a local adaptive coarse-mesh nonlinear diffusion...

A Local Adaptive Coarse-Mesh Nonlinear Diffusion AccelerationScheme for Neutron Transport Calculations

Sicong Xiao, Kangyu Ren, and Dean Wang *

University of Massachusetts Lowell, 1 University Avenue, Lowell, Massachusetts 01854

Received September 4, 2017Accepted for Publication October 14, 2017

Abstract — In order to improve the effectiveness and stability of the coarse-mesh finite difference method(CMFD), we developed a new nonlinear diffusion acceleration scheme for solving neutron transportequations. This scheme, called LR-NDA, employs a local refinement approach on the framework ofCMFD by solving a local boundary value problem of the scalar flux on the coarse-mesh structure toreplace the piecewise constant scalar flux obtained by CMFD. The refined flux is then used to update thescalar flux in the neutron transport source iteration. In this paper, a detailed convergence study of LR-NDAis carried out based on a two-dimensional fixed-source problem, and it shows that LR-NDA is much moreeffective and stable than CMFD for a wide range of optical thicknesses. In addition, we demonstrate thatLR-NDA is a local adaptive method. LR-NDA does not necessarily require local refinement for all thecoarse-mesh cells on the problem domain, i.e., it can be used only for relatively optically thick regionswhere the standard CMFD scheme would encounter the convergence problem.

Keywords — Neutron transport, CMFD, LR-NDA.

Note — Some figures may be in color only in the electronic version.

I. INTRODUCTION

The numerical solution of the neutron transport equa-tion often requires acceleration techniques to improve itscomputational efficiency. Various acceleration methodshave been developed in the last two decades.1–5 Of thesemethods, the coarse-mesh finite difference (CMFD) hasbecome one of the most popular methods due to its effi-ciency and simplicity. However, various numerical andtheoretical results show that CMFD will become unstablefor problems with large optical thickness.

There exist a number of stabilization techniques toimprove the stability of CMFD. For example, an ad-hoctechnique2 with an underrelaxation factor applied on thedrift flux term is used to stabilize CMFD. However, anissue with this technique is that the convergence willdeteriorate or even fail if a nonoptimal underrelaxationfactor is used. Reference 4 introduced a linear

prolongation from two neighboring coarse-mesh cellsinstead of the flat prolongation approach utilized in thestandard CMFD calculation to update the scalar flux forneutron transport iteration. It was shown that the linearprolongation technique is more effective than the under-relaxation factor applied on the drift flux term.Reference 5 proposed a variant of the CMFD method,called partial-current-based CMFD (pCMFD), whichwas found to be unconditionally stable for monoener-getic infinite homogenous problems, but it becomesslower than CMFD for intermediate and smallercoarse-mesh sizes.6 Most recently, Ref. 7 proposed anew optimally diffusive coarse-mesh finite difference(odCMFD) method, which generalizes pCMFD by add-ing an artificial term to the diffusion coefficient. Thecoefficient of the artificially diffusive term is a functionof optical thickness, which is optimized using Fourieranalysis. It was found that odCMFD is unconditionallystable and can slightly improve the convergence ofpCMFD. Reference 8 introduced a two-level pCMFD*E-mail: [email protected]

NUCLEAR SCIENCE AND ENGINEERING · VOLUME 189 · 272–281 · MARCH 2018© American Nuclear Society

DOI: https://doi.org/10.1080/00295639.2017.1394088

272

http://orcid.org/0000-0001-5387-9133

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acceleration scheme to overcome the slow convergenceproblem of pCMFD for optically thick coarse-meshcells. In this scheme, a fine-mesh-based accelerationwith a fixed-source is coupled with a coarse-mesh-based acceleration with power iteration. It was foundthat the two-level pCMFD scheme enhances the conver-gence speed of pCMFD for optically thick coarse-meshcells.

