differentiation and quantification of chlorine residuals ...

CONVERGENCE ANALYSIS OF THE COARSE MESH FINITE DIFFERENCE METHOD

A Thesis Submitted to the Faculty of Purdue University by Deokjung Lee

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

December 2003

ii

To Sunja, Sangdo, Sangjin, and Yuna.

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ACKNOWLEDGEMENTS

I would like to express my deepest gratitude to my major advisor, Professor Thomas J. Downar, for his guidance and support. Working with him here at Purdue has been a real pleasure to me. Sincere thanks are extended to advisory committee members, Professors Shripad T. Revankar, Tatjana Jevremovic, and Ananth Grama for their invaluable discussions and comments. I am indeed indebted to Professor Chang Hyo Kim at Seoul National University, Mr. Sang Hee Lee, Mr. Chang Sub Lee, and Mr. Sung Man Bae at KEPRI, and Dr. Yonghee Kim at KAERI for their support and trust in me. Without them, none of this would have been possible. A very special thank should go to Dr. Yonghee Kim for his really invaluable guidance, discussion, and warmhearted encouragement during the study. I would like to thank friends in our research group, Dr. Hyun Chul Lee, Tomasz, Matt, Yunlin, Jun, Zhaopeng, Justin, and Chris for their support and friendship. I also would like to thank our old group members, Dr. Changho Lee, Dr. T.K. Kim, and Dan Tinkler. I would like to express my sincere gratitude to my parents and my mother-in-law for their love. The loves from my wife Sunja, my children Sangdo, Sangjin, and Yuna have made my effort possible and worthwhile.

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TABLE OF CONTENTS

Page LIST OF TABLES............................................................................................................. vi LIST OF FIGURES .......................................................................................................... vii NOMENCLATURE ...................................................................................................... xivii ABSTRACT.................................................................................................................... xvii 1. INTRODUCTION .......................................................................................................... 1 2. CONVERGENCE ANALYSIS OF ONE-DIMENSIONAL ONE-GROUP PROBLEM ......................................................................................................................................... 4 2.1 Model Problem.................................................................................................. 4 2.2 Analytic Derivation of Convergence Rate ....................................................... 6 2.2.1 Partial Current Sweep ....................................................................... 6 2.2.2 Coarse Mesh Rebalance .................................................................. 13 2.2.3 Coarse Mesh Finite Difference with One-Node Kernel (I) ............ 21 2.2.4 Coarse Mesh Finite Difference with One-Node Kernel (II) ........... 37 2.2.5 Coarse Mesh Finite Difference with Two-Node Kernel ................. 44 2.3 Numerical Spectral Radius ............................................................................ 50 2.4 Effects of Scattering Ratio to Spectral Ratio ................................................. 51 2.5 Comparison of Acceleration Performance ..................................................... 52 2.6 Non-Dimensional Analysis ............................................................................ 53 2.7 Linear CMFD Algorithm With Two-Node Local Problem ........................... 54 2.8 Summary ........................................................................................................ 55 3. CONVERGENCE ANALYSIS OF TWO-DIMENSIONAL ONE-GROUP PROBLEM ....................................................................................................................................... 58 3.1 Model Problem ............................................................................................... 58 3.2 Analytic Derivation Of Convergence Rate .................................................... 59 3.2.1 Partial Current Sweep ..................................................................... 59 3.2.2 Coarse Mesh Rebalance .................................................................. 68 3.2.3 Coarse Mesh Finite Difference With One-Node Kernel ................. 75 3.2.4 Coarse Mesh Finite Difference With Two-Node Kernel ................ 91

v Page 3.3 Numerical Spectral Radius .......................................................................... 102 3.4 Summary ...................................................................................................... 102 4. CONVERGENCE ANALYSIS OF ONE-DIMENSIONAL TWO-GROUP PROBLEM ..................................................................................................................................... 104 4.1 Model Problem ............................................................................................. 104 4.2 Analytic Derivation Of Convergence Rate .................................................. 105 4.2.1 Partial Current Sweep ................................................................... 105 4.2.2 Coarse Mesh Rebalance ................................................................ 111 4.2.3 Coarse Mesh Finite Difference With One-Node Kernel (I) .......... 117 4.2.4 Coarse Mesh Finite Difference With One-Node Kernel (II) ........ 133 4.2.5 Coarse Mesh Finite Difference With Two-Node Kernel .............. 150 4.3 Summary ...................................................................................................... 157 5. NUMERICAL CONVERGENCE ANALYSIS FOR THREE-DIMENSIONAL TWO ENERGY GROUP PROBLEMS ............................................................................... 158 5.1 Three-Dimensional Two-Group Neutron Diffusion Equation ..................... 158 5.2 Two-Node ANM Kernel .............................................................................. 160 5.3 One-Node ANM Kernel ............................................................................... 162 5.4 Numerical Convergence Analysis with Three-Dimensional Two-Group Problem ........................................................................................................ 167 5.5 Numerical Convergence Analysis with NEACRP LWR Transient Benchmark ....................................................................................................................... 179 5.6 Summary ...................................................................................................... 191 6. SUMMARY AND CONCLUSIONS ........................................................................ 192 LIST OF REFERENCES................................................................................................ 194 VITA ............................................................................................................................... 198

vi

LIST OF TABLES

Table

Page

2.1 Problem Data of One-Group Model Problem .............................................................. 5 4.1 Cross Section Data for Two-Group Model Problem ............................................... 101 4.2 Dominant Factors to Convergence Rate for 1-D 2-G Model FSP ........................... 153 5.1 List of Test Cases (3-D Cubic Core with Uniform Meshes) ................................... 165 5.2 Parameters for 1-N CMFD Strategy ........................................................................ 153 5.3 Sensitivity of CCF Convergence Rate to the Iteration Control (3-D 2-G Model EVP, 1-N Kernel, Joo's CCF, 1 G-S Sweep, R-B Ordering, No Relaxation, 10x10 Meshes) ................................................................................................................................... 153 5.4 Summary of NEACRP A1 Calculation (Joo's CCF, G-S Jin Update, R-B Ordering) ................................................................................................................................... 178 5.5 Summary of NEACRP A1 Steady State with 1-Sweep 1-N CMFD (Joo's CCF, G-S Jin Update, R-B Ordering) ...................................................................................... 180 5.6 Summary of NEACRP A1 Steady State with 2-Sweep 1-N CMFD (Joo's CCF, G-S Jin Update, R-B Ordering) ...................................................................................... 180 5.7 Summary of NEACRP A1 Steady State with 3-Sweep 1-N CMFD (Joo's CCF, G-S Jin Update, R-B Ordering) ...................................................................................... 185 5.8 Summary of NEACRP Steady State Calculations (Joo's CCF, 3 Sweeps (1-N CMFD), G-S Jin Update, R-B Ordering, Under-relaxation 0.8) ........................................... 186 5.9 Summary of NEACRP Transient Calculations (Joo's CCF, 3 Sweeps (1-N CMFD), G-S Jin Update, R-B Ordering, Under-relaxation 0.8) ........................................... 186

