Lecture Notesin Computational Scienceand Engineering 48Editors

Timothy J. BarthMichael GriebelDavid E. KeyesRisto M. NieminenDirk RooseTamar Schlick

Frank GrazianiEditor

Computational Methodsin TransportGranlibakken 2004

ABC

With 196 Figures and 23 Tables

Editor

Frank GrazianiLawrence Livermore National LaboratoryEast Avenue 7000Livermore, 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 YorkISBN-13 978-3-540-28122-1 Springer Berlin Heidelberg New York

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Contents

Part I Astrophysics

Radiation Hydrodynamics in AstrophysicsChris L. Fryer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Defining Radiation Hydrodynamics Terms . . . . . . . . . . . . . . . . . . . . . . 32 Schemes Used in Astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Astrophysical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 SPH Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Radiative Transfer in Astrophysical ApplicationsI. Hubeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Description of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Absorption, Emission and Scattering Coefficients . . . . . . . . . . . . . . . . 174 Hierarchies of Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Exact Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Neutrino Transport in Core Collapse SupernovaeAnthony Mezzacappa, Matthias Liebendorfer, Christian Y. Cardall,O.E. Bronson Messer, Stephen W. Bruenn . . . . . . . . . . . . . . . . . . . . . . . . 351 The Core Collapse Supernova Paradigm . . . . . . . . . . . . . . . . . . . . . . . . 352 The O(v/c) Neutrino Transport Equation in Spherical Symmetry:

An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Finite Differencing of the O(v/c) Neutrino Transport Equation

in Spherical Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 The General Case: The Multidimensional Neutrino Transport

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Boltzmann Neutrino Transport: The Current State of the Art . . . . . 596 Previews of Coming Distractions: Neutrino Flavor Transformation . 637 Summary and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

VI Contents

Discrete-Ordinates Methods for Radiative Transferin the Non-Relativistic Stellar RegimeJim E. Morel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 The Approximate Radiation-Hydrodynamics Model . . . . . . . . . . . . . . 693 Discretization and Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 73References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Part II Atmospheric Science, Oceanography, and Plant Canopies

Effective Propagation Kernels in Structured Mediawith Broad Spatial Correlations, Illustrationwith Large-Scale Transport of Solar Photons ThroughCloudy AtmospheresAnthony B. Davis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 Extinction and Scattering Revisited,

and Some Notations Introduced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964 Multiple Scattering and Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 Large-Scale 3D RT Effects in Cloudy Atmospheres . . . . . . . . . . . . . . . 1226 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Mathematical Simulation of the Radiative Transferin Statistically Inhomogeneous CloudsEvgueni I. Kassianov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1412 Stochastic RT Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423 Statistically Inhomogeneous Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434 Ensemble Averaged Radiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Transport Theory for Optical OceanographyN.J. McCormick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1511 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1512 Aspects Requiring Special Computational Attention . . . . . . . . . . . . . 1563 Computational Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594 Computing Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Contents VII

Perturbation Technique in 3D Cloud Optics:Theory and ResultsIgor N. Polonsky, Anthony B. Davis, Michael A. Box . . . . . . . . . . . . . . . . 1651 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1652 Definition of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653 Variational Principe

to Derive the Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . 1664 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675 A Toy Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Vegetation Canopy Reflectance Modelingwith Turbid Medium Radiative TransferBarry D. Ganapol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1731 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1732 Description of the LCM2 Coupled Leaf/Canopy Radiative

Transfer (RT) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1803 LCM2 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Rayspread: A Virtual Laboratory for Rapid BRF SimulationsOver 3-D Plant CanopiesJean-Luc Widlowski, Thomas Lavergne, Bernard Pinty, MichelVerstraete, Nadine Gobron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111 Canopy Radiation Transfer Fundamentals . . . . . . . . . . . . . . . . . . . . . . 2122 The Rayspread Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2193 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Part III High Energy Density Physics

Use of the Space Adaptive Algorithm to Solve 2D Problemsof Photon Transport and Interaction with MediumA. V. Alekseyev, R. M. Shagaliev, I. M. Belyakov, A. V. Gichuk, V.V. Evdokimov, A. N. Moskvin, A. A. Nuzhdin, N. P. Pleteneva, andT. V. Shemyakina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2351 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2352 Statement of a 2D Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . 2363 Description of 2D Transport Equation Approximation Methods . . . 2384 Description of the Space Adaptive Computational Algorithm for

Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2385 Results of Computational Investigations of the Adaptive Method

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2406 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

VIII Contents

Accurate and Efficient Radiation Transport in OpticallyThick Media – by Means of the Symbolic Implicit MonteCarlo Method in the Difference FormulationAbraham Szoke, Eugene D. Brooks III, Michael Scott McKinley,and Frank C. Daffin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2551 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2552 Radiation Transport in LTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2583 The Difference Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614 Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685 Summary and Directions for Further Work . . . . . . . . . . . . . . . . . . . . . 277References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

An Evaluation of the Difference Formulationfor Photon Transport in a Two Level SystemFrank Daffin, Michael Scott McKinley, Eugene D. Brooks III, andAbraham Szoke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2831 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2832 The Equations for Line Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2853 Numerical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2894 Numerical Results in the Gray Approximation . . . . . . . . . . . . . . . . . . 2955 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Non-LTE Radiation Transport in High Radiation PlasmasHoward A. Scott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3071 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3072 Non-LTE Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3093 Radiation Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3114 Test Case: Radiation-driven Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 3165 Linear Response Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3226 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Finite-Difference Methods Implemented in SATURNComplex to Solve Multidimensional Time-DependentTransport ProblemsR.M. Shagaliev, A.V. Alekseyev, A.V. Gichuk, A.A. Nuzhdin,N.P. Pleteneva, and L.P. Fedotova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3271 Multiple-Group Transport Equation Approximation . . . . . . . . . . . . . 331

Implicit Solution of Non-Equilibrium Radiation DiffusionIncluding Reactive Heating Source in Material EnergyEquationDana E. Shumaker, Carol S. Woodward . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3531 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3532 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

Contents IX

3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3554 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3595 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Part IV Mathematics and Computer Science