Recently, we developed a new nonlinear diffusionacceleration scheme, called LR-NDA, which not onlycan stabilize CMFD, but also can greatly improve itseffectiveness.9,10 Based on the framework of CMFD,this scheme employs an additional computational levelwhich solves a local boundary value problem (BVP) ofthe scalar flux on the coarse-mesh structure of CMFD anduses the refined flux to update the source term of theneutron transport iteration. As compared with the two-level pCMFD scheme, it should be noted that LR-NDAdoes not need to solve a global fine-mesh nonlineardiffusion equation for the CMFD calculations. Thispaper extends our study of LR-NDA by focusing on itsacceleration effectiveness for two-dimensional (2-D) pro-blems and local adaptivity.

The paper is organized as follows. A brief overviewof the formulation and algorithm of LR-NDA is given inSec. II. Section III is devoted to a detailed study of theconvergence performance of LR-NDA and its compari-son with CMFD based on a 2-D fixed-source problem.Section IV discusses local adaptivity of LR-NDA with asimple 2-D k-eigenvalue problem. A brief summary anddiscussion is given in Sec. V.

II. LR-NDA FORMULATION AND ALGORITHM

In our previous study,10 a monoenergetic SN fixed-source neutron transport problem in slab geometry wasused to study the convergence behavior of LR-NDA. Inthis paper, we extend our study of the convergencebehavior of LR-NDA for 2-D problems by using amonoenergetic SN fixed-source neutron transport equa-tion with isotropic scattering and neutron source. Theflow chart of the LR-NDA algorithm is shown in Fig. 1.There are three levels of mesh structures employed inLR-NDA. The SN transport equation is first solved onthe fine-mesh grid, and the CMFD equation is thensolved on the coarse mesh. Finally, local refinement,solving a local BVP of the scalar flux, is carried outon the local refined mesh.

The l’th iteration cycle, which begins with the SNtransport equation with iteration indices, is expressed as

μqqx

ψlþ1=2 x; y; μ; ηð Þ þ ηqqy

ψlþ1=2 x; y; μ; ηð Þ

þ Σtψlþ1=2 x; y; μ; ηð Þ ¼ Σs

4ϕl x; yð Þ þ Q x; yð Þ

4; ð1Þ

where

ϕ = scalar flux

ψ = angular flux

Σs = scattering cross section

Σt = total cross section

μ, η = neutron angular directions

x, y = spatial positions

Q = external neutron source

l = source iteration index

l þ 1=2 = intermediate step.

During each SN source iteration, the coarse-mesh fluxis obtained by solving the CMFD equation

� � �1

3Σt;CM�þ D̂lþ1=2

CM

� �Φlþ1

þ Σt;CM � Σs;CM

� �Φlþ1 ¼ Q ; ð2Þ

where

Φlþ1 = coarse-mesh scalar flux

Q = external neutron source

Σt;CM = total cross section defined on the coarsemesh

Σs; CM = scattering cross section defined on thecoarse mesh

D̂lþ1=2CM = drift coefficient which is calculated using

the information from the l þ 1=2 step SNsource iteration.

For the 2-D coarse mesh, we define D̂lþ1=2CM in x- and

y-directions on the coarse-mesh edge as

D̂lþ1=2CM ;x ¼

ð1�1

μψlþ1=2 x; y; μ; ηð Þdμþ 1

3Σt;CM

qϕlþ1=2

qx

ϕlþ1=2

ð3Þ

and

ADAPTIVE NONLINEAR DIFFUSION ACCELERATION · XIAO et al. 273

NUCLEAR SCIENCE AND ENGINEERING · VOLUME 189 · MARCH 2018

D̂lþ1=2CM ;y ¼

ð1�1

ηψlþ1=2 x; y; μ; ηð Þdηþ 1

3Σt;CM

qϕlþ1=2

qy

ϕlþ1=2;

ð4Þ

where the denominator in Eqs. (3) and (4) is the aver-aged value of the scalar flux of two neighboring coarsemeshes relative to the coarse-mesh edge. After solving

Eq. (2), we employ two types of flux update. Forcoarse-mesh cells with small optical thickness, the SNsource iteration scalar flux for the l þ 1 cycle isupdated using the same scaling approach as in thestandard CMFD:

ϕlþ1 ¼ ϕlþ1=2 Φlþ1

�ϕlþ1=2

; ð5Þ

Fig. 1. Flowchart of the LR-NDA algorithm for fixed-source problems.

