STUDIES OF DETERMINISTIC TRANSPORT METHODS FOR THERMALRADIATION ON UNSTRUCTURED MESHES

R. P. Smedley-StevensonAtomic Weapons Establishment

Aldermaston, Reading, RG7 4PR, UKrsmedley-stevenson@awe.co.uk

ABSTRACT

Numerical simulations of Inertial Confinement Fusion (ICF) and plasma physics experi-ments on the National Ignition Facility, require the development of thermal radiation trans-port methods for both the 2D and 3D unstructured meshes used by hydrodynamics codes.Due to the computational complexities inherent in solving the transport equation, a solu-tion of the diffusion equation has been developed for these types of meshes. In combinationwith a flux limiting procedure this can provide reasonably accurate results for a wide classof problems. Details of this scheme are presented together with the results from examplecalculations of an ICF capsule implosion.

The diffusion approximation is adequate for qualitative modelling of capsule performance.However accurate solutions of the transport equation are required in order to model thedetails of the radiation drive from the hohlraum. Ideas for extending this work to provideaccurate solutions of the transport equation are considered in the remainder of the paper.Computational experiments in 1D slab geometry are used to study the performance of po-tential schemes based on the use of finite element methods in space and angle. Promisingresults are obtained for both the first and second order forms of the transport equation withthe intensity as the primitive variable.

1. INTRODUCTION

A brief introduction to ICF modelling is presented in order to highlight what can be achievedwith radiation diffusion simulations and the need for deterministic transport methods. ICF

targets can be imploded by direct illumination by a symmetric arrangement of beams froma high power laser or alternatively by focussing the beams of the laser onto the inside of thewalls of a gold enclosure or hohlraum with the capsule mounted inside, where the energyis converted into thermal X-rays which are absorbed in the outer layers of the capsule.

In either mechanism, the symmetry of the implosion needs to be carefully controlled inorder to optimise the neutron yield of the capsule, in particular in order to achieve igni-tion in the central gas cavity. The heated wall acts as a highly localised source of X-rayswhich either heat the capsule surface or are absorbed by another part of the hohlraum wall,which subsequently re-emits some fraction of this energy. The early time behaviour of thehohlraum can be modelled using radiosity techniques[1], with an effective albedo for thewall based on the results of experimental measurements and/or detailed simulations of thewall physics.

The implosion of an indirectly driven ICF capsule is sensitive to the directional and spatialdependence of radiation field in the hohlraum. The X-rays emitted by localised hot spotson the hohlraum wall produced by the laser have a strong directional dependence. As thehohlraum fills with plasma ablated from the hohlraum walls and the surface of the capsule,the laser energy is absorbed before it reaches the hohlraum wall.

Radiosity calculations are no longer accurate, as they do not take into account the motion ofthe hot spots and re-emission from the stagnation of the plasma filling the hohlraum at latetimes. A solution of the coupled system of hydrodynamic and radiation transport equationsis required in order to accurately predict the late time X-ray drive. The hohlraum has to bedesigned in such a way that the pressures exerted by the hot plasma do not disrupt the latetime implosion of the capsule.

1.1 USING DIFFUSION TO MODEL CAPSULE IMPLOSION DYNAMICS

While it has been stressed that accurate transport solutions are required in order to modelthe X-ray drive from the hohlraum, typically the details of the capsule implosion are mod-elled in a separate calculation. This calculation is restricted to modelling the capsule only.The results of experimental measurements of the radiation spectrum in the cavity and/orhohlraum simulations are used to determine the radiation field at the outer boundary of thecapsule, and this is used to drive the implosion calculation.

Rather than having to model the complex processes occuring in the hohlraum, the physicsincluded in these calculations can be simplified, allowing higher resolutions to be achievedin the hydrodynamics simulation. In particular, the radiation transport modelling can besimplified by using radiation diffusion to model the dynamics of the implosion.

The diffusion approximation replaces a hyperbolic partial differential equation for the in-tensity of the radiation in a particular direction, for photons travelling at the speed of light,

by a parabolic equation for the angle averaged intensity for which the speed of propagationis unbounded. A flux limiting procedure can be used to physically limit the speed of thethermal wave, but this is not able to accurately replicate the solution of the transport prob-lem. As a consequence the radiation flux imposed on the outer surface of the capsule mustbe reduced to compensate for these errors.

The amount of reduction required depends on the mechanism used to specify the boundarycondition for the diffusion equation. The net radiation flux entering the problem is sensi-tive to the optical depth of the cells next to the drive region as well as the details of the fluxlimiting procedure. This scaling factor is determined by attempting to match experimen-tal measurements of the implosion dynamics. This procedure is adequate for modellingsymmetric implosions, or those where there is a small asymmetry in the X-ray flux at thecapsule surface. Example results are presented for a high convergence NIF target, whichshow the quality of the solutions which can be obtained.

