center cell, which denotes the total number of system neutron.
center
dN
center
d
center
edge
dJ Ndt dt Jdt
Denotes the derivative of outflux current and total number of system neutron respectively.
The goals of discrete method and iterative method are:
•
numerical precision
•
computing time
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For some physical problems, the differential quantity of flux about time variable is very important.
theoretical solution or Analytic solution
center
dN Ndt
edge
dJ Numerical solution Jdt
sketch map The numerical solution can not give the exact maximum
point.
Therefore, we will focus on preserving physical nature based on keeping some numerical precision.
In the following sections, we will talk about numerical schemes which can suppress numerical oscillation .
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2 Time discrete scheme
Adaptive time step Change from 10 -3 to 10 -5 some magnitude difference Therefore we need to study more accurate numerical method to simulate complex transport equations.
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2.1 Typical time discrete scheme We focus on conservative equation for 1-D spherical geometry transport equations in the multi-group form: With the following initial and boundary conditions:
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To spherical transport equation, the finite volume method ( FVM ) is the typical method which involves the extrapolation of angular, time, space variables. These extrapolation can adopt the same form and also adopt different form for physical problems. The classical extrapolations are: (1) (2) exponential method(EM); diamond difference(DD).
The time step can be
large
at stage for physical progress The time step can be
small
at strenuous stage for physical progress.
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The modied exponential method(MEM) is The modied diamond difference(MDD) is
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2.2 Second-order time evolution scheme To consider the time step change in the whole physical progress evolution(SOTE) scheme to time-dependent spherical neutron transport equation by method. adequately, we apply the second-order time discrete ordinates(Sn) The SOTE considers the case of adaptive time step for the whole physical progress and needs not to introduce exponential extrapolation or diamond extrapolation .
We deduce the discrete scheme for neutron transport equation by SOTE. The SOTE take three-level backward difference and the equation is as followed:
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SOTE_EM: -1<μm<0:
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SOTE_DD: -1<μ m <0
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We get the SOTE_EM and SOTE_DD by combining SOTE for time variable with EM or DD for other variables.
The discrete equation for SOTE_EM is a nonlinear equation;
The discrete equation for SOTE_DD is a linear equation .
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3 Space linear discontinuous finite element method
Space LD
＋
time DD
＋
angular DD
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The primary function is
k
r
0
k
1
r k r else r k r k
1
f k
1
r
r k
0
r k r k else r k
1
Weight function is:
w r
2 ( )
r r k
1
r k
r
m
m
0 0
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The cells are as followed:
m
m
k
1,
m
b
r k a:
m
0 r k+1
k
1,
m
r k b:
m
0 r k+1
b k
1,
m
The discrete equations are:
1 [ 2
V k n
1 15
tr V g k n
1 ]
n
1 [ 2
V k n
1 1
tr V g k n
1 1 ]
k n
1 1,
m
2 (
n
1 2
V k n
1
k n
1, 2 1
V m k n
1 1 )
z
15
k
1,
m
k
1
k n
1
Q V k
1
k n
1 1 [ 2
z
8
v g
t
z
16
z
8
tr g
]
n
1 [
v
2
z
1
t g
z
16 1
tr g
]
n
1
k
1,
m
2
v g
t
(
z
8
n
1 2
z
1
k n
1 2 1,
m
)
z Q
8
k n
1
z Q
1
n
1
k
1 25
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m
0 [ 2
V k n v g
1
t
m
r A k k n
1
g tr V k n
1
k k n
1 1
A k n
1 )
m
1 2 2
m
]
n
1 [ 2
V k n
1 1
v g
t
g tr V k n
1 1
k
m
r A k k n
1 1
k
1,
m
1
k n
1 1
A k n
1 )
m
1 2 2
m
]
k n
1 1,
m
r k
(
A k n
1 1
A k n
1 )(
m
1 2 2
m
2
v g
t
(
n
1 2
V k n
1
k n
1 2 1,
m V k n
1 1 )
m
1 2 )
k n
1 1 2 ,
m
1 2
k n
1
Q V k n
1
k n
1
Q V
1
k n
1 1 [
v
2
z
8
t g
z
10
m z
8
g z
14
r k
m
1 2
m
]
n
1 [
v
2
z
1
t g
z
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m
2
v g
t
(
z
8
n
1 2
z
1
k n
1 2 1,
m
)
r z k
14 (
m
1 2
m
m
1 2 )
k n
1 1 2 ,
m
1 2
z
1
g z
14
r k
m
z Q
8
k n
1
z Q
1
k n
1 1
m
1 2 ]
k n
1 1,
m
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m
0 [ 2
V k n v g
1
t
g tr V k n
1
k
k k n
1 1
A k n
1 )
m
1 2 2
m
]
n
1 [ 2
V k n
1 1
v g
t k n
1 1
A k n
1 )
m
1 2 2
m
]
k n
1 1,
m
m
r A k k n
1 1
g tr V k n
1 1 2
v g
t
(
n
1 2
V k n
1
k n
1 2 1,
m V k n
1 1 )
m
r A k n
1 1
k
[
v
2
z
1
t g
k n
1 1
z
3
m A k n
1
z
1 )(
m
1 2 2
m
z
7
r k g
m
1 2
m
)
m
1 2
k
]
n
1 1 2 ,
m
n
1 1 2
k n
1
Q V k n
1 [
v
2
z
2
t g
z
4
m n
1
Q V k
1
k n
1 1
z
2
tr g
2
v g
t
(
z
1
n
1 2
z
2
k n
1 2 1,
m
)
r z k
7 (
m
1 2
m
1 2
m
)
k n
1 1 2 ,
m
1 2
z
7
r k
m m
z Q
1
k n
1
z Q
2
k n
1 1 1 2 ]
k n
1 1,
m
1 2
The progress of soving
（
t n
） 3
r
μ 1/2 = 1 μ 1 = 0.86
μ 2 =-0.34
μ 5/2 =0 μ 3 =0.34
μ 4 =0.86
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(boundary condition) (discrete scheme) (extrapolate form for DD or EM)
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μ The key problem is that the progress should be agreement with movement direction of neutron.