274 XIAO et al. · ADAPTIVE NONLINEAR DIFFUSION ACCELERATION


where �ϕlþ1=2 is obtained by averaging the calculatedtransport scalar flux on the coarse mesh.

For coarse-mesh cells with large optical thickness, alocal refinement calculation is performed on these coarse-mesh cells by solving the local neutron diffusion in Eq. (6)with the fixed boundary conditions on the local mesh:

� � �1

3Σt�þ D̂lþ1=2

FM

� �ϕlþ1local þ Σt � Σsð Þϕlþ1

local ¼ Q ;

ð6Þ

where ϕlþ1local is the scalar flux on the local mesh, and

D̂lþ1=2FM is the drift coefficient which is defined on the

local mesh and calculated using the information from

the l þ 1=2 step SN source iteration. ϕlþ1local in Eq. (6) is

the scalar flux on the local refinement mesh point,marked as green points in Fig. 2.

The D̂lþ1=2FM in Eq. (6) is defined on the fine-mesh

edges, including D̂lþ1=2FM ;x and D̂lþ1=2

FM ;y :

D̂lþ1=2FM ;x ¼

ð1�1

μψlþ1=2 x; y; μ; ηð Þdμþ 1

3Σt

qϕlþ1=2

qxϕlþ1=2

ð7Þ

and

D̂lþ1=2FM ;y ¼

ð1�1

ηψlþ1=2 x; y; μ; ηð Þdηþ 1

3Σt

qϕlþ1=2

qy

ϕlþ1=2;

ð8Þ

where the denominator in Eqs. (7) and (8) is the averagedvalue of the scalar flux of two neighboring fine meshes

relative to the fine-mesh edge. The boundary scalar fluxes(marked as the red points in Fig. 2) for the local BVP areobtained by weighting the transport flux values at themesh points on the coarse-mesh edges with the coarse-mesh flux ratio between the CMFD and SN transportresults. There are two types of mesh points, i.e., coarse-mesh corner points and coarse-mesh side points. Theboundary flux at each corner-mesh point is defined inEq. (9):

ϕlþ1BC; corner ¼

1

4

Φlþ1

�ϕlþ1=2

��L B

þ Φlþ1

�ϕlþ1=2

��R B

þ Φlþ1

�ϕlþ1=2

��L T

þ Φlþ1

�ϕlþ1=2

��R T

!ϕlþ1=2corner ; ð9Þ

and the boundary flux at each side-mesh point is definedin Eq. (10):

ϕlþ1BC;side ¼

1

2

Φlþ1

�ϕlþ1=2

��þþ Φlþ1

�ϕlþ1=2

��

!ϕlþ1=2side ; ð10Þ

where ϕlþ1=2corner and ϕlþ1=2

side are the transport flux at thecorner- and side-mesh point, respectively. The sub-scripts L_B, R_B, L_T, and R_T denote the left-bottom,right-bottom, left-top, and right-top coarse-mesh cellssurrounding the corner-mesh point, respectively. Thesubscripts þ and � denote the right and left sides ofthe coarse-mesh cell edge in the x-direction or the topand bottom sides of the coarse-mesh cell edge in they-direction. If local refinement is applied for coarse-mesh cells which share the boundary with the problemdomain, Eqs. (9) and (10) can be used to calculate the

Fig. 2. Local refinement mesh for 2-D problem.



boundary conditions for local refinement calculationswith appropriate simplification for the corner- or side-mesh points.