vii

LIST OF FIGURES

Figure

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2.1 Mesh Structure for 1-D Model Problem ....................................................................... 4 2.2 Convergence Rate vs. Phase Angle τ (PCS)................................................................. 9 2.3 Spectral Radius vs. (σh, 1/Dh) (PCS) .......................................................................... 10 2.4 Eigenvalues vs. Optical Mesh Size σ (c=0.95; PCS).................................................. 11 2.5 Spectral Radius vs. Optical Mesh Size σ (c=0.95; PCS) ........................................... 11 2.6 Sensitivity of Convergence Rate to Absorption Cross Section and Diffusion Coefficient (PCS)........................................................................................................ 12 2.7 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (PCS) ...... 13 2.8 Convergence Rate vs. Phase Angle τ (CMR) ............................................................. 17 2.9 Spectral Radius vs. (σh, 1/Dh) (CMR) ....................................................................... 18 2.10 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMR) ............................................. 18 2.11 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMR) ....................................... 19 2.12 Sensitivity of Convergence Rate to Absorption Cross Section and Diffusion Coefficient (CMR) ..................................................................................................... 20 2.13 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (CMR) .. 20 2.14 Convergence Rate vs. Phase Angle τ (CMFD1N-NR) ............................................ 26 2.15 Spectral Radius vs. (σh, 1/Dh) (CMFD1N-NR) ....................................................... 26 2.16 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR) ............................... 27 2.17 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR) ......................... 28 2.18 Sensitivity of the Convergence Rate to the Absorption Cross Section and Diffusion Coefficient (CMFD1N-NR) ....................................................................................... 29

viii Figure

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2.19 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (CMFD1NNR) ............................................................................................................................. 29 2.20 Eigenvalues vs. Relaxation Parameter α ................................................................. 30 2.21 Slopes m1 and m1 vs. (σh, 1/Dh) ........................................................................... 31 2.22 Eigenvalues vs. Relaxation Parameter α ................................................................. 32 2.23 Spectral Radius vs. Relaxation Parameter α ........................................................... 33 2.24 Sensitivity to Relaxation Parameter α of Spectral Radius ........................................ 33 2.25 Optimum Relaxation Parameter vs. Optical Mesh Size σ (c=0.95) ........................ 34 2.26 Spectral Radius vs. (σh, 1/Dh) (CMFD1N-OR) ....................................................... 35 2.27 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR/OR) .................. 36 2.28 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMFD1N with α=0.7345) ............. 37 2.29 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N with α=0.7345) ...... 37 2.30 Convergence Rate vs. Phase Angle τ (CMFD1N-DF/NR) ...................................... 42 2.31 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-DF/NR) ................... 43 2.32 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-DF with α=0.7345) . 43 2.33 Convergence Rate vs. Phase Angle τ (CMFD2N) ................................................... 47 2.34 Spectral Radius vs. (σh, 1/Dh) (CMFD2N) .............................................................. 47 2.35 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD2N) ................................ 48 2.36 Sensitivity of Convergence Rate to Absorption Cross Section and Diffusion Coefficient (CMFD2N) .............................................................................................. 49 2.37 Spectral Radius vs. Absorption Cross Section & Diffusion Coefficient (CMFD2N) .................................................................................................................................... 49 2.38 Numerical Spectral Radius vs. Number of Mesh (CMFD1N).................................. 50 2.39 Spectral Radius vs. Σ t h for Different Scattering Ratios .......................................... 51 2.40 Spectral Radius Depending on Algorithm ............................................................... 52 3.1 Mesh Structure for 2-D Model Problem .................................................................... 59 3.2 Spectral Radius vs. (σh, 1/Dh) (PCS) ......................................................................... 64 3.3 Eigenvalues vs. Optical Mesh Size σ (c=0.95; PCS) ................................................. 65

ix Figure

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3.4 Spectral Radius vs. optical mesh size σ (c=0.95; PCS).............................................. 66 3.5 Sensitivity of Convergence Rate to Absorption Cross Section and Diffusion Coefficient (PCS) ....................................................................................................... 67 3.6 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (PCS) ...... 67 3.7 Spectral Radius vs. (σh, 1/Dh) (CMR) ....................................................................... 72 3.8 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMR) ............................................... 72 3.9 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMR) .......................................... 73 3.10 Sensitivity of Convergence Rate............................................................................... 74 3.11 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (CMR) ... 74 3.12 Spectral Radius vs. (σh, 1/Dh) (CMFD1N-NR) ........................................................ 82 3.13 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR)................................ 82 3.14 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR).......................... 83 3.15 Sensitivity of Convergence Rate to Absorption Cross Section and Diffusion Coefficient (CMFD1N-NR) ....................................................................................... 83 3.16 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (CMFD1NNR) ............................................................................................................................. 84 3.17 Spectral Radius vs. Relaxation Parameter α ............................................................ 84 3.18 Eigenvalues at α=1 vs. (σh, 1/Dh) ........................................................................... 86 3.19 Sensitivity of Spectral Radius to Relaxation Parameter α (CMFD1N) .................... 86 3.20 Spectral Radius vs. Relaxation Parameter α Depending on Optical Mesh Size....... 87 3.21 Eigenvalues vs. Relaxation Parameter α .................................................................. 87 3.22 Optimum Relaxation Parameter vs. Optical Mesh Size σ (c=0.95) ......................... 89 3.23 Spectral Radius vs. (σh, 1/Dh) (CMFD1N-OR) ........................................................ 89 3.24 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR/OR) ................... 90 3.25 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMFD1N with α=0.5).................... 90 3.26 Spectral Radius vs. (σh, 1/Dh) (CMFD2N) ............................................................... 96 3.27 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMFD2N) ...................................... 96 3.28 Measured Convergence Rate vs. Iteration Index (CMFD2N) .................................. 97