Transport Approximations in Partially Diffusive MediaGuillaume Bal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3731 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3732 Variational Formulation for Transport . . . . . . . . . . . . . . . . . . . . . . . . . 3753 Transport-Diffusion Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3894 Generalized Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393A Local Second-Order Equation and Linear Corrector . . . . . . . . . . . . . . 398References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

High Order Finite Volume Nonlinear Schemesfor the Boltzmann Transport EquationBarna L. Bihari, Peter N. Brown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4011 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4033 Discretization of the 3-D Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4054 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4105 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Obtaining Identical Results on Varying Numbersof Processors in Domain Decomposed ParticleMonte Carlo SimulationsN.A. Gentile, Malvin Kalos, Thomas A. Brunner . . . . . . . . . . . . . . . . . . . 4231 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4232 Ensuring the Invariance of the Pseudo-Random Number Stream

Employed by Each Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4263 Ensuring That Addition is Commutative . . . . . . . . . . . . . . . . . . . . . . . 4274 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4305 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

KM-Method of Iteration Convergence Accelerationfor Solving a 2D Time-Dependent Multiple-Group TransportEquation and its ModificationsA.V. Gichuk, L.P. Fedotova, R.M. Shagaliev . . . . . . . . . . . . . . . . . . . . . . . 4351 Statement of a 2D Transport Problem . . . . . . . . . . . . . . . . . . . . . . . . . 4352 KM-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4373 MKM-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

X Contents

4 KM3-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4395 Test Computation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

A Regularized Boltzmann Scattering Operatorfor Highly Forward Peaked ScatteringAnil K. Prinja, Brian C. Franke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4451 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4452 Generalized Fermi Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4463 Regularized Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4494 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Implicit Riemann Solvers for the Pn EquationsRyan McClarren, James Paul Holloway, Thomas Brunner,Thomas Mehlhorn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4571 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4572 Pn Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4583 Solving the Riemann Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4594 High Resolution Flux from Linear Reconstruction . . . . . . . . . . . . . . . 4615 Time Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4626 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4637 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4638 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

The Solution of the Time–Dependent SN Equationson Parallel ArchitecturesF. Douglas Swesty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4691 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4692 A Brief Review of The Implicit Discrete Ordinates Discretization

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4703 Iterative Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4724 Speeding Up and Obtaining Convergence . . . . . . . . . . . . . . . . . . . . . . . 4755 Parallel Implementation

of the Full Linear System Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 4816 Parallel Scalability of a 2-D Test Problem . . . . . . . . . . . . . . . . . . . . . . 4837 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4848 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Different Algorithms of 2D Transport Equation Parallelizationon Random Non-Orthogonal GridsShagaliev R.M., Alekseev A.V., Beliakov I.M., Gichuk A.V., NuzhdinA.A., Rezchikov V.Yu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

Contents XI

Part V Neutron Transport

Parallel Deterministic Neutron Transport with AMRC.J. Clouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4991 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4992 Code Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5003 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5084 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

An Overview of Neutron Transport Problemsand Simulation TechniquesEdward W. Larsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5131 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5132 Physical and Mathematical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5133 Basics of Stochastic and Deterministic Methods . . . . . . . . . . . . . . . . . 5214 Stochastic (Monte Carlo) Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5225 Deterministic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5276 Automatic Variance Reduction (Hybrid) Methods . . . . . . . . . . . . . . . 5307 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Introduction

There exist a wide range of applications where a significant fraction of the mo-mentum and energy present in a physical problem is carried by the transportof particles. Depending on the specific application, the particles involved maybe photons, neutrons, neutrinos, or charged particles. Regardless of whichphenomena is being described, at the heart of each application is the factthat a Boltzmann like transport equation has to be solved.

The complexity, and hence expense, involved in solving the transportproblem can be understood by realizing that the general solution to the 3DBoltzmann transport equation is in fact really seven dimensional: 3 spatialcoordinates, 2 angles, 1 time, and 1 for speed or energy. Low-order approxi-mations to the transport equation are frequently used due in part to physicaljustification but many in cases, simply because a solution to the full trans-port problem is too computationally expensive. An example is the diffusionequation, which effectively drops the two angles in phase space by assumingthat a linear representation in angle is adequate. Another approximation isthe grey approximation, which drops the energy variable by averaging overit. If the grey approximation is applied to the diffusion equation, the expenseof solving what amounts to the simplest possible description of transport isroughly equal to the cost of implicit computational fluid dynamics. It is cleartherefore, that for those application areas needing some form of transport,fast, accurate and robust transport algorithms can lead to an increase inoverall code performance and a decrease in time to solution.

Besides the multi-dimensional nature of the transport equation, becauseof the coupling of particle transport to other phenomena the transport equa-tion can in fact be non-linear. Hence, except for a few simple benchmark an-swers, the transport problem is solvable only via numerical methods. Thesenumerical methods have developed and grown over the years and with theadvent of massively parallel architectures, new scalable methods are beingsought. Unfortunately, it is still true that in most computer codes, transportis the largest consumer of computational resources. In application areas thatuse transport, the computational time is usually dominated by the trans-port calculation. Therefore, there is a potential for great synergy; progressin transport algorithms could help quicken the time to solution for manyapplications.

XIV Introduction

Consider for the moment, the details of particular applications wheretransport plays a role and it is clear the impact of solving the transportequation has on a variety of fields. In astrophysics, the life cycle of the stars,their formation, evolution, and death all require transport. In star formationand evolution for example, the problem is a multi-physics one involving MHD,self-gravity, chemistry, radiation transport, and a host of other phenomena.Supernova core collapse is an example where 3D, multi-group, multi-anglephoton and neutrino transport are important in order to model the explosionmechanism. The spectra and light curves generated from a supernova havegenerated a wealth of data. In order to make a connection between simulationdata and observational data and in order to remove systematic errors insupernova standard candle determinations of cosmological parameters, 3D,multi-group, multi-angle radiation transport is required.

The simulation of nuclear reactor science poses a similar set of challenges.In order to move beyond the current state-of-the-art for such calculations,several requirements must be met: (1) a description based on explicit hetero-geneous geometry instead of homogenized assemblies; (2) dozens of energygroups instead of two; (3) the use of 3D high-order transport instead of dif-fusion. These requirements would allow for accurate real-time simulations ofnew reactor operating characteristics, creating a virtual nuclear reactor testbed. Such a virtual reactor would enable assessments of the impact of newfuel cycles on issues like proliferation and waste repositories. With a 1000-times increase in computer power, accurate virtual reactors could reduce theneed to build expensive prototype reactors.

