1 Zhenying Hong Time and space discrete scheme to suppress numerical solution oscillations for the neutron transport equations th International Conference on Hyperbolic Problems:Theory, Numerics, Applications, June 25–29, 2012, Padova, Italy Cooperated with Pro. Guangwei Yuan and Pro. Xuedong Fu

2 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

3 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.

4 There are seven variables to demonstrate the angular flux. Space variable ： x ， y ， z Angular variable ： μ ， η Energy variable ： E time variable ： t neutron transport equation The time-dependent neutron transport equation may be written as follows in multi- group form:

5 Ψ g is the angular flux of g- th group neutron; v g is the velocity of g-th group neutron; Σ Tr g is the total macroscopic cross section of g-th group neutron; Q g is the sum of scattering source(Q s g ), fission source(Q f g )and external source(S g ).

6 The determine methods for transport equation are: SN (simple) PN (complex) Nuclear pile Medicine region astrophysics Solve neutrality particle transport equations

7 We consider t he spherical neutron transport equations which the coordinate is as follows: Spherical system

8 time variable (1) Space variable(1) Energy variable(1) angular variable(1) If each space directions are the same of the spherical device, the equation can be changed to one-dimensional spherical transport equation.

9 This physical quantity J gives the information about outflux at outermost boundary,which denotes the outflux current of system particle. This physical quantity N gives the information about flux at center cell, which denotes the total number of system neutron. Denotes the derivative of outflux current and total number of system neutron respectively. We give the following definition for describe the physical progress.

10 The goals of discrete method and iterative method are: numerical precision computing time

11 Therefore, we will focus on preserving physical nature based on keeping some numerical precision. For some physical problems, the differential quantity of flux about time variable is very important. theoretical solution or Analytic solution Numerical solution The numerical solution can not give the exact maximum point. In the following sections, we will talk about numerical schemes which can suppress numerical oscillation. sketch map

12 2 Time discrete scheme Adaptive time step Change from to Therefore we need to study more accurate numerical method to simulate complex transport equations. some magnitude difference

13 With the following initial and boundary conditions: We focus on conservative equation for 1-D spherical geometry transport equations in the multi-group form: 2.1 Typical time discrete scheme

14 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).

15 The time step can be large at stage for physical progress The time step can be small at strenuous stage for physical progress.

16 The modied exponential method(MEM) is The modied diamond difference(MDD) is

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.

18 We deduce the discrete scheme for neutron transport equation by SOTE. The SOTE take three-level backward difference and the equation is as followed:

19 SOTE_EM: -1<μm<0:

20 SOTE_DD: -1<μ m <0

21 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.

22 Space LD ＋ time DD ＋ angular DD 3 Space linear discontinuous finite element method

23 The primary function is Weight function is:

24 The cells are as followed:

25 The discrete equations are:

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(boundary condition) (discrete scheme) (extrapolate form for DD or EM) The progress of soving （ t n ） The key problem is that the progress should be in agreement with movement direction of neutron.

29 4 Numerical results 4.1 Tests for time discrete scheme The problem includes two media. media 1 media 2

30 The isotropic scattering source is employed; The discrete angular takes S 4 ; 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 -4 µs)(EM)to this problem.

31 TABLE 1 Adaptive time step

32 Fig.1. Neutron number for EM,MEM,SOTE_EM Fig.2. for EM,MEM,SOTE_EM

33 Fig.3. Neutron current for EM,MEM,SOTE_EMFig.4. for EM,MEM,SOTE_EM Fig.5. Iteration for EM, MEM, SOTE_EM

34 Fig.6. Neutron number forDD,MDD,SOTE_DDFig.7. for DD,MDD,SOTE_Dd

35 Fig.8. Neutron current for DD,MDD,SOTE_DD Fig.9. for DD,MDD,SOTE_DD Fig.10. Iteration for DD, MDD, SOTE_DD

Test for space discrete scheme The results presented and discussed in this section are organized into three subsections.  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 time- dependent transport equation is studied, and anisotropic scattering source with P 5 spherical harmonic expansion is considered.

Time-independent transport problems

Transport problems with a small perturbation The radius is 0.5cm, and boundary condition is

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, Δ x = 0.1cm, Δ t=5 ×10 -5 μs ) 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.

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42  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).

43 5 Conclusions 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 ). We study the numerical solution oscillation from the aspect of time discrete and space discrete scheme. 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.

44 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.

45 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 time- dependent neutron transport equations. According the character of mult-media, we study different space discrete scheme:  Typical EM,DD;  LD.

46 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. Future work

47 Thank You!