After solving Eq. (6), the calculated local mesh pointscalar flux along with the BVP boundary mesh point fluxare averaged to obtain the center flux of each fine-meshcell which is used to update the scalar flux in the nexttransport sweeping:

ϕlþ1iþ1

2;jþ12¼ 1

4ϕlþ1i;j þ ϕlþ1

iþ1;j þ ϕlþ1i;jþ1 þ ϕlþ1

iþ1;jþ1

� �;

ð11Þ

where i and j are the indices of the mesh point in thelocal refinement mesh as shown in Fig. 2.

The source iteration will continue until the conver-gence criterion is satisfied.

The flowchart in Fig. 3 illustrates the LR-NDAscheme used for k-eigenvalue problems.

In Fig. 3, m stands for the power iteration index, ε1is the convergence criterion for the power iteration inthe CMFD calculation, and ε2 is the convergence cri-terion for the k-eigenvalue. A local refinement calcula-tion level is added into the two-level Transport/CMFDalgorithm. After the CMFD power iteration is done, thenew coarse-mesh flux and keff are obtained. For coarse-mesh cells with small optical thickness, Eq. (5) is stillused to update the SN transport sweep scalar flux forthe l þ 1 cycle. For coarse-mesh cells with large opticalthickness, a local refinement calculation is performedon these coarse-mesh cells by solving the local neutrondiffusion in Eq. (12) with the fixed boundary condi-tions on the local mesh obtained with Eqs. (9)and (10):

� � �1

3Σt�þ D̂lþ1=2

FM

� �ϕlþ1local þ Σt � Σsð Þϕlþ1

local ¼υΣf

keffϕlþ1local ;

ð12Þ

where

υ = mean number of neutrons produced perfission

Σf = fission cross section

keff = eigenvalue calculated from CMFD.

Similarly, after solving Eq. (12), the SN transport sweepscalar flux for the l þ 1 cycle is updated with Eq. (11).

In addition, it is worthwhile to mention that theLR-NDA method can be utilized to accelerate other

transport solvers, such as the Method of Characteristicsand Monte Carlo methods.

III. NUMERICAL CONVERGENCE STUDY

A numerical study of the LR-NDA convergenceperformance was carried out based on a 2-D modelproblem, which is a homogeneous 10 × 10-cm squarewith the vacuum boundary condition for four sides. Thedomain is discretized into 10 × 10 uniform coarse-meshcells. The fine-mesh number in each coarse-mesh cell is10 × 10. The numerical solutions were obtained usingthe Gauss-Legendre S12 quadrature set for angular dis-cretization and the diamond difference (DD) method forspatial discretization. Both CMFD and LR-NDA accel-eration schemes were implemented in the MATLABcode for the problem.

In order to characterize the convergence behavior, wecalculated the spectral radius numerically as defined by

ρ ¼ kϕlþ1 � ϕlkkϕl � ϕl�1k : ð13Þ

Figure 4 presents the spectral radius results for CMFDand LR-NDA as a function of coarse-mesh optical thick-ness (i.e., Σt;CMΔ; where Δ is the coarse-mesh size), forthe scattering ratios of 0.6, 0.8, 0.9, and 0.99.

Similar to our previous one-dimensional findings,8

the following observations can be drawn from the 2-Dresults:

1. For small scattering ratio, i.e., c = 0.6 or 0.8,CMFD is stable for the whole range of the opticalthickness.

2. When the scattering ratio increases to 0.99,CMFD is only effective for the optical thickness <1. Itbecomes unstable and fails to converge when the opticalthickness is >2.

3. The convergence performance of LR-NDA isalmost the same with CMFD for the optical thickness<1, and is more effective and stable than CMFD for theoptical thickness >1.

4. In addition, it is interesting to note that the spectralradius of LR-NDA first increases with the optical thicknessup to 10 and thereafter tends to decrease. The improvedperformance of LR-NDA at high optical thickness is due tothe fact that the diffusion solution becomes a better approx-imation to the SN solution at high optical thickness.