x Figure

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3.29 Convergence Rate vs. Optical Mesh Size σ (c=0.95; CMFD2N)............................. 98 3.30 Initial Guess for the CCFs (CMFD2N)..................................................................... 99 3.31 Effects of Initial CCFs to the Convergence (CMFD2N) .......................................... 99 3.32 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD2N)............................... 100 3.33 Convergence Rate Measurements and Flux Error Reduction (Optical Thickness=10, Flat Initial CCFs, c=0.95; CMFD2N) ...................................................................... 100 3.34 Sensitivity of Convergence Rate to Absorption Cross Section and Diffusion Coefficient (CMFD2N) ............................................................................................ 101 3.35 Numerical Spectral Radius vs. Number of Mesh (CMFD1N)................................ 102 4.1 Convergence Rate vs. Phase Angle τ (1-D 2-G PCS) .............................................. 108 4.2 Eigenvalues vs. Mesh Size h (1-D 2-G PCS) ........................................................... 110 4.3 Spectral Radius vs. Mesh Size h (1-D 2-G PCS)...................................................... 111 4.4 Convergence Rate vs. Phase Angle τ (1-D 2-G CMR)............................................. 115 4.5 Eigenvalues vs. Mesh Size h (1-D 2-G CMR).......................................................... 116 4.6 Spectral Radius vs. Mesh Size h (c=0.95; CMR) ..................................................... 116 4.7 Convergence Rate vs. Phase Angle τ (1-D 2-G CMFD1N, h=10 cm) ..................... 124 4.8 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N-NR) ........................................... 125 4.9 Spectral Radius vs. Mesh Size (1-D 2-G CMFD1N w/ α=1)................................... 127 4.10 Spectral Radius vs. Relaxation Parameter α Depending on Mesh Size ................. 128 4.11 Sensitivity of Spectral Radius to Relaxation Parameter α ...................................... 128 4.12 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N with α=0.5) ............................. 129 4.13 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N with Two group-wise Optimum

α’s for h=10 cm) ...................................................................................................... 130 4.14 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N with α Optimum for Thermal Group and h=10 cm) ................................................................................................ 130 4.15 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N with Group-wise Optimum Underrelaxation Parameters).............................................................................................. 131

xi Figure

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4.16 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N with Thermal Optimum Underrelaxation Parameter) ............................................................................................... 132 4.17 Spectral Radius vs. Mesh Size h (1-D 2-G CMFD1N with / without Optimum Under-relaxation) ..................................................................................................... 132 4.18 Convergence Rate vs. Phase Angle τ (1-D 2-G CMFD1N-C) ............................. 143 4.19 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N-C with no relaxation) .............. 144 4.20 Spectral Radius vs. Mesh Size (1-D 2-G CMFD1N-C with No Relaxation) ......... 144 4.21 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N-NR) ......................................... 145 4.22 Spectral Radius vs. Relaxation Parameter α Depending on Mesh Size ................. 147 4.23 Sensitivity of Spectral Radius to Relaxation Parameter α (1-D 2-G CMFD1N-C) 147 4.24 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD1N-C with α=0.7) ......................... 148 4.25 Optimum Relaxation Parameters vs. Mesh Size h Depending on Cross Section ... 149 4.26 Spectral Radii vs. Mesh Size h (1-D 2-G CMFD1N-C with Optimum α’s of Three Cross Section Sets: Fast Group, Thermal Group and Equilibrium 1-G Cross Section) .................................................................................................................................. 149 4.27 Convergence Rate vs. Phase Angle τ (CMFD2N, h=20 cm).................................. 155 4.28 Eigenvalues vs. Mesh Size h (1-D 2-G CMFD2N) ................................................ 156 4.29 Spectral Radius vs. Mesh Size h (1-D 2-G CMFD2N)........................................... 156 5.1 Sensitivity of Convergence Rate to Core Size (2-N CMFD; 3-D 2-G Model Eigenvalue Problem) ................................................................................................ 169 5.2 Convergence Rate vs. Mesh Size (2-N CMFD; 3-D 2-G Model Problem) .............. 170 5.3 Sensitivity of 1-N CMFD Convergence Rate to Mesh Size and Relaxation Parameter with 3-D 2-G Model EVP (Shin’s CCF; Jacobi-style Jin Update, Single Sweep) .. 172 5.4 Sensitivity of 1-N CMFD Convergence Rate to Sweep Strategy For the 3-D 2-G Model EVP (Shin’s CCF) ........................................................................................ 173 5.5 Sensitivity of 1-N CMFD Convergence Rate to CCF Form with 3-D 2-G Model EVP (2 G-S-style Sweep, R-B Ordering) ......................................................................... 174

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5.6 Sensitivity of 1-N CMFD Convergence Rate to CCF From with 3-D 2-G Model EVP(2 G-S-style Sweep, R-B Ordering; Shin’s Form B : CCF Update after Local Sweeps Finished) ...................................................................................................... 174 5.7 Sensitivity of 1-N CMFD Convergence Rate to Sweep Strategy ............................. 175 5.8 Sensitivity of 1-N CMFD Convergence Rate to Mesh Size and Under-relaxation with 3-D 2-G Model EVP (Joo’s CCF, R-B Ordering; Single Sweep, G-S Jin Update). 177 5.9 Sensitivity of 1-N CMFD Convergence Rate to Mesh Size and Under-relaxation with 3-D 2-G Model EVP (Joo’s CCF, R-B Ordering; 2 Sweeps, G-S Jin Update)........ 177 5.10 Transient Power of NEACRP A1 Case with 1-N/2-N CMFD ............................... 179 5.11 Jacobi vs. Gauss-Seidel Jin Update (NEACRP A1 Steady State: Joo's CCF, 3 Sweeps (1-N CMFD)) .............................................................................................. 180 5.12 Convergence Rate of 1-N/2-N CMFD (NEACRP A1 Steady State: , Joo's CCF , G-S Jin Update) ............................................................................................................... 180 5.13 Convergence of 1-Sweep 1-N CMFD for NEACRP A1 Steady State (Joo's CCF, GS Jin Update, R-B Ordering) .................................................................................... 183 5.14 Convergence of 2-Sweep 1-N CMFD for NEACRP A1 Steady State (Joo's CCF, GS Jin Update, R-B Ordering) .................................................................................... 185 5.15 Convergence of 3-Sweep 1-N CMFD for NEACRP A1 Steady State (Joo's CCF, GS Jin Update, R-B Ordering) .................................................................................... 186 5.16 Convergence of 3-Sweep 1-N CMFD for NEACRP A1 Steady State (Joo's CCF, GS Jin Update, R-B Ordering) .................................................................................... 187 5.17 Fast Group Convergence of 3-Sweep 1-N CMFD for NEACRP A1 Steady State (Joo's CCF, G-S Jin Update, R-B Ordering) ............................................................ 188 5.18 Convergence of 1-N CMFD for NEACRP A1 Steady State (Joo's CCF, G-S Jin Update, R-B Ordering)............................................................................................. 189

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NOMENCLATURE

ENGLISH ANM

analytic nodal method

c

scattering ratio

CCF

current correction factor

CMFD

coarse mesh finite difference

CMFD1N

coarse mesh finite difference with one-node kernel

CMFD1N-C coarse mesh finite difference with one-node kernel and using condensation / prolongation between CMFD and local kenel CMFD1N-DF coarse mesh finite difference with one-node kernel and using discontinuity factor instead of current correction factor CMFD1N-DF/NR

coarse mesh finite difference with one-node kernel and using discontinuity factor but without relaxation

