Graziani F. (editor) Computational Methods in Transport

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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 transport

of particles. Depending on the speciﬁc application, 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 approxi-

mations to the transport equation are frequently used due in part to physical

justiﬁcation but many in cases, simply because a solution to the full trans-

port problem is too computationally expensive. An example is the diﬀusion

equation, which eﬀectively 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 diﬀusion equation, the expense

of solving what amounts to the simplest possible description of transport is

roughly equal to the cost of implicit computational ﬂuid 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.

Besides the multi-dimensional nature of the transport equation, because

of 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. These

numerical methods have developed and grown over the years and with the

advent of massively parallel architectures, new scalable methods are being

sought. Unfortunately, it is still true that in most computer codes, transport

is the largest consumer of computational resources. In application areas that

use transport, the computational time is usually dominated by the trans-

port calculation. Therefore, there is a potential for great synergy; progress

in transport algorithms could help quicken the time to solution for many

applications.

XIV Introduction

Consider for the moment, the details of particular applications where

transport plays a role and it is clear the impact of solving the transport

equation has on a variety of ﬁelds. In astrophysics, the life cycle of the stars,

their formation, evolution, and death all require transport. In star formation

and 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-angle

photon and neutrino transport are important in order to model the explosion

mechanism. The spectra and light curves generated from a supernova have

generated a wealth of data. In order to make a connection between simulation

data and observational data and in order to remove systematic errors in

supernova 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 energy

groups instead of two; (3) the use of 3D high-order transport instead of dif-

fusion. These requirements would allow for accurate real-time simulations of

new reactor operating characteristics, creating a virtual nuclear reactor test

bed. Such a virtual reactor would enable assessments of the impact of new

fuel cycles on issues like proliferation and waste repositories. With a 1000-

times increase in computer power, accurate virtual reactors could reduce the

need 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 accurate

modeling of photon and charged particle transport. For ICF, whether one

is dealing with direct drive through photon or ion beams or dealing with

indirect drive via thermal photons in a hohlraum, the accurate transport

of energy around and into tiny capsules requires high-order transport solu-

tions for photons and electrons. For direct drive experiments, simple radiation

treatments suﬃce (i.e. laser ray tracing with multi-group diﬀusion). Although

the radiation treatment can be rather crude, direct drive experiments require

sophisticated 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 is

used to drive some target. In order to accurately treat the radiation drive in

the hohlraum and its attendant asymmetries will require a radiation trans-

port model with NLTE opacities for the hohlraum. The ability to generate

NLTE is a tremendous computational challenge. Currently, calculating such

opacities in-line comes at a great cost. Typically, the diﬀerence between an

LTE transport and NLTE transport calculation is a factor of 5. This fact

has sparked research into alternatives such as tabulating steady state NLTE

opacities 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 to

be 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 are

feasible.

In planetary atmospheres, cloud variability and radiative transfer play a

key role in understanding climate. For example coupling clouds to the radia-

tive transfer problem and representing their distribution and size accurately

is a diﬃcult problem where such methods as sub-grid scale methods are be-

ing applied. Tightly integrated to planetary atmospheres is the problem of

radiative transfer and plant canopies and the oceans. For example, the 3D

structure of vegetation is a key player in processes eﬀecting carbon seques-

tration, landscape dynamics and the exchanges of energy, water and trace

gases 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 in

a given discipline are communicated to other researchers in that discipline.

Rarely are those methods communicated outside of that speciﬁc ﬁeld. For

example, nuclear engineers and astrophysicists rarely attend the same meet-

ings. The seven-dimensional nature of transport means that factors of 100

or 1000 improvement in computer speed or memory are quickly absorbed in

slightly higher resolution in space, angle, and energy. Therefore, the biggest

advances in the last few years and in the next several years will be driven by

algorithms. Because transport is an implicit problem requiring iteration, the

biggest gains are to be made in ﬁnding faster techniques for acceleration to

convergence. Some of these acceleration methods are very application speciﬁc

because they are physics based; others are very general because they address

the 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 issues

could be addressed by providing a forum where computational transport re-

searchers in a variety of disciplines could communicate across disciplinary

boundaries their methods and their methods successes and failures. The goal

of the workshop was to open channels of communication and cooperation

between all members of the computational transport community so that (1)

existing methods used in one ﬁeld can be applied to other ﬁelds (2) greater

scientiﬁc resource can be brought to bear on the unsolved outstanding prob-

lems.

