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
1
Outline


1 Introduction
2 Time discrete scheme
2.1 Typical time discrete scheme
2.2 Second-order time evolution scheme



2
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
The time-dependent transport equation is
studied to comprehend the time behavior for
neutron, photon, charged particle.
3
neutron transport equation
The time-dependent neutron transport
equation may be written as follows in multigroup form:
Space variable：x，y，z
Angular variable：μ，η
Energy variable ：E
time variable ：t
There are seven variables to demonstrate the angular
flux.
4
Ψg is the angular flux of g- th group neutron;
v g is the velocity of g-th group neutron;
ΣTrg is the total macroscopic cross section of
g-th group neutron;
Qg is the sum of scattering source(Qsg),
fission source(Qfg )and external source(Sg).
5
The determine methods for transport equation
are:
SN
(simple)
Solve neutrality
particle transport
equations
6
PN (complex)
Nuclear pile
Medicine region
astrophysics
We consider the spherical neutron transport
equations which the coordinate is as follows:

e2


( ex ) x

(e y ) y



r
o

( ez ) z

er

e1
Spherical system
7
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)
8
We give the following definition for describe the
physical progress.
J 

0
 g (rJ ,  , t )d 
w
This physical quantity J gives the information about outflux at outermost
boundary,which denotes the outflux current of system particle.
N  4  dE  r 2 dr
E
V
 (r , E , t )
v
This physical quantity N gives the information about flux at center cell,
which denotes the total number of system neutron.
center 
dN
Ndt
 center
d center

dt
edge 
dJ
Jdt
Denotes the derivative of outflux current and total number of system
neutron respectively.
9
The goals of discrete method and iterative method are:
• numerical precision
• computing time
10
For some physical problems, the differential quantity of flux
about time variable is very important.
theoretical solution or
Analytic solution
center 
edge
dN
Ndt
dJ

Jdt
Numerical solution
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.
11
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.
12
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:
13
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) exponential method(EM);
(2) diamond difference(DD).
14
The time step can be large at stage for physical
progress
The time step can be small at strenuous stage for
physical progress.
15
The modied exponential method(MEM) is
The modied diamond difference(MDD) is
16
2.2 Second-order time evolution scheme
To consider the time step change in the whole physical
progress adequately, we apply the second-order time
evolution(SOTE) scheme to time-dependent spherical
neutron transport equation by discrete ordinates(Sn)
method.
The SOTE considers the case of adaptive time step for
the whole physical progress and needs not to introduce
exponential extrapolation or diamond extrapolation.
17
We deduce the discrete scheme for neutron transport
equation by SOTE. The SOTE take three-level backward
difference and the equation is as followed:
18
SOTE_EM:
-1<μm<0:
19
SOTE_DD:
-1<μm<0
20
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.
21
3 Space linear discontinuous finite
element method
Space LD＋time DD＋angular DD
22
The primary function is
 rk 1  r
rk  r  rk 1


r
f k (r )  
k
0
else

 r  rk
rk  r  rk 1


r
f k 1 (r )   k
0
else

Weight function is:
w1 (r )  1
 r  rk
w2 (r )  
 rk 1  r
23
m  0
m  0
The cells are as followed:
m
m
k ,m
 kb, m
rk
24
k 1,m
a: m  0
rk+1
kb1,m
k ,m
rk
k 1,m
b:
m  0
rk+1
The discrete equations are:
  1
n1
2Vkn1
2
V
2 n 12 n1 n 12 n1
tr n1 n1
tr n 1 n 1
k 1
[
 z15  gVk ]k ,m  [
 gVk 1 ]k 1,m 
(k ,m Vk  k 1,mVk 1 )  z15k(b),1,nm1  Qkn1Vkn1  Qkn11Vkn11
vg t
vg t
vg t
[
2 z8
2z
2
n 1
n 1
 z16  z8trg ]kn,m1  [ 1  z16  z1trg ]kn1,1 m 
( z8k ,m2  z1k 1,2 m )  z8Qkn1  z1Qkn11
vg t
vg t
vg t
25
m  0
n 1
n 1

r
(
A

A
) m  1 n 1
k
k

1
k
2Vkn 1
n 1
tr n 1
2
[
 m rk Ak   g Vk 
]k ,m 
vg t
2m
n 1
n 1

r
(
A

A
) m  1 n 1
k
k

1
k
2Vkn11
2
n 1
n 1
tr n 1
2
[
  g Vk 1 
] k 1,m 
( k ,m2Vkn 1  k 1,2 mVkn11 )
vg t
2m
vg t
  m rk Akn11 k( b),1,nm1 
rk ( Akn11  Akn 1 )( m  1   m  1 ) kn11 ,m  1
2
2m
2
2
2
 Qkn 1Vkn 1  Qkn11Vkn11
z14 rk m  1 n 1
z14 rk m  1 n 1
2 z8
2 z1
tr
tr
2
2
[
 z10 m  z8 g 
]k ,m  [
 z11m  z1 g 
] k 1,m
vg t
m
vg t
m
n 1