CMFD1N-NR coarse mesh finite difference with one-node kernel and without CCF relaxation CMFD1N-OR coarse mesh finite difference with one-node kernel and using optimum under-relaxation parameter CMFD2N

coarse mesh finite difference with two-node kernel

CMR

coarse mesh rebalance

D

diffusion coefficient

D

finite difference operator (= D / h for 2-node CMFD, 2 D / h for 1-node CMFD)

Dˆ

current correction factor

Dh

nondimensional diffusion coefficient (= D / h )

xiv Deq

one-group diffusion coefficient condensed from two-group constants using infinite spectrum

DF

discontinuity factor

EVP

eigenvalue problem

Fi ( k )

coarse mesh rebalancing factor of node i at kth iteration

f i ,l

discontinuity factor at the left surface of node i

f i ,r

discontinuity factor at the right surface of node i

FSP

fixed source problem

GET

generalized equivalence theory

h

mesh size (cm)

i

imaginary unit or the square root of -1

J

neutron net current

j

neutron partial current

Jin

neutron incoming partial current

L

diffusion length (= D / Σ ), cm

L

iteration matrix from Fourier analysis of the linearized algorithm

Lu

transverse leakage in the u- direction (u=x, y, or z)

NEM

nodal expansion method

NO

natural ordering

PCS

partial current sweep

Q

neutron source

R-B

red / black ordering

GREEK SYMBOLS

α

relaxation parameter of current correction factor in CMFD1N algorithm

α opt

optimum relaxation parameter of current correction factor

β

surface flux correction factor

xv

∆

mesh size measured in the metric of diffusion length (= h / L )

ε

perturbation

φ

neutron scalar flux

φ

neutron node average flux

ηi(,kl )

error of left surface current correction factor of node i at kth iteration

ηi(,kr )

error of right surface current correction factor of node i at kth iteration

λ

Fourier error mode

ρ

spectral radius

ρ num

numerically evalutated spectral radius

Σ

absorption cross section

Σt

total cross section

Σ12

scattering cross section from fast to thermal energy group

Σ r1

removal cross section of fast energy group

Σr 2

removal cross section of thermal energy group

σ

optical mesh size (= Σt h )

σh

nondimensional absorption cross section (= hΣ )

σ eq

one-group removal cross section condensed from two-group constants using infinite spectrum

τ

phase angle of Fourier mode (= λ h )

ω

amplification factor and / or convergence rate

ξi(,lk )

error of left surface partial current of node i at kth iteration

ξi(,rk )

error of right surface partial current of node i at kth iteration

Ψ

neutron energy spectrum of outgoing partial current

ς i( k )

error of node average flux of node i at kth iteration

xvi SUPERSCRIPTS

in

incoming to the node

k

iteration index

out

outgoing from the node

+

the direction of the positive x direction

-

the direction of the negative x direction

*

nondimensional quantity

SUBSCRIPTS

i

spatial node index

k

iteration index

l

left surface

opt

optimum relaxation

r

right surface

xvii

ABSTRACT

Deokjung Lee, Ph.D., Purdue University, December 2003. Convergence Analysis of the Coarse Mesh Finite Difference Method. Major Professor: Thomas J. Downar. The convergence rates of the nonnlinear coarse mesh finite difference (CMFD) and the coarse mesh rebalance (CMR) methods are derived analytically for one- and twodimensional geometries and one- and two- energy group solutions of the fixed source diffusion problem in a non-multiplying infinite homogeneous medium. The derivation is performed by linearizing the nonlinear algorithm and by applying Fourier error analysis to the linearized algorithm. The mesh size measured in units of the diffusion length was shown to be a dominant parameter for the convergence rate and for the stability of the iterative algorithms. For a small mesh size problem, CMFD is shown to be a more effective acceleration method than CMR. Both CMR and two-node CMFD algorithms are shown to be unconditionally stable. However, one-node CMFD becomes unstable for large mesh sizes. To remedy this instability, an under-relaxation of the current correction factor for the one-node CMFD method is successfully introduced and the domain of stability is significantly expanded. Furthermore, the optimum under-relaxation parameter is analytically derived and the one-node CMFD with the optimum relaxation is shown to be unconditionally stable. Additional numerical analysis was performed on

the

convergence of each algorithm with the U.S. NRC Neutron Kinetics code PARCS. It was confirmed that the insights about the CMFD and the CMR methods obtained for simple model problems in Chapters 2 ∼ 4 are valid for the realistic three-dimensional two-group eigenvalue and transient fixed source problems.

1

1. INTRODUCTION

In 1983, Smith introduced the nonlinear coarse mesh finite difference (CMFD) method in order to reduce the storage requirements for the analytic nodal method [1]. Since then it has become widely used to accelerate various diffusion solution methods [210]. Recently, CMFD has also been successfully applied to the acceleration of neutron transport solution methods [11-13]. The nonlinear CMFD method is based on generalized equivalence theory (GET) [14, 15], in which discontinuity factors (DFs) are introduced to reproduce any reference higher order solution. The specific role of the DFs is to preserve the reference net current at the node interface such that the reaction rates in each node of a lower order solution are the same as the higher order solution. Instead of the DFs of GET, CMFD utilizes a current correction factor (CCF) in order to preserve the net currents at each interface and thereby exactly reproduce any higher order solution in terms of the node-wise reaction rates and the global eigenvalue. In principle, there are no limitations in choosing the higher order method since the CCF guarantees the equivalence between the low order coarse-mesh finite-difference (CMFD) solution and whatever the higher order solution is chosen (e.g. diffusion or transport theory, nodal or fine-mesh spatial discretization, analytic method or polynomial expansion method, etc). Furthermore, the higher order solution can be obtained from either local one-node or twonode problems, or from a whole core solution. The PARCS code (Purdue Advanced Reactor Core Simulator) [4, 16], which was developed at Purdue University with the support of U.S. Nuclear Regulatory Commission, is one of the most advanced nuclear reactor transient analysis codes. PARCS takes advantage of the effectiveness of CMFD accelerations for all of its solution options. For higher order methods, PARCS uses one-node/two-node/whole-core geometry, and has options for both the analytic nodal method (ANM) [17] and the nodal expansion method

2 (NEM) [18, 19]. Recently a fine-mesh finite-difference (FMFD) option was introduced for either diffusion or simplified P3 transport. For most practical LWR problems, experience has shown PARCS to be numerically stable for all of the solution options. However, numerical instabilities have recently been encountered for some unusual configurations. For example, y

CMFD with a two-node local problem (CMFD2N) solver in which large and small mesh structures are mixed randomly,

y

CMFD with a one-node local problem (CMFD1N) solver with a very large mesh size.