The idea for the workshop was born at the SCaLeS (Scientiﬁc Case for

Large 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 charter

was to demonstrate, from an applications standpoint, the need for new ultra-

scale computing facilities for Oﬃce of Science missions. A variety of breakout

sessions were formed. One of these sessions, chaired by myself and Gordon

Olson of Los Alamos National Laboratory, dealt with particle transport. The

particle transport group consisted of experts from nuclear engineering, astro-

physics, combustion, atmospheres, mathematics, etc. The discussions were

lively and informative and it was soon realized what a wonderful oppor-

tunity this was to discuss transport issues with persons outside of speciﬁc

disciplines. Conversations ﬂowed between experts who would probably never

meet in the normal course of doing research. The Computational Methods

in Transport Workshop was created out of my hope that a forum could exist

where researchers could discuss successes and failures of their methods across

discipline boundaries in an invigorating and relaxing atmosphere that would

beneﬁt the transport community at large.

Both executive and scientiﬁc organizing committees were formed with

the goal of putting together a conference designed to stimulate discussion

and interaction among attendees. With this goal in mind, talks were selected

that gave attendees both an overview of transport issues from a wide variety

of ﬁelds along with talks giving specialized detail. In addition, the workshop

talks were purposefully chosen to be few in number yet fairly long in length so

that speakers could communicate to the audience in a more eﬀective fashion.

The papers selected for the present volume have hopefully met both of these

goals. Each author has put considerable work into writing a paper meant for

an audience who although interested in transport, might not be an expert in

plant canopies or supernova. My thanks to each and every author for writing

papers 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 provided

invaluable advice and guidance of workshop logistics and ﬁnances and I owe

them a sincere note of gratitude. The scientiﬁc committee focused on the

technical issues of the meeting and it was comprised of individuals drawn

from a variety of disciplines who had established themselves in the ﬁeld of

computational transport. My warmest thanks to Marv Adams (Texas A&M

University), John Castor (Lawrence Livermore National Laboratory), Frank

Evans (University of Colorado), Ivan Hubeny (University of Arizona), Tom

Manteuﬀel (University of Colorado), and Gordon Olson (Los Alamos National

Laboratory) for a job well done.

A workshop like the Computational Methods in Transport Workshop

would never get oﬀ the ground without ﬁnancial support. In particular, be-

cause of the generous support of our sponsors, fourteen students were able to

attend the meeting with all expenses paid. The workshop was jointly spon-

Introduction XVII

sored by Lawrence Livermore National Laboratory and the Institute of Pure

and Applied Mathematics at UCLA. Mark Green (IPAM, UCLA), Stanley

Osher (IPAM, UCLA), Jim McGraw (LLNL), and Rob Falgout (LLNL) de-

serve singular thanks for their guidance and ﬁnancial support. A special note

of thanks goes to the administrative and support people who have made

this workshop a reality. Leigh Faulk (LLNL), Anita Williams (LLNL), Linda

Becker (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. In

particular, Martin Peters and Thanh-Ha LeThi were of invaluable help and

I appreciate their support and patience through this project.

Lawrence Livermore National Laboratory Frank R. Graziani

July 2005

Part I

Astrophysics

Radiation Hydrodynamics in Astrophysics

Chris L. Fryer

1,2

Theoretical Astrophysics, T-6, Los Alamos National Laboratory, Los Alamos,

NM 87545

fryer@lanl.gov

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 less

common, radiation hydrodynamics codes are being used in a growing num-

ber of astrophysics problems from energetic out ﬂows of compact remnants

such as core-collapse supernovae and active galactic nuclei to the formation of

stars and planets to cosmological simulations of the ﬁrst stars. In this paper,

we review the current “state-of-the-art” radiation hydrodynamics techniques

used 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 techniques

are well-known in both the transport and hydrodynamics communities. The

only surprises may be the rudemenatry level at which most working astro-

physics codes do radiation transport. In Sect. 4, we will study in more detail

the 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 a

number of astrophysics problems. What astrophysicists mean when they say

they 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 restrict

ourselves to those codes that, in some manner or another, solve both the

Boltzmann equation and the hydrodynamics equations, coupling the results

of 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 includes

moment 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 almost

entirely limited to solving the inviscid (Euler) equations. Similarly, I will

loosely deﬁne “coupling these equations” to any scheme that combines the

solutions of both these equations. In practice, this is generally done using

operator split methods with a single or no iterations to enforce convergence.

Working Astrophysics Code: By working astrophysics code, I mean

codes that have been developed that have been used to solve an astrophysics

4C.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 in

papers describing the technique without actually being run on an astrophysics

problem.

Parallelized Code: By parallelized code, I mean those codes that actually

run on many processors, scaling reasonably well beyond 100 processors. This

excludes codes that are parallelized by solving a diﬀerent energy group on

each processor (which for most astrophysics problems, limits the code’s use

to 16-32 processors). We also exclude codes that can only be used on shared

memory machines.

