Computational Methods in Transport (eBook)

Contents 5
Introduction 12
I Astrophysics 17
Radiation Hydrodynamics in Astrophysics 18
1 De.ning Radiation Hydrodynamics Terms 18
2 Schemes Used in Astrophysics 19
3 Astrophysical Applications 21
4 SPH Radiation Transport 25
References 28
Radiative Transfer 30
in Astrophysical Applications 30
1 Introduction 30
2 Description of Radiation 31
3 Absorption, Emission and Scattering Coe.cients 32
4 Hierarchies of Approximations 35
5 General Problem 39
6 Exact Numerical Solution 41
7 Conclusions 47
References 47
Neutrino Transport 49
in Core Collapse Supernovae 49
1 The Core Collapse Supernova Paradigm 49
2 The 54
Neutrino Transport Equation 54
in Spherical Symmetry: An Illustrative Example 54
3 Finite Di.erencing of the 56
Neutrino Transport 56
Equation 56
in Spherical Symmetry 56
4 The General Case: The Multidimensional Neutrino 69
Transport Equations 69
5 Boltzmann Neutrino Transport: 73
The Current State of the Art 73
6 Previews of Coming Distractions: 77
Neutrino Flavor Transformation 77
7 Summary and Prospects 79
Acknowledgments 80
References 81
Discrete-Ordinates Methods 83
for Radiative Transfer 83
in the Non-Relativistic Stellar Regime 83
1 Introduction 83
2 The Approximate Radiation-Hydrodynamics Model 83
3 Discretization and Solution Techniques 87
References 94
II Atmospheric Science, Oceanography, and Plant Canopies 96
Effective Propagation Kernels in Structured Media with Broad Spatial Correlations, Illustration with Large-Scale Transport of Solar Photons Through Cloudy Atmospheres 97
1 Introduction and Overview 97
2 Extinction and Scattering Revisited, and Some Notations Introduced 100
3 Propagation 108
4 Multiple Scattering and Di.usions 126
5 Large-Scale 3D RT E.ects in Cloudy Atmospheres 134
6 Concluding Remarks 146
Acknowledgments and Dedication 148
References 148
Mathematical Simulation of the Radiative Transfer in Statistically Inhomogeneous Clouds 153
1 Introduction 153
2 Stochastic RT Equation 154
3 Statistically Inhomogeneous Model 155
4 Ensemble Averaged Radiance 156
5 Validation 158
6 Summary 159
Acknowledgments 160
References 160
Transport Theory for Optical Oceanography 162
1 Introduction 162
2 Aspects Requiring Special Computational Attention 167
3 Computational Programs 170
4 Computing Challenges 172
References 172
Perturbation Technique in 3D Cloud Optics: Theory and Results 175
1 Introduction 175
2 Definition of the Problem 175
3 Variational Principe to Derive the Radiative Transfer Equation 176
4 Perturbation 177
5 A Toy Example 178
References 180
Vegetation Canopy Re.ectance Modeling with Turbid Medium Radiative Transfer 182
1 Introduction 182
2 Description of the LCM2 Coupled Leaf/Canopy Radiative Transfer (RT) Model 189
3 LCM2 Demonstration 205
References 219
Rayspread: A Virtual Laboratory for Rapid BRF Simulations Over 3-D Plant Canopies 220
1 Canopy Radiation Transfer Fundamentals 221
2 The Rayspread Model 228
3 Conclusion 236
References 237
III High Energy Density Physics 241
Use of the Space Adaptive Algorithm to Solve 2D Problems of Photon Transport and Interaction with Medium 242
1 Introduction 242
2 Statement of a 2D Transport Equation 243
3 Description of 2D Transport Equation 245
Approximation Methods 245
4 Description of the Space Adaptive Computational 245
Algorithm for Transport Equation 245
5 Results of Computational Investigations 247
of the Adaptive Method Performance 247
6 Conclusion 258
References 261
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
1 Introduction 262
2 Radiation Transport in LTE 265
3 The Di.erence Formulation 268
4 Test Problems 275
5 Summary and Directions for Further Work 284
Acknowledgement 287
References 287
An Evaluation of the Di.erence Formulation for Photon Transport in a Two Level System* 290
1 Introduction 290
2 The Equations for Line Transport 292
3 Numerical Development 296
4 Numerical Results in the Gray Approximation 302
5 Concluding Remarks 311
References 312
Non-LTE Radiation Transport in High Radiation Plasmas 314
1 Introduction 314
2 Non-LTE Energetics 316
3 Radiation Transport 318
4 Test Case: Radiation-driven Cylinder 323
5 Linear Response Matrix 329
6 Summary 331
Acknowledgments 331
References 332
Finite-Difference Methods Implemented in SATURN Complex to Solve Multidimensional Time-Dependent Transport Problems 333
1 Multiple-Group Transport Equation Approximation 337
Implicit Solution of Non-Equilibrium Radiation Di.usion Including Reactive Heating Source in Material Energy Equation 359
1 Introduction 359
2 Mathematical Model 360
3 Numerical Methods 361
4 Results 365
5 Conclusions 374
Acknowledgements 375
References 375
IV Mathematics and Computer Science 377
Transport Approximations 378
Transport Approximations n Partially Diffusive Media 378
1 Introduction 378
2 Variational Formulation for Transport 380
3 Transport-Di.usion Coupling 394
4 Generalized Di.usion Models 398
Acknowledgments 402
A Local Second-Order Equation and Linear Corrector 403
References 404
High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation 406
1 Introduction 406
2 Background 408
3 Discretization of the 3-D Problem 410
4 Numerical Experiments 415
5 Discussion 425
Acknowledgements 426
References 426
Obtaining Identical Results on Varying Numbers of Processors in Domain Decomposed Particle Monte Carlo Simulations 428
1 Description of the Problem 428
2 Ensuring the Invariance of the Pseudo-Random Number Stream Employed by Each Particle 431
3 Ensuring That Addition is Commutative 432
4 Results 435
5 Conclusions 436
Appendix: Shave Algorithm 437
References 437
KM-Method of Iteration Convergence Acceleration for Solving a 2D Time-Dependent Multiple-Group Transport Equation and its Modi.cations 439
1 Statement of a 2D Transport Problem 439
2 KM-method 441
3 MKM-method 442
4 KM3-method 443
5 Test Computation Results 444
A Regularized Boltzmann Scattering Operator for Highly Forward Peaked Scattering 448
1 Introduction 448
2 Generalized Fermi Expansion 449
3 Regularized Collision Operator 452
4 Numerical Results 455
Conclusions 456
Acknowledgement 458
References 458
Implicit Riemann Solvers for the Pn Equations 459
1 Introduction 459
2 Pn Equations 460
3 Solving the Riemann Problem 461
4 High Resolution Flux from Linear Reconstruction 463
5 Time Integration 464
6 Implementation 465
7 Results 465
8 Conclusion 468
References 469
The Solution of the Time–Dependent Sn Equations on Parallel Architectures 470
1 Introduction 470
2 A Brief Review of The Implicit Discrete Ordinates 471
Discretization Method 471
3 Iterative Approaches 473
4 Speeding Up and Obtaining Convergence 476
5 Parallel Implementation 482
of the Full Linear System Approach 482
6 Parallel Scalability of a 2-D Test Problem 484
7 Conclusions and Future Directions 485
8 Acknowledgments 486
References 486
Different Algorithms of 2D Transport Equation Parallelization on Random Non-Orthogonal Grids 488
V Neutron Transport 498
Parallel Deterministic Neutron Transport with AMR 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 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
Lecture Notes in Computational Science and Engineering 537