Graziani Computational Methods in Transport

Granlibakken 2004
1. Auflage 2006
ISBN: 978-3-540-28125-2
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 48, 539 Seiten

Reihe: Lecture Notes in Computational Science and Engineering

Thereexistawiderangeofapplicationswhereasigni?cantfractionofthe- mentum and energy present in a physical problem is carried by the transport of particles. Depending on the speci?capplication, the particles involved may be photons, neutrons, neutrinos, or charged particles. Regardless of which phenomena is being described, at the heart of each application is the fact that a Boltzmann like transport equation has to be solved. The complexity, and hence expense, involved in solving the transport problem can be understood by realizing that the general solution to the 3D Boltzmann transport equation is in fact really seven dimensional: 3 spatial coordinates, 2 angles, 1 time, and 1 for speed or energy. Low-order appro- mations to the transport equation are frequently used due in part to physical justi?cation but many in cases, simply because a solution to the full tra- port problem is too computationally expensive. An example is the di?usion equation, which e?ectively drops the two angles in phase space by assuming that a linear representation in angle is adequate. Another approximation is the grey approximation, which drops the energy variable by averaging over it. If the grey approximation is applied to the di?usion equation, the expense of solving what amounts to the simplest possible description of transport is roughly equal to the cost of implicit computational ?uid dynamics. It is clear therefore, that for those application areas needing some form of transport, fast, accurate and robust transport algorithms can lead to an increase in overall code performance and a decrease in time to solution.

Autoren/Hrsg.