2020/5/1
in
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4 Numerical results
4.1
Tests for time discrete scheme
The problem includes two media.
media 2 media 1
The isotropic scattering source is employed; The discrete angular takes The end time is 0 .
1µ s ; S 4 ; The self adaptive time step is showed in table 1. We adopt the typical EM, DD and the modified time discrete scheme and second-order time evolution scheme. To study the computing effectiveness, we also take constant time step(10 -4 µs)(EM)to this problem.
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Fig.1. Neutron number for EM,MEM,SOTE_EM Fig.2. for EM,MEM,SOTE_EM
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Fig.3. Neutron current for EM,MEM,SOTE_EM Fig.4. for EM,MEM,SOTE_EM
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Fig.5. Iteration for EM, MEM, SOTE_EM
Fig.6. Neutron number forDD,MDD,SOTE_DD Fig.7. for DD,MDD,SOTE_Dd
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Fig.8. Neutron current for DD,MDD,SOTE_DD Fig.9. for DD,MDD,SOTE_DD
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Fig.10. Iteration for DD, MDD, SOTE_DD
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4.2 Test for space discrete scheme The results presented and discussed in this section are organized into three subsections.
① ② ③
we analyze the time-independent transport equation.
a kind of particular transport equation with a small perturbation is studied.
1-D spherical geometry multi-group time dependent transport equation is studied, and anisotropic scattering source with harmonic expansion is considered.
P 5 spherical
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4.2.1 Time-independent transport problems
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4.2.2 Transport problems with a small perturbation The radius is 0 .
5 cm , and boundary condition is
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4.2.3 Spherical geometry multi-group time dependent transport problem
• • •
This test is about spherical geometry multi-group time-dependent problem including two media. The four-group cross sections are considered and the anisotropic scattering source with P 5 is employed.
•
There is no analytic solution for this problem, therefore the numerical solution of exponential method by fine cell(S16,
Δ
x = 0 .
1 cm , Δ t=5
×10 -5 μs
) is used by reference solution.
The solutions of coarse cell(S4, max Δ x = 0 .
97 cm , Δ t=2
×
μs ) for different scheme are contrasted with that of fine cell.
10 -4
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The cell center flux and cell edge flux for numerical solution oscillation near different media interface.
scheme(Larsen and Morel, 1983; Klar,1998, Jin, 1999).
However, the LD is EM, DD asymptotic preserving exit The cell center flux and cell edge flux for LD are very smooth and approach to benchmark solution(fine cell).
5 Conclusions
We study the numerical solution oscillation aspect of time discrete and space discrete from the scheme.
According the character of time discrete for adaptive time step , we study:
Typical EM,DD;
Modifed time discrete scheme; Second-order time evolution method (construct SOTE_EM; SOTE _DD ).
Advantage
(MDD,MEM): The modifed scheme is simple and the iteration number is lower than others. The
neutron number
are
smooth
.
Weakness
(MDD,MEM): There has
oscillating
for
out current at outermost boundary
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The second-order time evolution scheme associated exponential method has some good properties. Advantage (SOTE_EM): The differential curves including out-current at outermost boundary are more smooth than that of EM,DD,MEM,MDD.
Weakness (SOTE_EM): The iteration number is more than other.
According the character of mult-media, we study different space discrete scheme:
Typical EM,DD;
LD.
The LD method yields more accurate results, especially for the flux on edge of cell, and can reduce the oscillation effectively. Therefore the LD provide accurate numerical solutions for time dependent neutron transport equations.
method can
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Future work The shortcoming of SOTE EM and iterative number is more than other schemes and we will take acceleration method such as taking effective iterative initial value(Hong,Yuan and Fu,2008) to decrease the iterative number.
LD is that the
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