In the broad area of plasma physics, ICF (Inertial Confinement Fusion)and to a lesser extent MFE (Magnetic Fusion Energy) require the accuratemodeling of photon and charged particle transport. For ICF, whether oneis dealing with direct drive through photon or ion beams or dealing withindirect drive via thermal photons in a hohlraum, the accurate transportof energy around and into tiny capsules requires high-order transport solu-tions for photons and electrons. For direct drive experiments, simple radiationtreatments suffice (i.e. laser ray tracing with multi-group diffusion). Althoughthe radiation treatment can be rather crude, direct drive experiments requiresophisticated models of electron transport. In indirect drive such as at NIF,laser energy is converted into thermal x-rays via a hohlraum which in turn isused to drive some target. In order to accurately treat the radiation drive inthe hohlraum and its attendant asymmetries will require a radiation trans-port model with NLTE opacities for the hohlraum. The ability to generateNLTE is a tremendous computational challenge. Currently, calculating suchopacities in-line comes at a great cost. Typically, the difference between anLTE transport and NLTE transport calculation is a factor of 5. This facthas sparked research into alternatives such as tabulating steady state NLTEopacities or by simplifying the electron population rate equations so that

Introduction XV

their calculation is fast. However, all of the alternatives suffer from draw-backs which inhibit their widespread use.

In the coming years, simulations of NIF (National Ignition Facility) exper-iments will be crucial in attaining the goal of ignition. The simulations need tobe predictive rather than after-experiment fits; therefore, high-order trans-port coupled self-consistently to other nonlinear physics is a requirement.With a 1000-fold increase in computer power, these types of simulations arefeasible.

In planetary atmospheres, cloud variability and radiative transfer play akey role in understanding climate. For example coupling clouds to the radia-tive transfer problem and representing their distribution and size accuratelyis a difficult problem where such methods as sub-grid scale methods are be-ing applied. Tightly integrated to planetary atmospheres is the problem ofradiative transfer and plant canopies and the oceans. For example, the 3Dstructure of vegetation is a key player in processes effecting carbon seques-tration, landscape dynamics and the exchanges of energy, water and tracegases with the atmosphere.

These examples are just a small subset of the applications where an ac-curate and fast determination of particle transport is required.

Typically, the numerical methods used to solve the transport equation ina given discipline are communicated to other researchers in that discipline.Rarely are those methods communicated outside of that specific field. Forexample, nuclear engineers and astrophysicists rarely attend the same meet-ings. The seven-dimensional nature of transport means that factors of 100or 1000 improvement in computer speed or memory are quickly absorbed inslightly higher resolution in space, angle, and energy. Therefore, the biggestadvances in the last few years and in the next several years will be driven byalgorithms. Because transport is an implicit problem requiring iteration, thebiggest gains are to be made in finding faster techniques for acceleration toconvergence. Some of these acceleration methods are very application specificbecause they are physics based; others are very general because they addressthe mathematics of the transport equation.

In September of 2004, the Computational Methods in Transport Work-shop was held at Granlibakken, California with the hope that these issuescould be addressed by providing a forum where computational transport re-searchers in a variety of disciplines could communicate across disciplinaryboundaries their methods and their methods successes and failures. The goalof the workshop was to open channels of communication and cooperationbetween all members of the computational transport community so that (1)existing methods used in one field can be applied to other fields (2) greaterscientific resource can be brought to bear on the unsolved outstanding prob-lems.

The idea for the workshop was born at the SCaLeS (Scientific Case forLarge Scale Simulation) meeting held in Washington D.C. in June of 2003

XVI Introduction

and chaired by David Keyes. In attendance were a group of scientists, engi-neers, and computer scientists from a wide variety of disciplines whose charterwas to demonstrate, from an applications standpoint, the need for new ultra-scale computing facilities for Office of Science missions. A variety of breakoutsessions were formed. One of these sessions, chaired by myself and GordonOlson of Los Alamos National Laboratory, dealt with particle transport. Theparticle transport group consisted of experts from nuclear engineering, astro-physics, combustion, atmospheres, mathematics, etc. The discussions werelively and informative and it was soon realized what a wonderful oppor-tunity this was to discuss transport issues with persons outside of specificdisciplines. Conversations flowed between experts who would probably nevermeet in the normal course of doing research. The Computational Methodsin Transport Workshop was created out of my hope that a forum could existwhere researchers could discuss successes and failures of their methods acrossdiscipline boundaries in an invigorating and relaxing atmosphere that wouldbenefit the transport community at large.

Both executive and scientific organizing committees were formed withthe goal of putting together a conference designed to stimulate discussionand interaction among attendees. With this goal in mind, talks were selectedthat gave attendees both an overview of transport issues from a wide varietyof fields along with talks giving specialized detail. In addition, the workshoptalks were purposefully chosen to be few in number yet fairly long in length sothat speakers could communicate to the audience in a more effective fashion.The papers selected for the present volume have hopefully met both of thesegoals. Each author has put considerable work into writing a paper meant foran audience who although interested in transport, might not be an expert inplant canopies or supernova. My thanks to each and every author for writingpapers of such high quality.

The executive committee consisted of David Keyes (Columbia University),James McGraw (Lawrence Livermore National Laboratory), and Stanley Os-her (Institute of Pure and Applied Mathematics, UCLA). They providedinvaluable advice and guidance of workshop logistics and finances and I owethem a sincere note of gratitude. The scientific committee focused on thetechnical issues of the meeting and it was comprised of individuals drawnfrom a variety of disciplines who had established themselves in the field ofcomputational transport. My warmest thanks to Marv Adams (Texas A&MUniversity), John Castor (Lawrence Livermore National Laboratory), FrankEvans (University of Colorado), Ivan Hubeny (University of Arizona), TomManteuffel (University of Colorado), and Gordon Olson (Los Alamos NationalLaboratory) for a job well done.