The instability of the first case was reported even for a “nodal method only” without the so-called “nonlinear iteration” involving the CCFs [17]. It was suspected that the conventional quadratic polynomial approximation for the transverse leakage (TL) was inadequate and was the cause for the instability. The problem was remedied by taking the differences of the mesh size into account for the TL approximation. In the second case, two remedies were investigated in which the mesh size was reduced and multiple sweeps of local one-node solutions were introduced. Even though CMFD has acquired considerable popularity as an acceleration technique, there has not been a methodical analysis of the stability of the CMFD algorithm. There have been some studies on the convergence or uniqueness of the solutions of the higher order nodal method itself [20-25]. But once CCF is involved, the algorithm is nonlinear and the convergence analysis becomes complicated. In 1990, Cefus and Larsen introduced an innovative method based on Fourier analysis to analyze the stability of the nonlinear CMR acceleration for the transport SN calculation [26]. Since then, this approach has become a standard technique for the convergence analysis of nonlinear acceleration methods for various transport equations [27-30]. The essential idea was to linearize the nonlinear algorithm around the exact solution and then apply Fourier analysis [31, 32] to the linearized algorithm. Fourier error

3 analysis is the classic technique for the stability analysis of linear methods [33-35]. For nonlinear methods, the convergence rate of the linearized algorithm can be considered as the limiting value of the practical nonlinear algorithm applied to the finite dimensional problem. Previous researchers have consistently reported that the numerical convergence rate in the asymptotic region of the realistic nonlinear iteration compares favorably with the theoretical convergence rate of the linearized algorithm [26-28]. In our study of the CMFD algorithm, we take essentially the same approach proposed by Cefus and Larsen, i.e. linearizing the nonlinear algorithm and applying Fourier analysis to the linearized algorithm. For consistency, we will also apply the same method to the convergence analysis of the conventional partial current sweep (PCS) and the coarse mesh rebalance (CMR) algorithm. As a remedy for the instability of the onenode CMFD algorithm, we will introduce a method to under-relax the CCFs and show that with the analytically derived optimum relaxation parameter, the CMFD1N is unconditionally stable. The objective of this study is to perform a robust and reliable analysis on the stability and rate of convergence for the CMR and CMFD methods as an acceleration of PCS. In the following chapters, the convergence analyses for one-dimensional one-group model problem, two-dimensional one-group model problem, and one-dimensional twogroup model problem are presented in Chapters 2, 3, and 4, respectively. In Chapter 5, the numerical convergence analysis with realistic three-dimensional two-group eigenvalue and transient problem is presented. Summary and conclusions are provided in Chapter 6.

4

2. CONVERGENCE ANALYSIS OF ONE-DIMENSIONAL ONE-GROUP PROBLEM

2.1 Model Problem The model problem is a one-dimensional (1-D) one-group (1-G) neutron diffusion problem with a flat fixed source for a non-multiplying infinite homogeneous medium. Under these conditions, a neutron flux φ(x) satisfies the neutron diffusion equation

−D

d2 dx 2

φ ( x ) + Σφ ( x ) = Q

(2.1a)

where D is a diffusion coefficient, Σ is an absorption cross section, and Q is a fixed source. The interface conditions are the conventional flux and current continuity

φi (h ) = φi +1 (0) and J i ( h ) = J i +1 (0)

(2.1b)

where J ( x ) ( ≡ − D dφ ( x ) / dx ) is a neutron current. The mesh structure of the 1-D model problem is shown in Fig. 2.1. As indicated, a uniform mesh spacing of h is assumed.

•••

(node i)

x i -1 ∆ xi = h

••• xi

Figure 2.1 Mesh Structure for 1-D Model Problem

5 The required parameters to specify the model problem are D, Σ, h, and Q. The data listed in Table 2.1 were taken from OECD NEACRP LWR benchmark problem and collapsed using a fast to thermal flux ratio of 6 which is a typical value for LWR. The data set A are adequate to fully specify the problem, and the data set B are derived from data A using the additional assumption

D=

1 . 3Σt

(2.2)

Also, for the purpose of evaluating the numerical spectral radius, a finite dimension problem is considered with the total number of meshes as 40 and with zero flux boundary conditions.

Table 2.1 Problem Data of One-Group Model Problem

Data Set A

Data Set B

Finite Dimension Problem

Diffusion Coefficient D (cm)

0.8333333

Absorption Cross Section Σ (cm-1)

0.02

Mesh Size h (cm)

20

Fixed Source Q (cm-3 sec-1 )

1

Total Cross Section Σt (cm-1)

0.40

Scattering Ratio c

0.95

Optical Mesh Size Σt h

8.0

Number of Meshes

40

6 2.2 Analytic Derivation of Convergence Rate For the model problem, the convergence rates are derived for the conventional PCS algorithm and CMR / CMFD acceleration methods. The centerpiece of the convergence analysis in this study is the Fourier error analysis which provides a theoretical convergence rate for a linear iterative algorithm. Since the PCS algorithm is linear, the Fourier analysis is directly applicable. However, for CMR and CMFD, a linearization of the nonlinear algorithm is necessary before Fourier analysis can be applied. The formulation of the spectral radius for each of the methods will be provided in the following subsections.

2.2.1 Partial Current Sweep An analytic solution for node (i) in Fig. 2.1 can be written as

φi ( x ) = Ai cosh( x / L) + Bi sinh( x / L) +

Q Σ

(2.3)

where Ai and Bi are coefficients to be determined, and L ( = D / Σ ) is the diffusion

length. Two partial incoming currents from the left and right interfaces are used to determine the coefficients Ai and Bi. y

ji+−(1k,r) : the outgoing partial current of node (i-1) through right interface of the

node which is the incoming partial current to node (i) from the left interface of node (i), at the kth iteration, y

ji−+(1k,l) : the outgoing partial current from node (i+1) through the left interface

of the node which is the incoming partial current to node (i) from the right interface of node (i), at the kth iteration.

