Computational Methods in Transport: Granlibakken 2004

537 Pages·2006·22.504 MB·English

by  Chris L. Fryer (auth.), Frank Graziani (eds.)

Lecture Notes
48
in Computational Science
and Engineering
Editors
Timothy J.Barth
MichaelGriebel
DavidE.Keyes
RistoM.Nieminen
Dirk Roose
TamarSchlick
Frank Graziani
Editor
Computational Methods
in Transport
Granlibakken 2004
With196Figuresand23Tables
ABC
Editor
FrankGraziani
LawrenceLivermoreNationalLaboratory
EastAvenue7000
Livermore,CA94550,U.S.A.
email:[email protected]
LibraryofCongressControlNumber:2005931994
MathematicsSubjectClassification:P19005,M1400X
ISBN-10 3-540-28122-3SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-28122-1SpringerBerlinHeidelbergNewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liableforprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springeronline.com
(cid:1)c Springer-VerlagBerlinHeidelberg2006
PrintedinTheNetherlands
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage
Coverdesign:design&productionGmbH,Heidelberg
Printedonacid-freepaper SPIN:11360605 46/TechBooks 543210
Contents
Part I Astrophysics
Radiation Hydrodynamics in Astrophysics
Chris L. Fryer.................................................. 3
1 Defining Radiation Hydrodynamics Terms ...................... 3
2 Schemes Used in Astrophysics................................. 4
3 Astrophysical Applications ................................... 6
4 SPH Radiation Transport .................................... 10
References ..................................................... 13
Radiative Transfer in Astrophysical Applications
I. Hubeny ...................................................... 15
1 Introduction ................................................ 15
2 Description of Radiation ..................................... 16
3 Absorption, Emission and Scattering Coefficients ................ 17
4 Hierarchies of Approximations ................................ 20
5 General Problem ............................................ 24
6 Exact Numerical Solution .................................... 26
7 Conclusions................................................. 32
References ..................................................... 32
Neutrino Transport in Core Collapse Supernovae
Anthony Mezzacappa, Matthias Liebend¨orfer, Christian Y. Cardall,
O.E. Bronson Messer, Stephen W. Bruenn ........................ 35
1 The Core Collapse Supernova Paradigm........................ 35
2 The O(v/c) Neutrino Transport Equation in Spherical Symmetry:
An Illustrative Example...................................... 40
3 Finite Differencing of the O(v/c) Neutrino Transport Equation
in Spherical Symmetry ....................................... 42
4 The General Case: The Multidimensional Neutrino Transport
Equations .................................................. 55
5 Boltzmann Neutrino Transport: The Current State of the Art..... 59
6 Previews of Coming Distractions: Neutrino Flavor Transformation. 63
7 Summary and Prospects...................................... 65
References ..................................................... 67
VI Contents
Discrete-Ordinates Methods for Radiative Transfer
in the Non-Relativistic Stellar Regime
Jim E. Morel................................................... 69
1 Introduction ................................................ 69
2 The Approximate Radiation-Hydrodynamics Model.............. 69
3 Discretization and Solution Techniques......................... 73
References ..................................................... 80
Part II Atmospheric Science, Oceanography, and Plant Canopies
Effective Propagation Kernels in Structured Media
with Broad Spatial Correlations, Illustration
with Large-Scale Transport of Solar Photons Through
Cloudy Atmospheres
Anthony B. Davis............................................... 85
1 Introduction and Overview ................................... 85
2 Extinction and Scattering Revisited,
and Some Notations Introduced ............................... 88
3 Propagation ................................................ 96
4 Multiple Scattering and Diffusions............................. 114
5 Large-Scale 3D RT Effects in Cloudy Atmospheres............... 122
6 Concluding Remarks......................................... 134
References ..................................................... 136
Mathematical Simulation of the Radiative Transfer
in Statistically Inhomogeneous Clouds
Evgueni I. Kassianov............................................ 141
1 Introduction ................................................ 141
2 Stochastic RT Equation ...................................... 142
3 Statistically Inhomogeneous Model ............................ 143
4 Ensemble Averaged Radiance ................................. 144
5 Validation .................................................. 146
6 Summary................................................... 147
References ..................................................... 148
Transport Theory for Optical Oceanography
N.J. McCormick ................................................ 151
1 Introduction ................................................ 151
2 Aspects Requiring Special Computational Attention ............. 156
3 Computational Programs..................................... 159
4 Computing Challenges ....................................... 161
References ..................................................... 161
Contents VII
Perturbation Technique in 3D Cloud Optics:
Theory and Results