Moment Closure vs. Direct Discretization vs. Stochastic Methods:

The transport schemes in astrophysics can be loosely classiﬁed into three

diﬀerent categories:

1: Moment closure methods where the Boltzmann equation is represented as

a series of angular moments and these moments are closed to form a solu-

tion. Both Flux-limited diﬀusion and variable Eddington factor techniques

ﬁt into this class.

2: Direct discretization of the Boltzmann equation. Probably the most com-

mon technique in this category used in astrophysics is the S

method.

This is often called “Boltzmann transport”.

3: Stochastic methods or Monte Carlo Methods. In astrophysics, the use of

Monte Carlo techniques has, in the past, been limited to time-independent

calculations (e.g. post-process).

2 Schemes Used in Astrophysics

Although these deﬁnitions don’t seem restrictive, they exclude a lot of the

techniques used in astrophysics: hydrodynamics schemes that put in radiation

as a cooling term or use a “leakage” scheme to control the radiation transport

through a dense medium (there are a number of problems, for example, in

cosmology, where such schemes are suﬃcient to solve the current questions

being addressed); and transport schemes that add in material motion, but

do not actually solve the hydrodynamics equations (some very sophisticated

transport techniques have been used to solve supernova spectra and stellar

atmospheres where hydrodynamics plays a secondary role, and hence has been

neglected in past studies). Transport Schemes coupled with hydrodynamics

schemes include:

1: Pure diﬀusion schemes. These schemes are generally solved as 1-T calcula-

tions (one temperature describes both the radiation and matter). Although

technically a simpliﬁed solution to the Boltzmann equation, we will not

consider codes using these schemes further.

Radiation Hydrodynamics in Astrophysics 5

2: Flux-limited diﬀusion schemes. These schemes are among the simplest of

the 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 implemented

using 2-T (one temperature describing the radiation ﬁeld and one temper-

ature describing the matter) and have been solved both using single and

multigroup energy discretization. This technique dominates what is used

in astrophysics, especially in multi-dimension.

3: Variable Eddington factor schemes. This technique also uses moments of

the Boltzmann equation, but closes at one higher moment using Edding-

ton factors. The technique used to solve for the Eddington factors diﬀers

from code to code. Although initially used in pure transport problems, this

scheme has been used in both 1-dimensional and 1.5-dimensional (trans-

port along rays) radiation hydrodynamics schemes.

4: S

transport. In general, most working codes using S

techniques have

been limited to 1-dimensional hydrodynamics codes, but at least 1 paper

has 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 these

codes at a later time. Although this trend is slowly reversing (hydrodynam-

ics schemes are now being coupled to transport codes) because the transport

scheme dominates the computing time, at this time, most of the working as-

trophysics codes arise from hydrodynamics schemes with transport schemes

added on top. The current range of hydrodynamics schemes coupled with

transport techniques include most of the hydrodynamics techniques used in

astrophysics:

1: Lagrangian techniques. In astrophysics, Lagrangian grid codes are used

in 1-dimension, but most multi-dimensional Lagrangian codes use smooth

particle hydrodynamics. A number of transport schemes have been cou-

pled to 1-dimensional Lagrangian codes. Although coupling transport with

smooth particle hydrodynamics takes some thought, it can, and has been

done 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 been

coupled 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 speciﬁc

codes applied to two of the current astrophysics problems best-studied using

radiation-hydrodynamics codes.

6C.L.Fryer

3 Astrophysical Applications

Computational astrophysicists work using small steps, gradually increasing

the level of sophistication on a given problem. Early models simulate only

the 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 the

computer models. Now, to study the formation of the ﬁrst stars, modelers

must include the eﬀects of radiation transport. Although most cosmological

studies today can make progress with simplistic radiation routines like leak-

age schemes, there will come a time when the observational detail requires

radiation hydrodynamics techniques.

A number of astrophysical problems have already reached this coding re-

quirement. Stellar atmosphere studies, which in the past have done some

of the most sophisticated transport schemes in astrophysics using static

atmospheres 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 phenomena

using radiation (usually ﬂux-limited diﬀusion) hydrodynamic codes from neu-

tron star accretion [Murray et al. (1995), Klein et al. (1996)] to molecular

clouds [Sandford et al. (1982)]. However, we will focus our attention on the

two ﬁelds that appear to be most actively pursuing radiation hydrodynamics

in 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 down

onto the compact object, centrifugal forces become increasingly important

and the behavior of these phenomena is dominated by the disk of material

that forms prior to the accretion phase. The wide variety of applications has

led 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 path

much 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 used

to 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