7 Using the analytic solution of node (i), a traditional response matrix form of the equations can be formulated as ⎡ ji−,l( k +1) ⎤ 1 ⎡δ 1 ⎢ + ( k +1) ⎥ = ⎢ ⎣ ji , r ⎦ χ ⎣δ 2

δ 2 ⎤ ⎡ ji+−(1k,r) ⎤ ⎡1⎤ ⎥ ⎢ −( k ) ⎥ + d ⎢ ⎥ δ 1 ⎦ ⎣ ji +1,l ⎦ ⎣1⎦

where δ 1 = (1 − δ 22 / 4) sinh[∆ ] , δ 2 = 4 LΣ ,

(2.4)

χ = δ 2 cosh[∆ ] + (1 + δ 22 / 4) sinh[∆ ] ,

d = κLQ , κ = 1 /(coth[∆ / 2)] + δ 2 / 2) , and ∆( ≡ h / L) is a mesh size measured in the metric of the diffusion length. The PCS algorithm is to sweep node by node updating the outgoing partial currents of each node using Eq. (2.4). This algorithm can be thought of as a variant of the nodal integration method [36] for the one-group one-dimensional fixed source problem. Let us introduce first order errors into Eq. (2.4), i.e., express the partial current at the kth iteration as a combination of exact solutions and first order error terms:

ji−,l( k ) =

Q Q (1 + εξ i−,l( k ) ) and ji+,r( k ) = (1 + εξ i+,r( k ) ) , ε 1

1- N CMFD H1- D 1- G L

»Eigenvalue »

2.5 2 1.5

ω2

1 0.5

ω1 0

5

10 St h

15

20

Figure 2.16 Eigenvalues vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR)

In Fig. 2.17, the analytically derived spectral radius is compared against the numerically evaluated spectral radius. As indicated, there is excellent agreement between the two spectral radii. Similar to the CMR algorithm, even though CMFD1N is a nonlinear algorithm, the analytic expectation of the spectral radius is very accurate. This fact also demonstrates that the spectral radius of the linearized 1-N CMFD represents the convergence behavior of the nonlinear 1-N CMFD with very high reliability, at least in the asymptotic region of the iteration.

28

1- N CMFD H1- D 1- G L

Spectral Radius r

2.5 2 1.5 1 0.5 0

5

10 St h

15

20

Figure 2.17 Spectral Radius vs. Optical Mesh Size σ (c=0.95; CMFD1N-NR)

The sensitivity of the convergence behavior to the absorption cross section and diffusion coefficient is shown in Fig. 2.18. The perturbation of the absorption cross section and the diffusion coefficient have been performed by multiplying a factor f as in PCS using Eq. (2.16). The multiplication factor f is varied from 0.l to 10 both for the absorption cross section and for the diffusion coefficient. As the absorption cross section becomes larger, the convergence of CMFD1N becomes slower. Conversely, as the diffusion coefficient becomes larger, the convergence of CMFD1N becomes faster. Fig. 2.19 illustrates the above observations: As D becomes smaller, the spectral radius becomes larger, and if D is small enough, the algorithm becomes unstable. As Σ becomes larger, the spectral radius becomes larger very quickly. Fig. 2.19 is for a mesh size of h=20 cm as given in Table 2.1. The behavior shown in Fig. 2.19 is consistent with Fig. 2.18.

29

1-NCMFDH1-D1-GL

1

Spectral Radius r

Spectral Radius r

1-NCMFDH1-D1-GL

0.8 f =0.1 0.6

f =1

0.4 f =10

0.2 0

10

20 h HcmL

30

1 0.8

f =10

0.6

f =1

0.4

f =0.1

0.2

40

0

(a) Absorption cross section Σ

10

20 h HcmL

30

40

(b) Diffusion coefficient D

Figure 2.18 Sensitivity of the Convergence Rate to the Absorption Cross Section and Diffusion Coefficient (CMFD1N-NR)

1-NCMFD H1-D1-GL

1

Spectral Radius r

Spectral Radius r

1-NCMFD H1-D1-GL 0.8 0.6 0.4 0.2 0

0.2

0.4 0.6 S Hcm-1L

(a) ρ vs. Σ

0.8

1

1 0.8 0.6 0.4 0.2 0

0.5

1 D HcmL

(b) ρ vs. D

Figure 2.19 Spectral Radius vs. Absorption Cross Section and Diffusion Coefficient (CMFD1N-NR)

1.5

2

30

Eigenvalue

1 (1 + m1 )

ω1

0.5 0

ω2

- 0.5

(1 + m 2 )

-1

0

0.5 1 1.5 Relaxation Parameter a

2

Figure 2.20 Eigenvalues vs. Relaxation Parameter α

2

1.5 hS 1

0.5 0

-0.8

m1 -0.9 -1

100 60 80 40 20 hêD (a) Slope m1 of the first eigenvalue

-1 m2- 2 -3 -4 -5

2 1.5 20

1 hS 40 hê D 60

0.5 80 100

(b) Slope m2 of the second eigenvalue

Figure 2.21 Slopes m1 and m1 vs. (σh, 1/Dh)

31 The m1 and m2 in Eq. (2.56) represent the slopes of the line in the graph of corresponding eigenvalues in (α, ω) plane as shown in Fig. 2.20. Both m1 and m2 are functions of ( σ h , D h ) and have the following properties as indicated in Fig. 2.21.

lim m1 = lim m2 = −1

σ h →0

σ h →0

lim m1 = lim m2 = −1

Dh →0

,

(2.57)

Dh →0

and − 1 < m1 < 0 , m2 < −1, and 1 + m1 < 1 + m2 .

(2.58)

Also, because of the limiting characteristics of Eq. (2.57), we can determine the limiting value of the spectral radius as the mesh size becomes small, i.e., lim ω i = lim ω i = 1 − α , i=1 and 2 h →0

σ h →0

(2.59)

Dh →0

Eq. (2.59) implies that for the case with no relaxation (α=1), as the mesh size becomes zero, the spectral radius becomes zero. But with a relaxation of α, the spectral radius tends to (1-α) as the mesh size goes to zero. Useful insights about the convergence behavior and the dependence of the spectral radius on the relaxation parameter are provided by the plots of the absolute value of the eigenvalues in Fig. 2.22, the spectral radius vs. the relaxation parameter depending on the mesh sizes in Fig. 2.23,

and the spectral radius vs. the optical mesh size

depending on the relaxation parameter in Fig. 2.24. Š

Without any relaxation (α=1), the second eigenvalue determines the spectral radius, i.e., 1 + m2 which can be larger than 1 depending on the mesh size and material property. (Figs. 2.22 and 2.23)

32 Š

By simply increasing the mesh size, CMFD1N can be unstable. (Fig. 2.23)

Š

Over-relaxation (α>1) always has an adverse effect on the convergence behavior. (Fig. 2.23)

Š

By applying the under-relaxation, the domain of stability can be expanded substantially. (Fig. 2.24) There exists an optimum under-relaxation (α 50 cm), either CMR or CMFD1N does not accelerate PCS. Note that for the comparisons in this section, the number of calculations per iteration for each method is ignored and the comparison is performed in terms of the convergence rate only.

Spectral Radius r

1.4 1.2 1

CMFD1N

0.8

PCS

0.6 CMR

0.4

CMFD1N w/ opt. relax.