Graziani, Frank

Inhaltsverzeichnis

1;Contents;5
2;Introduction;12
3;I Astrophysics;17
3.1;Radiation Hydrodynamics in Astrophysics;18
3.1.1;1 De.ning Radiation Hydrodynamics Terms;18
3.1.2;2 Schemes Used in Astrophysics;19
3.1.3;3 Astrophysical Applications;21
3.1.4;4 SPH Radiation Transport;25
3.1.5;References;28
3.2;Radiative Transfer;30
3.3;in Astrophysical Applications;30
3.3.1;1 Introduction;30
3.3.2;2 Description of Radiation;31
3.3.3;3 Absorption, Emission and Scattering Coe.cients;32
3.3.4;4 Hierarchies of Approximations;35
3.3.5;5 General Problem;39
3.3.6;6 Exact Numerical Solution;41
3.3.7;7 Conclusions;47
3.3.8;References;47
3.4;Neutrino Transport;49
3.5;in Core Collapse Supernovae;49
3.5.1;1 The Core Collapse Supernova Paradigm;49
3.5.2;2 The;54
3.5.3;Neutrino Transport Equation;54
3.5.4;in Spherical Symmetry: An Illustrative Example;54
3.5.5;3 Finite Di.erencing of the;56
3.5.6;Neutrino Transport;56
3.5.7;Equation;56
3.5.8;in Spherical Symmetry;56
3.5.9;4 The General Case: The Multidimensional Neutrino;69
3.5.10;Transport Equations;69
3.5.11;5 Boltzmann Neutrino Transport:;73
3.5.12;The Current State of the Art;73
3.5.13;6 Previews of Coming Distractions:;77
3.5.14;Neutrino Flavor Transformation;77
3.5.15;7 Summary and Prospects;79
3.5.16;Acknowledgments;80
3.5.17;References;81
3.6;Discrete-Ordinates Methods;83
3.7;for Radiative Transfer;83
3.8;in the Non-Relativistic Stellar Regime;83
3.8.1;1 Introduction;83
3.8.2;2 The Approximate Radiation-Hydrodynamics Model;83
3.8.3;3 Discretization and Solution Techniques;87
3.8.4;References;94
4;II Atmospheric Science, Oceanography, and Plant Canopies;96
4.1;Effective Propagation Kernels in Structured Media with Broad Spatial Correlations, Illustration with Large-Scale Transport of Solar Photons Through Cloudy Atmospheres;97
4.1.1;1 Introduction and Overview;97
4.1.2;2 Extinction and Scattering Revisited, and Some Notations Introduced;100
4.1.3;3 Propagation;108
4.1.4;4 Multiple Scattering and Di.usions;126
4.1.5;5 Large-Scale 3D RT E.ects in Cloudy Atmospheres;134
4.1.6;6 Concluding Remarks;146
4.1.7;Acknowledgments and Dedication;148
4.1.8;References;148
4.2;Mathematical Simulation of the Radiative Transfer in Statistically Inhomogeneous Clouds;153
4.2.1;1 Introduction;153
4.2.2;2 Stochastic RT Equation;154
4.2.3;3 Statistically Inhomogeneous Model;155
4.2.4;4 Ensemble Averaged Radiance;156
4.2.5;5 Validation;158
4.2.6;6 Summary;159
4.2.7;Acknowledgments;160
4.2.8;References;160
4.3;Transport Theory for Optical Oceanography;162
4.3.1;1 Introduction;162
4.3.2;2 Aspects Requiring Special Computational Attention;167
4.3.3;3 Computational Programs;170
4.3.4;4 Computing Challenges;172
4.3.5;References;172
4.4;Perturbation Technique in 3D Cloud Optics: Theory and Results;175
4.4.1;1 Introduction;175
4.4.2;2 Definition of the Problem;175
4.4.3;3 Variational Principe to Derive the Radiative Transfer Equation;176
4.4.4;4 Perturbation;177
4.4.5;5 A Toy Example;178
4.4.6;References;180
4.5;Vegetation Canopy Re.ectance Modeling with Turbid Medium Radiative Transfer;182
4.5.1;1 Introduction;182
4.5.2;2 Description of the LCM2 Coupled Leaf/Canopy Radiative Transfer (RT) Model;189
4.5.3;3 LCM2 Demonstration;205
4.5.4;References;219
4.6;Rayspread: A Virtual Laboratory for Rapid BRF Simulations Over 3-D Plant Canopies;220
4.6.1;1 Canopy Radiation Transfer Fundamentals;221
4.6.2;2 The Rayspread Model;228
4.6.3;3 Conclusion;236
4.6.4;References;237
5;III High Energy Density Physics;241
5.1;Use of the Space Adaptive Algorithm to Solve 2D Problems of Photon Transport and Interaction with Medium;242
5.1.1;1 Introduction;242
5.1.2;2 Statement of a 2D Transport Equation;243
5.1.3;3 Description of 2D Transport Equation;245
5.1.4;Approximation Methods;245
5.1.5;4 Description of the Space Adaptive Computational;245
5.1.6;Algorithm for Transport Equation;245
5.1.7;5 Results of Computational Investigations;247
5.1.8;of the Adaptive Method Performance;247
5.1.9;6 Conclusion;258
5.1.10;References;261
5.2;Accurate and E.cient Radiation Transport in Optically Thick Media – by Means of the Symbolic Implicit Monte Carlo Method in the Di.erence Formulation*;262
5.2.1;1 Introduction;262
5.2.2;2 Radiation Transport in LTE;265
5.2.3;3 The Di.erence Formulation;268
5.2.4;4 Test Problems;275
5.2.5;5 Summary and Directions for Further Work;284
5.2.6;Acknowledgement;287