A workshop like the Computational Methods in Transport Workshopwould never get off the ground without financial support. In particular, be-cause of the generous support of our sponsors, fourteen students were able toattend the meeting with all expenses paid. The workshop was jointly spon-

Introduction XVII

sored by Lawrence Livermore National Laboratory and the Institute of Pureand Applied Mathematics at UCLA. Mark Green (IPAM, UCLA), StanleyOsher (IPAM, UCLA), Jim McGraw (LLNL), and Rob Falgout (LLNL) de-serve singular thanks for their guidance and financial support. A special noteof thanks goes to the administrative and support people who have madethis workshop a reality. Leigh Faulk (LLNL), Anita Williams (LLNL), LindaBecker (LLNL), Janice Amar (UCLA), Karen Lee (UCLA), Linda Oribello(LLNL), Dave Parker (LLNL), Linda Null (LLNL) and Fred Allen (LLNL)performed nothing short of miracles.

Finally, a word of thanks to the editors and staff at Springer-Verlag. Inparticular, Martin Peters and Thanh-Ha LeThi were of invaluable help andI appreciate their support and patience through this project.

Lawrence Livermore National Laboratory Frank R. GrazianiJuly 2005

Radiation Hydrodynamics in Astrophysics

Chris L. Fryer1,2

1 Theoretical Astrophysics, T-6, Los Alamos National Laboratory, Los Alamos,NM 87545fryer@lanl.gov

2 Physics Department, University of Arizona, Tucson, AZ 85721

1 Defining Radiation Hydrodynamics Terms

Hydrodynamics codes are used to study nearly every astrophysical phenom-ena observed. Although coupling radiation and hydrodynamics is much lesscommon, radiation hydrodynamics codes are being used in a growing num-ber of astrophysics problems from energetic out flows of compact remnantssuch as core-collapse supernovae and active galactic nuclei to the formation ofstars and planets to cosmological simulations of the first stars. In this paper,we review the current “state-of-the-art” radiation hydrodynamics techniquesused in astrophysics.

This review will cover both the techniques used in astrophysics (Sect. 2)and the problems that have been studied (Sect. 3). Most of the techniquesare well-known in both the transport and hydrodynamics communities. Theonly surprises may be the rudemenatry level at which most working astro-physics codes do radiation transport. In Sect. 4, we will study in more detailthe method used to do radiation hydrodynamics with smooth particle hydro-dynamics, as it is less well-known and may prove a powerful tool to solve anumber of astrophysics problems. What astrophysicists mean when they saythey have a “radiation hydrodynamics” code varies from person to person.So before we begin this review, we must go through a few definitions.

Radiation Hydrodynamics: By “radiation hydrodynamics” we restrictourselves to those codes that, in some manner or another, solve both theBoltzmann equation and the hydrodynamics equations, coupling the resultsof both to determine the state of matter in a problem. By solving the Boltz-mann equation, we include any technique, no matter how crude. This includesmoment closure techniques as well as stochastic and direct discretization tech-niques. For most applications in astrophysics, this means flux-limited diffu-sion. By solving the hydrodynamics equations, in astrophysics this is almostentirely limited to solving the inviscid (Euler) equations. Similarly, I willloosely define “coupling these equations” to any scheme that combines thesolutions of both these equations. In practice, this is generally done usingoperator split methods with a single or no iterations to enforce convergence.

Working Astrophysics Code: By working astrophysics code, I meancodes that have been developed that have been used to solve an astrophysics

4 C.L. Fryer

problem and the results of these simulations have been published in the as-trophysics literature. We do not consider codes that have only appeared inpapers describing the technique without actually being run on an astrophysicsproblem.

Parallelized Code: By parallelized code, I mean those codes that actuallyrun on many processors, scaling reasonably well beyond 100 processors. Thisexcludes codes that are parallelized by solving a different energy group oneach processor (which for most astrophysics problems, limits the code’s useto 16-32 processors). We also exclude codes that can only be used on sharedmemory machines.

Moment Closure vs. Direct Discretization vs. Stochastic Methods:The transport schemes in astrophysics can be loosely classified into threedifferent categories:

1: Moment closure methods where the Boltzmann equation is represented asa series of angular moments and these moments are closed to form a solu-tion. Both Flux-limited diffusion and variable Eddington factor techniquesfit into this class.

2: Direct discretization of the Boltzmann equation. Probably the most com-mon technique in this category used in astrophysics is the Sn method.This is often called “Boltzmann transport”.

3: Stochastic methods or Monte Carlo Methods. In astrophysics, the use ofMonte Carlo techniques has, in the past, been limited to time-independentcalculations (e.g. post-process).

2 Schemes Used in Astrophysics

Although these definitions don’t seem restrictive, they exclude a lot of thetechniques used in astrophysics: hydrodynamics schemes that put in radiationas a cooling term or use a “leakage” scheme to control the radiation transportthrough a dense medium (there are a number of problems, for example, incosmology, where such schemes are sufficient to solve the current questionsbeing addressed); and transport schemes that add in material motion, butdo not actually solve the hydrodynamics equations (some very sophisticatedtransport techniques have been used to solve supernova spectra and stellaratmospheres where hydrodynamics plays a secondary role, and hence has beenneglected in past studies). Transport Schemes coupled with hydrodynamicsschemes include:

1: Pure diffusion schemes. These schemes are generally solved as 1-T calcula-tions (one temperature describes both the radiation and matter). Althoughtechnically a simplified solution to the Boltzmann equation, we will notconsider codes using these schemes further.

Radiation Hydrodynamics in Astrophysics 5

2: Flux-limited diffusion schemes. These schemes are among the simplest ofthe moment closure techniques. A number of recipes (a.k.a. flux-limiters)are used in astrophysics to connect the diffusion and free-streaming lim-its of the transport equation. These schemes are generally implementedusing 2-T (one temperature describing the radiation field and one temper-ature describing the matter) and have been solved both using single andmultigroup energy discretization. This technique dominates what is usedin astrophysics, especially in multi-dimension.

3: Variable Eddington factor schemes. This technique also uses moments ofthe Boltzmann equation, but closes at one higher moment using Edding-ton factors. The technique used to solve for the Eddington factors differsfrom code to code. Although initially used in pure transport problems, thisscheme has been used in both 1-dimensional and 1.5-dimensional (trans-port along rays) radiation hydrodynamics schemes.