0.2

CMFD2N

0

5

10 Sth

15

Figure 2.40 Spectral Radius Depending on Algorithm

20

53 2.6 Non-dimensional Analysis A non-dimensional analysis was performed in order to understand what is the important parameters for the convergence rate of in the algorithms considered here for the model problem,. Introducing non-dimensional quantities

x* =

x J Σφ , φ* = , and J * = L Q LQ

(2.98)

into Eqs. (2.1) leads to

−

d2 dx

*2

φ * ( x* ) + φ * ( x* ) = 1

(2.99)

with the interface conditions

φi* ( h * ) = φi*+1 (0) and J i* ( h * ) = J i*+1 (0)

where J * ( x * ) = −

(2.100)

d * * φ ( x ) is a non-dimensional current and h * = h / L is a non* dx

dimensional mesh size. From Eqs. (2.98) through (2.100), it is clear that the nondimensional mesh size h* is the only parameter required to specify the problem completely. Therefore, the analytic solution of the model problem will be determined by

h*. If a solution algorithm uses the quantities in Eqs. (2.98) through (2.100), i.e., flux and net current, the convergence rate also will be determined by h*. CMFD2N algorithm uses the flux and the net currents, but not the partial currents. Therefore, the spectral radius of CMFD2N is a function of h* (or ∆) only, as in Eq. (2.94). Using the non-dimensional quantities of Eq. (2.98), the partial current can be written

54

j ±* ( h * ) =

where j ±* = (Σ / Q ) j ±

1 * 1 φ s ± (LΣ )J s* 4 2

(2.101)

are non-dimensional partial currents. An additional non-

dimensional quantity (LΣ) is introduced here for the partial currents. Since PCS, CMR, and CMFD1N use the partial currents in addition to the flux and net currents, the convergence rates of these algorithms are functions of both (LΣ) and ∆(=h*). This is consistent with the analytically derived spectral radius for each algorithm in the previous sections.

2.7 Linear CMFD Algorithm with Two-Node Local Problem The two-node CMFD algorithm is a nonlinear procedure in which the nonlinearity is introduced from the definition of the CCFs in Eq. (2.81). A linear CMFD algorithm with a two-node problem (LCMFD2N) can be formulated by using a different definition of the CCF. Specifically, if the CCF is written as: 2Q ˆ ~ J i ,i +1 = − D (φ i +1 − φ i ) + Di ,i +1 Σ

(2.102)

where Dˆ i ,i +1 is CCF, then the nodal balance equation can be written as

( )

(

)

( )

(

~ ~ ~ − D φi −( 1k ) + Σh + 2 D φi ( k ) − D φi +( 1k ) = Qh + (2Q / Σ ) Dˆ i(−k1),i − Dˆ i(,ki+)1

)

(2.103)

The CCF is then updated using higher order interface currents of Eq. (2.84) as

Dˆ

( k +1) i ,i +1

~ J i(,ki++11) + D (φi +( 1k ) − φi ( k ) ) = . 2Q / Σ

(2.104)

55

Inserting Eq. (2.84) into Eq. (2.103) yields

Σh (1 + 2∆−2 (1 − cosh[∆ ]))csch[ ∆ / 2]2 (φi +( 1k ) − φi ( k ) ) + 8(Q / Σ) Dˆ i(,ki++11) = 0 .

(2.105)

The LCMFD2N algorithm consists of Eqs. (2.103) and (2.105). The convergence rate of LCMFD2N will be the same as the linearized CMFD2N since the Eqs. (2.103) and (2.105) can be converted into the linearized CMFD2N of Eqs. (2.87) and (2.88), by the linear transformation

⎡ ς i( k ) ⎤ ⎡Σ / Q 0⎤ ⎡ φi ( k ) ⎤ ⎡1⎤ ⎢ (k ) ⎥ = ⎢ ⎥ ⎢ Dˆ ( k ) ⎥ − ⎢0⎥ 0 1 η ⎦ ⎣ i ,i +1 ⎦ ⎣ ⎦ ⎣ i ,i +1 ⎦ ⎣

(2.106)

The linear CMR algorithm and the linear CMFD method for the one-node local problem can be constructed in a similar manner for the model problem.

2.8 Summary For the one-dimensional one-group diffusion fixed source problem, we have performed theoretical studies on the convergence rates of the PCS, CMR, and CMFD iterative methods. The five algorithms, PCS, CMR, CMFD1N, CMFD1N-DF, and CMFD2N, were constructed for the model problem and the nonlinear algorithms of CMR and CMFD were linearized for Fourier analysis. Fourier error analysis was then applied to each method and the convergence rates were derived analytically. The numerical spectral radius of each method was evaluated for the model problems with a finite number of meshes. Non-dimensional equations were formulated for the model problem and the dominant factors affecting the rate of convergence were identified. A linear version of the CMFD algorithm with a 2-node local problem was also formulated. Finally, an under-relaxation was applied to mitigate the instability of the CMFD1N

56 algorithm. The optimum value of the relaxation parameter was derived analytically and the CMFD1N algorithm with the optimum under-relaxation was shown to be unconditionally stable. The conclusions of this chapter can be summarized as follows y

The method proposed by Cefus and Larsen [26] to apply Fourier analysis to the linearized equations effectively predicts the convergence rates of nonlinear methods (CMR, CMFD1N, and CMFD2N) in the vicinity of the exact solutions.

y

The theoretical and experimental convergence rates agree well with each other.

y

The convergence rates are dependent on the error mode frequency τ, the nondimensional mesh size, and the non-dimensional diffusion constant. Assuming

D=1/(3Σt) allows that the convergence rates are expressed also as functions of the optical mesh size σ(=hΣt), and the scattering ratio c. The convergence rate of the CMFD1N with under-relaxation also depends on the relaxation parameter α. y

The PCS, CMR, and CMFD2N algorithms are always stable. However, the CMFD1N algorithm becomes unstable for large mesh sizes.

y

The stability domain of CMFD1N can be expanded substantially by underrelaxation. Furthermore, with the analytically derived optimum relaxation parameter, the algorithm is shown to be unconditionally stable.

y

Each iterative method has its own value of the error mode frequency τ0, which gives the slowest convergence rate, i.e., the spectral radius.

y

The CMFD2N algorithm has a corresponding linear algorithm LCMFD2N which is equivalent to the linearized CMFD2N in terms of convergence rate.