r
z
(

1   m  1 ) k  1 , m  1
1
1
k
14
2
m

n
n
2
2
2
2

( z8 k , m2  z1 k 1,2 m ) 
 z8Qkn 1  z1Qkn11
vg t
m
26
m  0
n 1
n 1

r
(
A

A
) m  1 n 1 2V n 1
k
k

1
k
2Vkn 1
tr n 1
2
[
  g Vk 
]k ,m  [ k 1  m rk Akn11  trg Vkn11
vg t
2m
vg t


rk ( Akn11  Akn 1 ) m  1
2
2m
]kn1,1 m 
2
n 1
n 1
(k ,m2Vkn 1  k 1,2 mVkn11 )  m rk Akn 1k(b,m),n 1
vg t
rk ( Akn11  Akn 1 )( m  1   m  1 ) kn11 ,m  1
2
2m
2
2
2
 Qkn 1Vkn 1  Qkn11Vkn11
z7 rk m  1 n 1
z7 rk m  1 n 1
2 z1
2 z2
tr
tr
2
2
[
 z3 m  z1 g 
]k ,m  [
 z4  m  z2  g 
] k 1,m
vg t
m
vg t
m
n 1

r
z
(



)

1
1
1
1
k
7
2
m 2
m 2
k  12 , m  12
n
n

( z1 k ,m2  z2k 1,2 m ) 
 z1Qkn 1  z2Qkn11
vg t
m
27
The progress of soving （tn）
1
2
μ1/2=1



μ1=0.86
3
r
μ2=-0.34μ5/2=0 μ34=0.34
4
μ4=0.86 μ
5
(boundary condition)
(discrete scheme)
(extrapolate form for DD or EM)
The key problem is that the progress should be in
agreement with movement direction of neutron.
2015/4/8
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4 Numerical results
4.1 Tests for time discrete scheme
The problem includes two media.
media 2
media 1
29
 The isotropic scattering source is employed;
 The discrete angular takes S4 ;
 The end time is 0.1µs;
 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(10-4µs)(EM)to this
problem.
30
TABLE 1
Adaptive time step
31
Fig.1. Neutron number for EM,MEM,SOTE_EM
32
Fig.2.
for EM,MEM,SOTE_EM
Fig.3. Neutron current for EM,MEM,SOTE_EM
33
Fig.4.
for EM,MEM,SOTE_EM
Fig.5. Iteration for EM, MEM, SOTE_EM
Fig.6. Neutron number forDD,MDD,SOTE_DD
34
Fig.7.
for DD,MDD,SOTE_Dd
Fig.8. Neutron current for DD,MDD,SOTE_DD
35
Fig.9.
for DD,MDD,SOTE_DD
Fig.10. Iteration for DD, MDD, SOTE_DD
4.2 Test for space discrete scheme
The results presented and discussed in this section
are organized into three subsections.
① we analyze the time-independent transport
equation.
② a kind of particular transport equation with a small
perturbation is studied.
③ 1-D spherical geometry multi-group timedependent transport equation is studied, and
anisotropic scattering source with P5 spherical
harmonic expansion is considered.
36
4.2.1 Time-independent transport problems
37
4.2.2 Transport problems with a small perturbation
The radius is 0.5cm, and boundary condition is
38
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 P5 is employed.
• There is no analytic solution for this problem, therefore the
numerical solution of exponential method by fine cell(S16, Δx =
0.1cm, Δ t=5×10-5μs) is used by reference solution.
• The solutions of coarse cell(S4, max Δ x = 0.97cm, Δ t=2×10-4
μs) for different scheme are contrasted with that of fine cell.
39
40
41
The cell center flux and cell edge flux for EM, DD exit
numerical solution oscillation near different media
interface.
However, the LD is asymptotic preserving
scheme(Larsen and Morel, 1983; Klar,1998, Jin, 1999).
The cell center flux and cell edge flux for LD are very
smooth and approach to benchmark solution(fine cell).
42
5 Conclusions
We study the numerical solution oscillation from the
aspect of time discrete and space discrete 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 outcurrent at outermost boundary .
43
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.
44
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 method can
provide accurate numerical solutions for timedependent neutron transport equations.
45
Future work
The shortcoming of SOTE EM and LD is that the
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.
46
Thank You!
47