4: Sn transport. In general, most working codes using Sntechniques havebeen limited to 1-dimensional hydrodynamics codes, but at least 1 paperhas been published showing results of a 2-dimensional simulation.

For many astrophysics phenomena, pure hydrodynamics codes can pro-vide a first order answer to a problem. Codes have been developed first mod-eling hydrodynamics only with transport schemes being added into thesecodes at a later time. Although this trend is slowly reversing (hydrodynam-ics schemes are now being coupled to transport codes) because the transportscheme dominates the computing time, at this time, most of the working as-trophysics codes arise from hydrodynamics schemes with transport schemesadded on top. The current range of hydrodynamics schemes coupled withtransport techniques include most of the hydrodynamics techniques used inastrophysics:

1: Lagrangian techniques. In astrophysics, Lagrangian grid codes are usedin 1-dimension, but most multi-dimensional Lagrangian codes use smoothparticle hydrodynamics. A number of transport schemes have been cou-pled to 1-dimensional Lagrangian codes. Although coupling transport withsmooth particle hydrodynamics takes some thought, it can, and has beendone in (at least at the level of flux-limited diffusion).

2: Eulerian techniques. These techniques include both fixed grid and adap-tive grid refinement methods.A number of transport schemes have beencoupled to both 1-dimensional and 2-dimensional Eulerian codes. Flux-limited diffusion has been coupled to 3-dimensional Eulerian codes.

Astrophysicists have applied a number of these schemes in 1-dimension.In 2- and 3-dimensions, most of the work is limited to a range of hydro-dynamics codes coupled to flux-limited diffusion. Below, we discuss specificcodes applied to two of the current astrophysics problems best-studied usingradiation-hydrodynamics codes.

6 C.L. Fryer

3 Astrophysical Applications

Computational astrophysicists work using small steps, gradually increasingthe level of sophistication on a given problem. Early models simulate onlythe most important physics to compare to broad features in the observations.But as the observations become more detailed, so must the codes. For exam-ple, early cosmological structure models modeled only the effects of gravity,focusing on the motion of the dark matter. But as observations of the bary-onic matter became more detailed, hydrodynamical effects were added to thecomputer models. Now, to study the formation of the first stars, modelersmust include the effects of radiation transport. Although most cosmologicalstudies today can make progress with simplistic radiation routines like leak-age schemes, there will come a time when the observational detail requiresradiation hydrodynamics techniques.

A number of astrophysical problems have already reached this coding re-quirement. Stellar atmosphere studies, which in the past have done someof the most sophisticated transport schemes in astrophysics using staticatmospheres are now studying the effects of convection by coupling a 3-dimensional hydrodynamics code with flux-limited diffusion [Stein & Nord-lund 2003]. Klein and collaborators have studied a number of phenomenausing radiation (usually flux-limited diffusion) hydrodynamic codes from neu-tron star accretion [Murray et al. (1995), Klein et al. (1996)] to molecularclouds [Sandford et al. (1982)]. However, we will focus our attention on thetwo fields that appear to be most actively pursuing radiation hydrodynamicsin astrophysics: accretion disks and core-collapse supernovae.

3.1 Accretion Disks

Many astrophysical phenomena are powered by the gravitational energy re-leased when matter accretes onto a compact object: ranging from active galac-tic nuclei to X-ray bursts to gamma-ray bursts to planet and star formation.Even if the initial velocity asymmetries are small, as this matter falls downonto the compact object, centrifugal forces become increasingly importantand the behavior of these phenomena is dominated by the disk of materialthat forms prior to the accretion phase. The wide variety of applications hasled to a number of studies of disk accretion with increasing levels of sophis-tication.

Portions of most of these disks are indeed optically thick (mean free pathmuch less than disk scale height) and modeling the radiation transport cor-rectly is important to many accretion disk problems. Although transport tech-niques ranging from leakage schemes through Monte-Carlo have been usedto post-process disk calculations, the current state-of-the-art for production,radiation-hydrodynamics calculations in disks is limited to flux limited dif-fusion [Kley 1989, Turner & Stone (2001), Turner et al. (2003)]. These codes

Radiation Hydrodynamics in Astrophysics 7

have mostly been run on shared memory machines with few nodes so it isunclear how well they scale to many processors with limited bandwidth.

3.2 Core-Collapse Supernovae

In the past, the supernovae “transport experts” have worked on supernovalight-curves. However, the transport schemes used in light-curve calculationsare almost entirely limited to solutions of a steady state or uniformly varyingmedium. Only recently have radiation hydrodynamics codes (generally 1-dimensional flux-limited diffusion hydrodynamics codes) been applied to thisproblem [Hoflich et al. (1993), Blinnikov et al. (2000)]. The results from thesecalculations are then post-processed to get detailed spectra.

The supernova “radiation-hydrodynamics experts” are actually the mod-ellers of stellar collapse (here the neutrino is the transport particle). Core-collapse supernova theorists came to radiation hydrodynamics by first fo-cusing on hydrodynamics and including increasingly sophisticated transporttechniques. These simulations have been at the cutting edge of computationalastrophysics due to a boost from codes built at national laboratories in thesixties [Colgate & Johnson (1960), Colgate & White (1966)]. Core collapsesupernovae are powered by the potential energy released during the collapseof the iron core of a massive star down to a ∼50 km proto-neutron star [Fryer(2003)]. This energy leaks out of the proto-neutron star in the form of neutri-nos. The neutrino mean free path ranges from a few cm in the proto-neutronstar to >100 km in the convective region where neutrino energy is deposited.Figure 1 shows a slice from a 3-dimensional simulation of a core-collapse su-pernovae. The three circles show the position where the optical depth outof the star is roughly 0.05 for the µ/τ , anti-electron and electron neutrinos(the innermost circle denotes µ/τ ; the outermost circle denotes the electronneutrinos).