57 y

Non-dimensional analysis reveals that the only control parameter of the model problem is the non-dimensional mesh size scaled on the diffusion length.

y

The convergence rates of the algorithms which use the partial current, such as PCS, CMR, and CMFD1N, are determined by two factors: a non-dimensional mesh size (h/L) and a non-dimensional quantity (LΣ). The spectral radius of CMFD2N algorithm, which does not use a partial current, can be expressed as a function of the mesh size in the metric of diffusion length.

y

Overall, CMFD accelerates PCS more effectively than CMR. As the mesh size becomes small, CMFD is even more effective than CMR. In summary, the study in this chapter provides a robust and reliable method to

analyze the stability and the rate of convergence for both the CMR and CMFD methods for accelerating the Partial Current Sweep (PCS) algorithm.

This analysis will be

extended to two energy groups and two-dimensional geometry problems in the following chapters.

58

3. CONVERGENCE ANALYSIS OF TWO-DIMENSIONAL ONE-GROUP PROBLEM

3.1 Model Problem The physical characteristics of the model problem for two dimensions is the same as the one-dimensional model problem, i.e., one-group neutron diffusion problem with a flat fixed source for a non-multiplying infinite homogeneous medium. The neutron diffusion equation for the model problem can be written

⎛ d2 d2 ⎞ − D ⎜ 2 + 2 ⎟φ ( x, y ) + Σφ ( x, y ) = Q ⎜ dx dy ⎟⎠ ⎝

(3.1)

where D is a diffusion coefficient, Σ is an absorption cross section, and Q is a fixed source. The node interface conditions are the conventional flux and current continuity same as 1-D model problem. The problem data D, Σ, h, and Q are the same as 1-D, i.e., Table 2.1 is used. In Fig. 3.1, the mesh structure to be used is shown and a uniform mesh spacing of ∆x i = ∆y j = h is assumed for all the numerical methods to be analyzed. For the purpose of evaluating the numerical spectral radius, a finite dimension problem is considered with the number of meshes as 40 for both x- and y- directions, and with zero flux boundary conditions.

59

(node i,j + 1) yj

(node i - 1,j)

∆y j = h

(node i, j)

(node i + 1, j)

x i-1

∆ xi = h

xi

y j-1

(node i, j - 1)

Figure 3.1 Mesh Structure for 2-D Model Problem

3.2 Analytic Derivation Of Convergence Rate For the 2-D model problem, the convergence rates are derived for the conventional PCS algorithm and CMR / CMFD acceleration methods. The approach is the same as 1-D model problem. The iteration matrix obtained from the Fourier analysis of each algorithm is 4x4 (2x2 for CMFD2N) compared to 2x2 (1x1 for CMFD2N) in 1-D. A similarity transformation of the matrix is used to derive the eigenvalues of the 4x4 matrices.

3.2.1 Partial Current Sweep The most popular approach for using the nodal method to solve Eq. (3.1) is to convert the one equation of the 2-D problem into a coupled system of two equations for the 1-D flux by using a transverse integration. The transverse integrated 1-D neutron diffusion equation can be written

60

−D

where φu (u ) =

1 hv

1 Lu (u ) = hv

∫

hv

0

∫

hv

0

d2 du 2

φ u (u ) + Σφu (u ) = Q − Lu (u )

(3.2)

φ (u, v )dv is a 1-D flux in u- direction, ⎛ ⎞ d2 ⎜ − D 2 φ (u, v ) ⎟dv is a u- directional transverse leakage, ⎜ ⎟ dv ⎝ ⎠

u=x,y A u- directional 1-D current can be expressed using the 1-D flux as

J u (u ) = − D

d φ u (u ) du

(3.3)

The average transverse leakage can then be written

Lu =

where Lu =

1 hu

∫

hu

0

J vR − J vL hv

(3.4)

Lu (u )du is an average transverse leakage in u- direction,

J vs is a net current at s (s=l,r) surface in v- direction, u= x, y and v=y, x. The transverse leakage is assumed to be flat for all the analyses in this study

Lu (u ) = Lu .

(3.5)

With this assumption, an analytic solution of the 1-D equation of (3.2) for node (i,j) in Fig. 3.1 can be written as

61

φu ,i , j (u ) = Au ,i , j cosh(u / L) + Bu ,i , j sinh(u / L) +

Q − Lu ,i , j

(3.6)

Σ

where Au ,i , j and Bu ,i , j (u = x, y ) are four coefficients to be determined, and L ( = D / Σ ) is the diffusion length. Four partial incoming currents at both interfaces are used to determine the coefficients. Using the analytic solution of node (i,j), a traditional response matrix form of the equations can be formulated as ,( k +1) jout = Π jini , j,( k ) + d i, j

(3.7)

where

[

, ( k +1) jout = ji−, jx,,xl( k +1) i, j

ji+, jy,,yr( k +1)

ji+, jx,,xr( k +1)

ji−, jy,,yl( k +1)

]

T

is a vector of the out-going partial currents,

[

jini , j,( k ) = ji+, jx,,xl( k )

ji−, jy,,yr( k )

ji−, jx,,xr( k )

ji+, jy,,yl( k )

]

T

is a vector of incoming partial currents,

⎡a ⎢b Π= ⎢ ⎢c ⎢ ⎣b

a=

b c b⎤ ⎡1⎤ ⎢1⎥ ⎥ a b c ⎥ , d = Q (I − Π ) ⋅ 1, 1 = ⎢ ⎥ , b a b⎥ ⎢1⎥ 4Σ ⎢⎥ ⎥ c b a⎦ ⎣1⎦

⎛ sinh( ∆ / 2)( ∆2 + 4 − 4h 2 Σ 2 + ( −4 + ∆2 (1 − 4 L2 Σ 2 )) cosh[∆ ] ⎞ ⎜ ⎟ ⎜ + 2∆2 LΣ(1 − 4 L2 Σ 2 ) sinh[∆ ]) ⎟ ⎝ ⎠ Γ1 Γ2 b=

c=

8hΣ sinh[∆ / 2] 2 Γ2

,

4 LΣ(( ∆2 + 1) cosh[∆ / 2] + ( − cosh[3∆ / 2] + 2∆hΣ sinh[∆ / 2])) Γ1 Γ2

,

,

62

Γ1 = 2 LΣ cosh[∆ / 2] + sinh[∆ / 2] , Γ2 = ∆2 + 4 − 4h 2 Σ 2 + ( −4 + ∆2 (1 + 4 L2 Σ 2 )) cosh[∆ ] + 4∆2 LΣ sinh[∆ ] ,

and ∆( ≡ h / L) is a mesh size measured in the metric of the diffusion length.

The PCS algorithm is to sweep node by node updating the outgoing partial currents of each node using Eq. (3.7). First order errors are introduced into Eq. (3.7), by expressing the partial current at the kth iteration as a combination of exact solutions and first order error terms:

, ( k +1) jout = i, j

j

in ,( k ) i, j

Q ,( k +1) (1 + ε ξ out ) i, j 4Σ ;

Q = (1 + ε ξ ini , ,j( k ) ) 4Σ

ε