Igor N. Polonsky, Anthony B. Davis, Michael A. Box................ 165
1 Introduction ................................................ 165
2 Definition of the Problem..................................... 165
3 Variational Principe
to Derive the Radiative Transfer Equation...................... 166
4 Perturbation................................................ 167
5 A Toy Example ............................................. 168
References ..................................................... 170
Vegetation Canopy Reflectance Modeling
with Turbid Medium Radiative Transfer
Barry D. Ganapol............................................... 173
1 Introduction ................................................ 173
2 Description of the LCM2 Coupled Leaf/Canopy Radiative
Transfer (RT) Model......................................... 180
3 LCM2 Demonstration........................................ 196
References ..................................................... 210
Rayspread: A Virtual Laboratory for Rapid BRF Simulations
Over 3-D Plant Canopies
Jean-Luc Widlowski, Thomas Lavergne, Bernard Pinty, Michel
Verstraete, Nadine Gobron ....................................... 211
1 Canopy Radiation Transfer Fundamentals ...................... 212
2 The Rayspread Model........................................ 219
3 Conclusion ................................................. 227
References ..................................................... 228
Part III High Energy Density Physics
Use of the Space Adaptive Algorithm to Solve 2D Problems
of Photon Transport and Interaction with Medium
A. V. Alekseyev, R. M. Shagaliev, I. M. Belyakov, A. V. Gichuk, V.
V. Evdokimov, A. N. Moskvin, A. A. Nuzhdin, N. P. Pleteneva, and
T. V. Shemyakina............................................... 235
1 Introduction ................................................ 235
2 Statement of a 2D Transport Equation......................... 236
3 Description of 2D Transport Equation Approximation Methods ... 238
4 Description of the Space Adaptive Computational Algorithm for
Transport Equation.......................................... 238
5 Results of Computational Investigations of the Adaptive Method
Performance ................................................ 240
6 Conclusion ................................................. 251
References ..................................................... 254
VIII Contents
Accurate and Efficient Radiation Transport in Optically
Thick Media – by Means of the Symbolic Implicit Monte
Carlo Method in the Difference Formulation
Abraham Sz˝oke, Eugene D. Brooks III, Michael Scott McKinley,
and Frank C. Daffin............................................. 255
1 Introduction ................................................ 255
2 Radiation Transport in LTE .................................. 258
3 The Difference Formulation................................... 261
4 Test Problems .............................................. 268
5 Summary and Directions for Further Work ..................... 277
References ..................................................... 280
An Evaluation of the Difference Formulation
for Photon Transport in a Two Level System
Frank Daffin, Michael Scott McKinley, Eugene D. Brooks III, and
Abraham Sz˝oke ................................................. 283
1 Introduction ................................................ 283
2 The Equations for Line Transport ............................. 285
3 Numerical Development ...................................... 289
4 Numerical Results in the Gray Approximation .................. 295
5 Concluding Remarks......................................... 304
References ..................................................... 305
Non-LTE Radiation Transport in High Radiation Plasmas
Howard A. Scott ................................................ 307
1 Introduction ................................................ 307
2 Non-LTE Energetics ......................................... 309
3 Radiation Transport ......................................... 311
4 Test Case: Radiation-driven Cylinder .......................... 316
5 Linear Response Matrix...................................... 322
6 Summary................................................... 324
References ..................................................... 325
Finite-Difference Methods Implemented in SATURN
Complex to Solve Multidimensional Time-Dependent
Transport Problems
R.M. Shagaliev, A.V. Alekseyev, A.V. Gichuk, A.A. Nuzhdin,
N.P. Pleteneva, and L.P. Fedotova ................................ 327
1 Multiple-Group Transport Equation Approximation ............. 331
Implicit Solution of Non-Equilibrium Radiation Diffusion
Including Reactive Heating Source in Material Energy
Equation
Dana E. Shumaker, Carol S. Woodward............................ 353
1 Introduction ................................................ 353
2 Mathematical Model......................................... 354
Contents IX
3 Numerical Methods.......................................... 355
4 Results..................................................... 359
5 Conclusions................................................. 368
References ..................................................... 369
Part IV Mathematics and Computer Science
Transport Approximations in Partially Diffusive Media
Guillaume Bal.................................................. 373
1 Introduction ................................................ 373
2 Variational Formulation for Transport ......................... 375