A number of techniques have been used to study radiation hydrodynamicsin core-collapse supernovae. Tony Mezzacappa has developed some very nicediagrams that differentiate these codes. Using a simplified version of such adiagram dividing codes by spatial hydrodynamics dimension and tranpsorttechnique (Fig. 2), we describe some of the key codes developed and theirrelevant citation (recall that we limit our discussion to codes that have beenapplied to real problems):

1: In one dimension, a range of transport techniques have been added toboth Lagrangian and Eulerian codes. Multi-group, flux-limited diffusionhas long been used in stellar collapse [Bruenn et al. (1978)].

2: Very sophisticated have been used in 1-dimensional codes. Discrete tech-niques (Sn method) have been around for over a decade and are said to beparallel although I have not seen scaling numbers [Mezzacappa & Bruenn(1993)]. The variable Eddington factor technique has also been used in

Radiation Hydrodynamics in Astrophysics 9

6Gray Flux−

Limited

Diffusion

Fully Parallel

I II III

Variable

Discrete

Eddington

Limited Diffusion

Gray Flux−

1.5 Multi−Group

Flux−Limited

Diffusion Along

Sin

gle

En

ergy

Mu

lti

En

ergy

Mu

lti

An

gle

Mu

lti

En

ergy

Neutrino/Spatial Dimension

Multi−Group,

Flux−Limited

Diffusion

1.5 Variable Eddington Factor

Discrete

Rays

Factor

Methods

Along Rays

Methods

1

2 5

4

3

Fig. 2. Simplified Mezzacappa diagram of the radiation-hydrodynamics modelsused in core-collapse studies. The y axis shows the transport technique (single-energy flux-limited diffusion, multi-group flux-limited diffusion, and more sophisti-cated techniques that include angular effects). The x axis denotes level of spatial(and purportedly neutrino dimension). Note that some 2-dimensional techniquesonly follow transport along rays. The numbers correspond to more detailed de-scriptions in the text

5: Some of the most intense research has been focused on sophisticated tech-niques intwo-dimensions. A variable Eddington technique has been devel-oped [Scheck et al. (2004)] that follows transport along rays (no lateraltransport). This technique is not extremely parallelizable (different nodesare given different energy groups to model). A discrete method has alsobeen developed [Livne et al. (2004)]. Similarly, it has only been parallelizedby putting different energy groups on different processors. Also, the onepaper using this technique did not appear to run the simulation for manytimesteps, and I suspect this technique will not be usable to do the fullcollapse problem (requiring 100,000 timesteps).

6: Three-dimensional, single-energy flux limited diffusion modelling the col-lapse through explosion was done using SNSPH [Fryer & Warren (2002)].This code is fully parallel and has scaled nearly linearly up to 512 proces-sors.

This list of techniques far outstrips any other astrophysics field in radiation-hydrodynamics codes. But the limitations are already evident. First note thatthere are few working codes able to model transport in more than 1 spatial

10 C.L. Fryer

dimension. Those that have passed our relaxed criterion for a “working code”are either not parallelizable, not truly multi-dimensional transport schemes orboth. Indeed, the only true multi-dimensional radiation hydrodynamics codeis limited to single-energy flux-limited transport. Astrophysicists have de-scribed much more sophisticated techniques (even at this meeting), but theyhave not put them into working codes, let alone parallelizable codes. Thereis a lot astrophysicists can gain from the transport community in makingworking codes.

4 SPH Radiation Transport

One of the few working multi-dimensional radiation hydrodynamics codes incore-collaspe is a technique coupling smooth particle hydrodynamics (SPH)to flux-limited diffusion. The coupling, developed in 2-dimensions in 1994[Herant et al. (1994)] and parallelized in 3-dimensions in 2002 [Fryer & War-ren (2002)] has some twists to it and it is worth describing it in some detail.Much of this description is taken from a paper submitted to the AstrophysicsJournal by Fryer, Rockefeller, and Warren (2005).

4.1 Transport Scheme

In core-collapse supernovae, we model the transport of 3 neutrino species(l = νe, νe, νx where νx corresponds to the τ, µ neutrinos and their anti-particles that are all treated equally). Because neutrino number is the con-served quantity, we transport neutrino number and then determine the en-ergy transport by using the mean neutrino energies. The radiation transportscheme in our SPH code is modeled after the technique to calculate forcesin SPH: we calculate symmetric interactions between all neighbor particles.Hence, our flux-limited diffusion scheme calculates the radiation diffusion inor out of a particle by summing the transport over all neighbors (the equiv-alent of all bordering cells in a grid calculation). The neutrino transport forparticle i is given by:

dniνl/dt =

∑

j

Λijνl

(

niνlb

i→jνl − nj

νlbj→iνl

)

∇W ijmj/ρj (1)

and the corresponding energy transport is:

deiνl/dt =∑

j

Λijνl

(

ǫiνlniνlb

i→jνl − ξj→iǫjνln

jνlb

j→iνl

)

∇W ijmj/ρj (2)

where niνl, e

iνl are, respectively, the neutrino density and energy density in

particle i for species νl, ǫiνl is the mean neutrino energy, and ξj→i is the

redshift correction for ǫjνl as seen by particle i.

Radiation Hydrodynamics in Astrophysics 11

Λijνl is the limiter for the flux-limited transport scheme. The simplest such

scheme for 3-dimensions is:

Λijνl = min(c,Dij

νl/rij) (3)

where c is the speed of light, Dijνl = 2Di

νlDjνl/(D

iνl + Dj

νl) is the harmonicmean of the diffusion coefficients for the species νl of particles i and j and rij

is the distance between particles i and j. This limiter was used by [Herantet al. (1994)] and, for comparison with that work, by [Fryer & Warren (2002)],but we have used a number of other flux-limiters, all of which are valid underthis transport scheme [Fryer et al. (1999)].

Beyond some radius in a core-collapse simulation, neutrinos are essentiallyin the free-streaming regime where transport is not necessary (unless onewants to truly follow the radiation wave as it progresses through the star). Wedo not model transport beyond this “trapping” radius. Instead we sum up allneutrinos that transport beyond this radius and emit them using a lightbulbapproximation. That is, the material beyond this radius sees a constant fluxand we can determine the amount of energy a particle gains (dEi/dt) fromneutrino interactions simply by using the free-streaming limit:

dEi/dt = Lν (1.0 − e−∆τi) (4)

where Lν is the neutrino luminosity and ∆τi is the optical depth of a parti-cle i. This assumption is only valid if the total amount of energy imparted(∑

i dEi/dt) from the neutrinos onto the matter is much less than the to-tal neutrino flux (Lν). To guarantee this, we determine this trapping radiusby evolving it with time such that (

∑

i dEi/dt)/Lν is always less than somevalue. This value was originally set to 0.1 by Herant et al. (1994), but inrecent calculations, we use 0.05.

