Graziani F. (editor) Computational Methods in Transport

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Lecture Notes

in Computational Science

and Engineering

Editors

Timothy J. Barth

Michael Griebel

David E. Keyes

Risto M. Nieminen

Dirk Roose

Tamar Schlick

Frank Graziani

Editor

Computational Methods

in Transport

Granlibakken 2004

ABC

With 196 Figures and 23 Tables

Editor

Frank Graziani

Lawrence Livermore National Laboratory

East Avenue 7000

Livermore, CA 94550, U.S.A.

email: graziani1@llnl.gov

Library of Congress Control Number: 2005931994

Mathematics Subject Classiﬁcation: P19005,M1400X

ISBN-10 3-540-28122-3 Springer Berlin Heidelberg New York

ISBN-13 978-3-540-28122-1 Springer Berlin Heidelberg New York

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Contents

Part I Astrophysics

Radiation Hydrodynamics in Astrophysics

Chris L. Fryer .................................................. 3

1 DeﬁningRadiationHydrodynamicsTerms...................... 3

2 SchemesUsedin Astrophysics................................. 4

3 Astrophysical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 SPHRadiationTransport .................................... 10

References ..................................................... 13

Radiative Transfer in Astrophysical Applications

I. Hubeny ...................................................... 15

1 Introduction ................................................ 15

2 Description of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Absorption,Emission and ScatteringCoeﬃcients ................ 17

4 Hierarchies of Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 GeneralProblem ............................................ 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 TheCoreCollapse Supernova Paradigm........................ 35

2TheO(v/c) Neutrino Transport Equation in Spherical Symmetry:

An IllustrativeExample...................................... 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 SummaryandProspects...................................... 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 TheApproximateRadiation-HydrodynamicsModel.............. 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 RTEquation ...................................... 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 ComputationalPrograms..................................... 159

4 ComputingChallenges ....................................... 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 Deﬁnitionof the Problem..................................... 165

3 Variational Principe

to Derive the Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . 166

4 Perturbation................................................ 167

5 AToyExample ............................................. 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 TheRayspreadModel........................................ 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

TransportEquation.......................................... 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 RadiationTransportinLTE .................................. 258

3 TheDiﬀerence Formulation................................... 261

4 Test Problems .............................................. 268

5 SummaryandDirections 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 TheEquationsforLineTransport ............................. 285

3 NumericalDevelopment...................................... 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-LTEEnergetics ......................................... 309

3 RadiationTransport ......................................... 311

4 Test Case: Radiation-drivenCylinder .......................... 316

5 Linear ResponseMatrix...................................... 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 NumericalMethods.......................................... 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 GeneralizedDiﬀusion 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 NumericalExperiments ...................................... 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 byEachParticle ................................... 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 GeneralizedFermiExpansion ................................. 446

3 Regularized Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

4 NumericalResults........................................... 452

References ..................................................... 455

Implicit Riemann Solvers for the P

Equations

Ryan McClarren, James Paul Holloway, Thomas Brunner,

Thomas Mehlhorn ............................................... 457

1 Introduction ................................................ 457

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

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

on Parallel Architectures

F. Douglas Swesty .............................................. 469

1 Introduction ................................................ 469

2 A Brief Review of The Implicit Discrete Ordinates Discretization

Method .................................................... 470

3 IterativeApproaches......................................... 472

4 SpeedingUp and Obtaining Convergence....................... 475

5 Parallel Implementation

of theFullLinear SystemApproach ........................... 481

6 Parallel Scalability of a 2-D Test Problem . . . . . . . . . . . . . . . . . . . . . . 483

7 ConclusionsandFuture Directions............................. 484

8 Acknowledgments ........................................... 485

References ..................................................... 485

Diﬀerent Algorithms of 2D Transport Equation Parallelization

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 CodeOverview.............................................. 500

3 NumericalResults........................................... 508

4 FutureWork................................................ 511

References ..................................................... 512

An Overview of Neutron Transport Problems

and Simulation Techniques

Edward W. Larsen .............................................. 513

1 Introduction ................................................ 513

2 Physical and MathematicalBasics............................. 513

3 Basics of Stochastic and Deterministic Methods . . . . . . . . . . . . . . . . . 521

4 Stochastic (MonteCarlo) Methods............................. 522

5 DeterministicMethods....................................... 527

6 Automatic Variance Reduction (Hybrid) Methods . . . . . . . . . . . . . . . 530

7 Discussion.................................................. 531

References ..................................................... 533