3 Transport-Diffusion Coupling ................................. 389
4 Generalized Diffusion Models ................................. 393
A Local Second-Order Equation and Linear Corrector.............. 398
References ..................................................... 399
High Order Finite Volume Nonlinear Schemes
for the Boltzmann Transport Equation
Barna L. Bihari, Peter N. Brown ................................. 401
1 Introduction ................................................ 401
2 Background ................................................ 403
3 Discretization of the 3-D Problem ............................. 405
4 Numerical Experiments ...................................... 410
5 Discussion.................................................. 420
References ..................................................... 421
Obtaining Identical Results on Varying Numbers
of Processors in Domain Decomposed Particle
Monte Carlo Simulations
N.A. Gentile, Malvin Kalos, Thomas A. Brunner ................... 423
1 Description of the Problem ................................... 423
2 Ensuring the Invariance of the Pseudo-Random Number Stream
Employed by Each Particle ................................... 426
3 Ensuring That Addition is Commutative ....................... 427
4 Results..................................................... 430
5 Conclusions................................................. 431
References ..................................................... 432
KM-Method of Iteration Convergence Acceleration
for Solving a 2D Time-Dependent Multiple-Group Transport
Equation and its Modifications
A.V. Gichuk, L.P. Fedotova, R.M. Shagaliev ....................... 435
1 Statement of a 2D Transport Problem ......................... 435
2 KM-method ................................................ 437
3 MKM-method .............................................. 438
X Contents
4 KM3-method ............................................... 439
5 Test Computation Results .................................... 440
A Regularized Boltzmann Scattering Operator
for Highly Forward Peaked Scattering
Anil K. Prinja, Brian C. Franke .................................. 445
1 Introduction ................................................ 445
2 Generalized Fermi Expansion ................................. 446
3 Regularized Collision Operator................................ 449
4 Numerical Results ........................................... 452
References ..................................................... 455
Implicit Riemann Solvers for the P Equations
n
Ryan McClarren, James Paul Holloway, Thomas Brunner,
Thomas Mehlhorn............................................... 457
1 Introduction ................................................ 457
2 P Equations ............................................... 458
n
3 Solving the Riemann Problem................................. 459
4 High Resolution Flux from Linear Reconstruction ............... 461
5 Time Integration ............................................ 462
6 Implementation ............................................. 463
7 Results..................................................... 463
8 Conclusion ................................................. 466
References ..................................................... 467
The Solution of the Time–Dependent S Equations
N
on Parallel Architectures
F. Douglas Swesty .............................................. 469
1 Introduction ................................................ 469
2 A Brief Review of The Implicit Discrete Ordinates Discretization
Method .................................................... 470
3 Iterative Approaches......................................... 472
4 Speeding Up and Obtaining Convergence....................... 475
5 Parallel Implementation
of the Full Linear System Approach ........................... 481
6 Parallel Scalability of a 2-D Test Problem ...................... 483
7 Conclusions and Future Directions............................. 484
8 Acknowledgments ........................................... 485
References ..................................................... 485
DifferentAlgorithmsof2DTransportEquationParallelization
on Random Non-Orthogonal Grids
Shagaliev R.M., Alekseev A.V., Beliakov I.M., Gichuk A.V., Nuzhdin
A.A., Rezchikov V.Yu............................................ 487
Contents XI
Part V Neutron Transport
Parallel Deterministic Neutron Transport with AMR
C.J. Clouse .................................................... 499
1 Introduction ................................................ 499
2 Code Overview.............................................. 500
3 Numerical Results ........................................... 508
4 Future Work................................................ 511
References ..................................................... 512
An Overview of Neutron Transport Problems
and Simulation Techniques
Edward W. Larsen .............................................. 513
1 Introduction ................................................ 513
2 Physical and Mathematical Basics ............................. 513
3 Basics of Stochastic and Deterministic Methods ................. 521
4 Stochastic (Monte Carlo) Methods............................. 522
5 Deterministic Methods ....................................... 527
6 Automatic Variance Reduction (Hybrid) Methods ............... 530
7 Discussion.................................................. 531
References ..................................................... 533