5.2.7;References;287
5.3;An Evaluation of the Di.erence Formulation for Photon Transport in a Two Level System*;290
5.3.1;1 Introduction;290
5.3.2;2 The Equations for Line Transport;292
5.3.3;3 Numerical Development;296
5.3.4;4 Numerical Results in the Gray Approximation;302
5.3.5;5 Concluding Remarks;311
5.3.6;References;312
5.4;Non-LTE Radiation Transport in High Radiation Plasmas;314
5.4.1;1 Introduction;314
5.4.2;2 Non-LTE Energetics;316
5.4.3;3 Radiation Transport;318
5.4.4;4 Test Case: Radiation-driven Cylinder;323
5.4.5;5 Linear Response Matrix;329
5.4.6;6 Summary;331
5.4.7;Acknowledgments;331
5.4.8;References;332
5.5;Finite-Difference Methods Implemented in SATURN Complex to Solve Multidimensional Time-Dependent Transport Problems;333
5.5.1;1 Multiple-Group Transport Equation Approximation;337
5.6;Implicit Solution of Non-Equilibrium Radiation Di.usion Including Reactive Heating Source in Material Energy Equation;359
5.6.1;1 Introduction;359
5.6.2;2 Mathematical Model;360
5.6.3;3 Numerical Methods;361
5.6.4;4 Results;365
5.6.5;5 Conclusions;374
5.6.6;Acknowledgements;375
5.6.7;References;375
6;IV Mathematics and Computer Science;377
6.1;Transport Approximations;378
6.2;Transport Approximations n Partially Diffusive Media;378
6.2.1;1 Introduction;378
6.2.2;2 Variational Formulation for Transport;380
6.2.3;3 Transport-Di.usion Coupling;394
6.2.4;4 Generalized Di.usion Models;398
6.2.5;Acknowledgments;402
6.2.6;A Local Second-Order Equation and Linear Corrector;403
6.2.7;References;404
6.3;High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation;406
6.3.1;1 Introduction;406
6.3.2;2 Background;408
6.3.3;3 Discretization of the 3-D Problem;410
6.3.4;4 Numerical Experiments;415
6.3.5;5 Discussion;425
6.3.6;Acknowledgements;426
6.3.7;References;426
6.4;Obtaining Identical Results on Varying Numbers of Processors in Domain Decomposed Particle Monte Carlo Simulations;428
6.4.1;1 Description of the Problem;428
6.4.2;2 Ensuring the Invariance of the Pseudo-Random Number Stream Employed by Each Particle;431
6.4.3;3 Ensuring That Addition is Commutative;432
6.4.4;4 Results;435
6.4.5;5 Conclusions;436
6.4.6;Appendix: Shave Algorithm;437
6.4.7;References;437
6.5;KM-Method of Iteration Convergence Acceleration for Solving a 2D Time-Dependent Multiple-Group Transport Equation and its Modi.cations;439
6.5.1;1 Statement of a 2D Transport Problem;439
6.5.2;2 KM-method;441
6.5.3;3 MKM-method;442
6.5.4;4 KM3-method;443
6.5.5;5 Test Computation Results;444
6.6;A Regularized Boltzmann Scattering Operator for Highly Forward Peaked Scattering;448
6.6.1;1 Introduction;448
6.6.2;2 Generalized Fermi Expansion;449
6.6.3;3 Regularized Collision Operator;452
6.6.4;4 Numerical Results;455
6.6.5;Conclusions;456
6.6.6;Acknowledgement;458
6.6.7;References;458
6.7;Implicit Riemann Solvers for the Pn Equations;459
6.7.1;1 Introduction;459
6.7.2;2 Pn Equations;460
6.7.3;3 Solving the Riemann Problem;461
6.7.4;4 High Resolution Flux from Linear Reconstruction;463
6.7.5;5 Time Integration;464
6.7.6;6 Implementation;465
6.7.7;7 Results;465
6.7.8;8 Conclusion;468
6.7.9;References;469
6.8;The Solution of the Time–Dependent Sn Equations on Parallel Architectures;470
6.8.1;1 Introduction;470
6.8.2;2 A Brief Review of The Implicit Discrete Ordinates;471
6.8.3;Discretization Method;471
6.8.4;3 Iterative Approaches;473
6.8.5;4 Speeding Up and Obtaining Convergence;476
6.8.6;5 Parallel Implementation;482
6.8.7;of the Full Linear System Approach;482
6.8.8;6 Parallel Scalability of a 2-D Test Problem;484
6.8.9;7 Conclusions and Future Directions;485
6.8.10;8 Acknowledgments;486
6.8.11;References;486
6.9;Different Algorithms of 2D Transport Equation Parallelization on Random Non-Orthogonal Grids;488
7;V Neutron Transport;498
7.1;Parallel Deterministic Neutron Transport with AMR;499
7.1.1;1 Introduction;499
7.1.2;2 Code Overview;500
7.1.3;3 Numerical Results;508
7.1.4;4 Future Work;511
7.1.5;References;512
7.2;An Overview of Neutron Transport Problems and Simulation Techniques;513
7.2.1;1 Introduction;513
7.2.2;2 Physical and Mathematical Basics;513
7.2.3;3 Basics of Stochastic and Deterministic Methods;521
7.2.4;4 Stochastic (Monte Carlo) Methods;522
7.2.5;5 Deterministic Methods;527
7.2.6;6 Automatic Variance Reduction (Hybrid) Methods;530
7.2.7;7 Discussion;531
7.2.8;References;533
8;Lecture Notes in Computational Science and Engineering;537