IV. LOCAL ADAPTATION OF LR-NDA

In this section, we study the local adaptivity of the LR-NDA scheme based on a 2-D monoenergetic k-eigenvalueproblem with large cross-section variations in the domain.Themodel problem considered is a 5 × 5-cm square with thereflective boundary condition on the four sides. The domain

is divided into 25 uniform coarse-mesh cells. The fine-mesh number in each coarse mesh is 10 × 10 as shownin Fig. 5. Similar to the 2-D fixed-source problem inSec. III, the numerical solutions for the SN neutron trans-port are obtained using the DD method for spatialdiscretization and the Gauss-Legendre S12 quadratureset for angular discretization.

Fig. 3. Flowchart of the LR-NDA algorithm for k-eigenvalue problems.



In this problem, there are three local regionsshown in color in Fig. 5, which have very large totalcross sections, i.e., the local optical thickness is verylarge. It should be pointed out that the cross sectionsare arbitrarily given to make it a very challengingproblem for numerical solution. For this problem, thestandard CMFD scheme fails to converge the SNiteration.

To study the local adaptivity of LR-NDA, we considerthree types of local refinement in Fig. 6. The first case isthat local refinement is only applied for the three diagonalcoarse-mesh cells. In the second case, the local refinementcalculation is applied for the 3 × 3 coarse-mesh cells

containing the three optically thick cells. The last case isthat local refinement is applied for all the coarse-meshcells in the domain (i.e., 5 × 5).

Figure 6a shows the normalized converged scalarflux. The flux changes significantly in the three opti-cally thick regions. The keff relative error is used asthe convergence performance index in Fig. 6b. Theerror criterion ε2 is 10�8 (Fig. 3). The error criterionε1 for the power iteration in CMFD is 10�12. It isshown that the second case with local refinement forthe 3 × 3 cells is similar to the case where localrefinement is applied for the whole domain (5 × 5).It is noteworthy to point out that the first case, where

Fig. 4. Convergence performance comparison between CMFD and LR-NDA.

Fig. 5. Specifications of 2-D k-eigenvalue problem.



local refinement is only applied on the three diagonalcoarse-mesh cells, is still very effective, althoughrequires more iterations. This study demonstratesthat LR-NDA is a local adaptive method and can beeasily implemented for any region of the problemdomain, which means that it can be used onlyfor optically thick regions where CMFD could havea convergence problem.

The computing time for each case is summarized inTable I. It is shown that LR-NDA can significantly reducethe number of transport sweeps. Compared to the CPU timespent on the transport sweeps and CMFD calculations, thetime spent on the local refinement calculation is much less. Inaddition, local refinement calculations for each coarse-meshcell can be parallelized to make the computational costnegligible.

(a) Normalized converged scalar flux

(b) keff relative error vs. iteration number

0

0.5

60

1

1.5

2

Nor

mal

ized

Sca

lar

Flu

x

2.5

40

3

Y

2D K-eigenvalue ProblemS12 Solution Accelerated with LR-NDA

20

X

50454035300 2520151050

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1 10 100

Ke

ff R

ela

tiv

e E

rro

r

Transport Sweep #

LR-NDA Local Adaptivity

3

3 x 3

5 x 5

Fig. 6. Numerical results of LR-NDA for 2-D k-eigenvalue problem.



V. CONCLUSIONS

This paper presents our latest work on the develop-ment and assessment of the nonlinear LR-NDA accelera-tion scheme for neutron transport calculations. LR-NDAincorporates a local refinement solution on the coarse-mesh structure based on the CMFD framework. Theconvergence study of LR-NDA based on the 2-D SNfixed-source problem demonstrates that LR-NDA cangreatly improve the stability and effectiveness of CMFD.

In addition, we demonstrate in this paper thatLR-NDA is a local adaptive method and that it can beeasily implemented for any region of the problem domainwhere the standard CMFD method would become inef-fective or unstable. It should be pointed out that thecomputational cost of local refinement is negligible ascompared with the CMFD cost because of its local com-pactness and efficient parallel implementation. This novelfeature will make it very computationally attractive forlarge 2-D/three-dimensional neutron transport problems.