Such a scheme can be easily converted into multi-group, but such modifi-cations have not yet been done. The scheme scales reasonably well on multipleprocessors (for a 5 million particle run, the code has scaled nearly linearlyup to 256 processors on the Space Simulator Beowulf cluster and up to 512processors on the ASC Q machine at Los Alamos National Laboratory). Inpart, this scalability is due to the explicit nature of the transport scheme. Ingeneral, explicit transport schemes strongly limits timesteps as the speed oflight, not sound, constrains the duration of the timestep. In core-collapse su-pernovae, this constraint is not too onerous because the sound speed is nearlya third the speed of light anyway, so the explicit transport scheme leads toonly a factor of 3 decrease in the timestep. But this explicit flux-limitedtransport scheme can be used in a much wider variety of problems where themean free path is very short for the smallest particles. Such scenarios occurin many astrophysics problems.

12 C.L. Fryer

4.2 First Test

Testing radiation transport schemes in general would, and has, comprisedmany papers in itself. Here we focus on a simple test comparing the SPH flux-limited scheme to a 1-dimensional grid-based flux-limited transport scheme.This test does not prove the applicability of flux-limited diffusion to thesupernova problem, but it does show that our scheme for putting flux-limiteddiffusion into SPH does work.

For the initial conditions of this test, we use a spherically-symmetric neu-tron star atmosphere (material below 1014 g cm−3) for a neutron star roughly130 ms after bounce. We map this structure onto our SPH particles, using theshell setup described in Fryer & Warren (2002). We make a similar setup in a1-dimensional grid using one zone per shell of SPH particles. With this setup,we minimize the differences between the density and temperature structureof our 1- and 3-dimensional models. We determine the trapping radius to be25 km. Below this radius, we set the electron neutrino fraction (Yνe) to 0.15with a mean energy of 10 MeV. Above this radius, Yνe is set to zero.

Our test focuses on the neutrino transport alone; we evolve only the elec-tron neutrino fraction with time and hold the density, temperature, and elec-tron fraction fixed. We allow no new neutrino emission. The flux arising from

Fig. 3. Neutrino luminosity versus time for a cooling neutron star using a simple 1-dimensional flux-limited diffusion scheme and the technique discussed here couplingflux-limited diffusion to smooth particle hydrodynamics. The initial discrepancyarises from the fact that diffusion occurs over all SPH neighbors, allowing transportacross the effective (in a 1-dimensional case) of several radial zones (instead of 1zone in the 1-dimensional transport scheme)

Radiation Hydrodynamics in Astrophysics 13

our neutrino trapping radius versus time for both our 1-dimensional gridsimulation and our 3-dimensional SPH calculation is plotted in Fig. 3. Theinitial flux of the SPH calculation is higher, since the neighbors of any par-ticle extend beyond the equivalent of an adjacent cell for the 1-dimensionalcalculation. In general, the luminosity for both these calculations agree tobetter than 3%.

References

[Blinnikov et al. (2000)] Blinnikov, S., Lundqvist, P., & Bartunov, O., Nomoto, K.,Iwamoto, K. Radiation Hydrodynamics of SN 1987A. I. Global Analysis of theLight Curve for the First 4 Months ApJ, 532, 1132 (2000)

[Bruenn et al. (1978)] Bruenn, S.W., Buchler, J. R., & Yueh,W. R. Neutrino Trans-port in Supernova Models – A Multigroup, Flux-Limited diffusion Scheme Ap&SS, 59, 261–284 (1978).

[Burrows et al. (1995)] Burrows, A., Hayes, J., & Fryxell, B. A. On the Nature ofCore-Collapse Supernova Explosions, ApJ, 450, 830–850 (1995).

[Colgate & Johnson (1960)] Colgate, S.A., Johnson, H.J., Phys Rev. Letters, 5,573–576 (1960).

[Colgate & White (1966)] Colgate, S.A., White, R.H. ApJ, 143, 626–681 (1966)[Fryer et al. (1999)] Fryer, C. L., Benz, W., Herant, M., Colgate, S.A. What Can

the Accretion-induced Collapse of White Dwarfs Really Explain? ApJ, 516,892 (1999)

[Fryer & Warren (2002)] Fryer, C. L., & Warren, M. S. Modeling Core-CollapseSupernovae in Three Dimensions ApJ, 574, L65 (2002)

[Fryer (2003)] Fryer, C. L. Stellar Collapse, International Journal of ModernPhysics D, 12, 1795–1835

[Fryer & Warren (2004)] Fryer, C. L., & Warren, M. S. The Collapse of RotatingMassive Stars in Three Dimensions, ApJ, 601, 391–404

[Herant et al. (1994)] Herant, M., Benz, W., Hix, W. R., Fryer, C. L., & Colgate, S.A. Inside the Supernova: A Powerful Convective Engine ApJ, 435, 339 (1994)

[Hoflich et al. (1993)] Hoflich, Muller, E., & Khokhlov, A. Light Curve Models forType Ia Supernovae - Physical Assumptions, their Influence and Validity A&A,268, 570–590 (1993)

[Klein et al. (1996)] Klein, R.I., Arons, J., Garrett, J., Hsu, J.J.L.: Photon bubbleOscillations in Accretion-powered Pulsars. ApJ, 457, L85–89 (1996).

[Kley 1989] Kley, W. Radiation Hydrodynamics of the Boundary Layer in Accre-tion Disks. I- Numberical models A&A, 208, 98–110 (1989).

[Livne et al. (2004)] Livne, E., Burrows, A., Walder, R., Lichtenstadt, I., & Thomp-son, T. A. Two-Dimension, Time-Dependent, Multigroup, Multiangle Radia-tion Hydrodynamics in the Core-Collapse Supernova Context, ApJ, 609, 277–287

[Mezzacappa & Bruenn (1993)] Mezzacappa, A., & Bruenn, S. W. A NumericalMethod for Solving the Neutrino Boltzmann Equation Coupled to SphericallySymmetric Stellar Core Collapse ApJ, 405, 669–684 (1993).