Acknowledgments

This research was supported by the U.S. Department ofEnergy’s Nuclear Energy University Program.

ORCID

Dean Wang http://orcid.org/0000-0001-5387-9133

References

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2. K. P. KEADY and E. W. LARSEN, “Stability of SNk-Eigenvalue Iterations Using CMFD Acceleration,” Proc.M&C, Nashville, Tennessee, April 19–23, 2015, AmericanNuclear Society (2015).

3. M. JARRETT, et al., “Analysis of Stabilization Techniquesfor CMFD Acceleration of Neutron Transport Problems,”Nucl. Sci. Eng., 184, 2, 208 (2016); https://doi.org/10.13182/NSE16-51.

4. L. LI, K. SMITH, and B. FORGET, “Techniques forStabilizing Coarse-Mesh Finite Difference (CMFD) inMethod of Characteristics (MOC),” Proc. M&C,Nashville, Tennessee, April 19–23, 2015, AmericanNuclear Society (2015).

5. N. Z. CHO et al., “Partial Current-Based CMFD Accelerationof the 2D/1D Fusion Method for 3D Whole-Core TransportCalculations,” Trans. Am. Nucl. Soc., 88, 594 (2003).

6. N. Z. CHO, “Partial Current-Based CMFD (p-CMFD)Method Revisited,” Trans. Kor. Nucl. Soc., Gyeongju,Korea, October 25–26, 2012, Korean Nuclear Society(2012).

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TABLE I

Computational Performance Comparison of the 2-D k-Eigenvalue Problem*

No Acceleration(Transport Sweep)

LR-NDA3ð Þ

LR-NDA(3 × 3)

LR-NDA(5 × 5)

Number of unknownsa M2 � P2 � N þ 2ð ÞN=2ð Þ ðP� 1Þ2 � 3 ðP� 1Þ2 � 32 ðP� 1Þ2 � 52

52 � 102 � 84� �

92 � 3� �

92 � 32� �

92 � 52� �

Number of transport sweep 1483 51 36 32Transport calculation time (s) 1895.6 64.5 45.2 40.4CMFD power iteration number 0 6495 4597 4078CMFD calculation time (s) 0 2.2 1.5 1.4Total local refinement calculation time (s) 0 6:9� 10�2 1:8� 10�1 3:2� 10�1

Local refinement calculation time for one coarsemesh (s)

0 2:3� 10�2 2:0� 10�2 1:3� 10�2

Total calculation time (s) 1895.6 66.8 46.9 42.1Speedup 1 28.4 40.4 45.0

*Computation with a single CPU [Intel (R) Xeon (R) E5-2630 v3 at 2.40 GHz].aM = 5 (coarse-mesh number in the x- or y-direction in the problem domain); P = 10 (fine-mesh number in the x- or y-direction ineach coarse-mesh); and N = 12 (the S12 quadrature set).



https://doi.org/10.1016/S0149-1970(01)00023-3

https://doi.org/10.1016/S0149-1970(01)00023-3

https://doi.org/10.13182/NSE16-51

https://doi.org/10.13182/NSE16-51

https://doi.org/10.1016/j.anucene.2016.05.004

https://doi.org/10.1016/j.anucene.2016.05.004

8. N. Z. CHO and S. YUK, “Two-Level Speedup Schemes forp-CMFD Acceleration in Neutron Transport Calculation,”Nucl. Sci. Eng., 188, 1, 1 (2017); https://doi.org/10.1080/00295639.2017.1332891.

9. D. WANG, S. XIAO, and R. MAGRUDER, “A Coarse-Mesh Nonlinear Diffusion Acceleration Schemewith Local Refinement for Neutron Transport

Calculation,” Trans. Am. Nucl. Soc., Las Vegas,Nevada, November 6–10, 2016, American NuclearSociety (2016).

10. S. XIAO et al., “Convergence Study of LR-NDA UsingFourier Analysis,” Trans. Am. Nucl. Soc., San Francisco,California, June 11–15, 2017, American Nuclear Society(2017).



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