[Murray et al. (1995)] Murray, S.D., Woods, D.T., Castor, J.I., Klein, R.I., McKee,C.F.: Radiation Hydrodynamic Models of Eclipsing Low-Mass X-ray BinariesApJ, 454, L133–136 (1995).

14 C.L. Fryer

[Sandford et al. (1982)] Sandford, M.T., Whitaker, R.W., Klein, R.I.: Radiation-Driven Implosions in Molecular Clouds ApJ 260, 183–201 (1982).

[Scheck et al. (2004)] Scheck, L., Plewa, T., Janka, H.-Th., Kifonidis, K., & Muller,E. Pulsar Recoil by Large-Scale Anisotropies in Supernova Explosions PRL, 92,0111031–011034

[Stein & Nordlund 2003] Stein, R.F., Nordlund, A.: Stellar Atmosphere Modeling,ASP Conference Proceedings, Vol 288, Aug. 8–12, 2002, Hubeny, Mihalas,Werner (eds.). San Francisco: Astronomical Society of the Pacific (2003) p.519

[Thompson et al. (2003)] Thompson, T. A., Burrows, A., Pinto, P. A. Shock Break-out in Core-Collapse Supernovae and Its Neutrino Signature, ApJ, 592, 434–456 (2003).

[Turner & Stone (2001)] Turner, N. J., & Stone, J. M. 2001, A Module for Radia-tion Hydrodynamic Calculations with ZEUS-2D Using Flux-Limited Diffusion,ApJS, 135, 95–107

[Turner et al. (2003)] Turner, N. J., Stone, J. M., Krolik, J. H. & Sano, T. LocalThree-dimensional Simulations of Magnetorotational Instability in Radiation-Dominated Accretion Disks ApJ, 593, 992 (2003).

Radiative Transferin Astrophysical Applications

I. Hubeny

Department of Astronomy, University of Arizona, Tucson, AZ 85721, USA

1 Introduction

Radiative transfer is particularly important in astrophysics. One reason isquite understandable: radiation is in most cases the only information we have(and will ever have) about distant objects (exceptions are detected neutrinosfrom the Sun and supernova SN1987a, and in a near future the gravitationalwaves). However, there is an even more compelling reason for the a need todeal with detailed radiation transport in astrophysics: In many astronomicalobjects the radiation is so strong that it significantly contributes to the energyand momentum budget of the medium. Therefore, radiation is not only aprobe of the physical state, but is in fact an important constituent. In otherwords, radiation in fact determines the structure of the medium, yet themedium is probed only by this radiation.

Unlike laboratory physics, where one can change a setup of the experimentin order to examine various aspects of the studied structures separately, we donot have this luxury in astrophysics: we are stuck with the observed spectrumso we should better make a very good use of it.

All that puts heavy demands on proper radiation transfer techniques inastrophysics. Also, there are many different situations and physical conditions(e.g., temperature ranges from a few K to more than 109 K; there is a hugerange of microscopic processes that give rise to an absorption or emissionof a photon), so consequently there is a wide range of numerical techniquesused for treating radiation transport in astrophysics. This paper is not aimedto provide a review of all or even the most important numerical techniques;instead it aims at outlining the outstanding problems and ideas behind recentnumerical approaches. I also stress that although I talk about transport ofphotons, many methods can be used for treating transport of neutrinos, whichis important for instance in numerical simulations of core-collapse supernovae.

Typical objects for which the radiation transport is important are aboveall stellar atmospheres. In this field, most numerical techniques were devel-oped that are now being used in other parts of astrophysics. Other objects areextrasolar planetary atmospheres (the solar system planets as well, but herewe have a lot of direct information thanks to various planetary missions).Other important radiation-dominated objects are accretion disks of variouskinds—around supermassive black holes (quasars and active galactic nuclei);

16 I. Hubeny

neutron stars (X-ray binaries); white dwarfs (cataclysmic variables), and evenmain-sequence close binaries. Transport of photons (and neutrinos at earlystages!) is important in supernovae, and radiative transfer is important incurcumstellar structures like planetary nebulae and H II regions. Finally,radiation transport is also important in global cosmological simulations oflarge-scale structure of the Universe.

The basic textbook of astrophysical radiative transfer is [1] but it doesnot cover modern and efficient numerical transport techniques. Another text-books, with emphasis more on hydrodynamical aspect of the problems, are [2],and recent textbook [3]; both contain excellent discussions of radiation trans-port techniques. Recent proceedings that contains a number of review papers,and to which the reader is referred to for more information about modernmethods used in astrophysical radiative transfer, is [4].

2 Description of Radiation

In astrophysical formalism, the basic quantity describing the radiation fieldis the specific intensity of radiation, I(r, t,n, ν), defined such that it is theenergy transported by radiation at position r, in a unit frequency range at thefrequency ν, across a unit area perpendicular to the direction of propagation,n, into a unit solid angle, and in a unit time interval. The specific intensityprovides a complete description of the unpolarized radiation field from themacroscopic point of view. The specific intensity is related to the photondistribution function, f , as

I = hνc f (1)

where h and c are the Planck constant and the speed of light, respectively.We note that the same quantity is called angular flux, and is usually denotedas ψ(r, t,n, ν), in the neutron transport theory.

The transport (or transfer) equation is essentially the Boltzmann equationfor photons, which can be generally written as

∂f

∂t+ (v · ∇)f + (F · ∇P )f =

(

Df

Dt

)

coll

, (2)

where v is the velocity, F is the external force, ∇P is the nabla operator inthe momentum space, and the term (Df/Dt)coll is the collision term. In caseof photons, v = cn, and F = 0 (in the absence of general relativistic effects).The transfer equation can then be written as

(

1

c

∂

∂t+ n · ∇

)

I(ν, r,n, t) = η(ν, r,n, t) − χ(ν, r,n, t) I(ν, r,n, t) . (3)

where we write the collisional term in terms of the usual absorption coefficientχ and emission coefficient η. The equation looks simple, but this is deceiving.