In reality the target fabrication process can produce significant perturbations to the capsule.Specifically the mechanism used to join the two hemispheres of a capsule together canintroduce a perturbation in either the geometry or the density of the ablator. The growth ofthis feature is more sensitive to the details of the transport model than the dynamics of theimplosion itself, and flux limited diffusion may not be adequate to model this phenomenon.

Monte Carlo transport calculations can accurately reproduce the observed implosion be-haviour without the need for a scaling factor on the imposed X-ray drive, however thestatistical noise present in the simulations couples to the hydro motion. This makes it diffi-cult to perform accurate calculations for high convergence ratio targets. The elimination ofnumerical noise from these solutions is the motivation for developing a deterministic trans-port scheme. In particular, the effect of non-uniformities in the drive, or imperfections inthe target and growth rate for the mixing processes at the unstable interfaces are all issuesof concern in being able to achieve ignition.

2. THE RADIATION DIFFUSION SCHEME

The first part of this paper describes the radiation diffusion algorithm which has been cou-pled to the 2D ALE (Arbirary Lagrange Eulerian) hydrocode CORVUS[2][3][4], while thesecond part provides an overview of preliminary investigations which have been carried outon deterministic solution methods for the transport equation. The aim is to use a consis-tent spatial discretisation in order to retain the same diffusion discretisation in the opticallythick limit, combining this with a suitable treatment for the angular variable. In this way,issues of mesh resolution can be addressed by performing diffusion calculations rather thanthe more computationally intensive transport solution.

CORVUS solves the Euler equations on an unstructured mesh of quadrilateral elements inboth planar and axisymmetric geometries, using a staggered grid formulation for the hy-drodynamics, where pressure and internal energy are cell centered and piecewise constantand the velocity is node centered using bilinear isoparametric finite elements. The ther-mal radiation diffusion solution is based on the scheme proposed by Shestakov, Harte andKershaw[5], with both the radiation field and the temperature centered at the nodes. In theirwork the mesh cells are subdivided into triangles.

By using the same quadrilateral elements as the hydrodynamics the scheme in CORVUShas important additional symmetry properties. In particular it can be made to preservesymmetry for equiangular radial meshes in both planar and more importantly axisymmetricgeometries. This has important implications for modelling high convergence ICF targets.It is harder to investigate the affect of small perturbations due to the drive, or the capsulefabrication if these have to be separated from the effects of asymmetries arising from thescheme itself.

2.1 SOLUTION OF THE RADIATION DIFFUSION EQUATION

In the multi-group formulation, the zeroth moment of the radiation transport equation canbe written as

∂Eg

∂t+ ∇ ·Fg = cσag(Bg−Eg) (1)

whereEg is the energy density,Fg is the flux, Bg is the integral of the Planck functionat the material temperature andσag is the absorption cross section for groupg. LocalThermodynamic Equilibrium (LTE) has been assumed in this expression, so that Kirchoff’slaw can been used to replace the emissivity.

Fick’s lawFg =−Dg∇(cEg) is used to eliminate the flux from this expression, to arrive at adiffusion equation for the energy density. The diffusion coefficientDg depends on the totalcross sectionσg and is given byDg = 1/(3σg) for isotropic diffusion theory. However, ingeneral a flux limiting prescription is used in order to improve the accuracy of the diffusionapproximation.

These set of group equations are coupled to the material energy equation

∂E∂t

= ∑g

cσag(En+1g −Bn+1

g )+ S (2)

and the rate of change of the material energyE is replaced by the product of the heat capac-ity CV and the rate of change of the material temperatureT. S is a generalised source termrepresenting contributions to the energy equation from sources other than energy exchangewith the radiation field.

In general this equation also includes an electron conduction term. However this term isomitted from the source linearisation procedure. Rather than solving the material energyequation directly, with the inclusion of the conduction term, this equation is solved usingthe same techniques as the single group radiation diffusion equation.

2.2 SOLUTION OF THE SINGLE GROUP EQUATION

A continuous finite element representation is used for the mean intensity in each photongroup,Eg = ∑Eg(r i)φi(r), and the emission source term,Bg = ∑Bg(r i)φi(r). The diffusioncoefficient and the absorption opacity are both treated as piecewise constant in each cell.

The Petrov-Galerkin (or “weak” form) of the single group radiation diffusion equation isformed by expanding the intensityψ in terms of the finite element basis functionsφi(r),multiplying by a set of arbitrary weight functionswi(r) (with a one-to-one correspondenceto the basis functions), and integrating over the support ofwi (volumeVi with surface∂Vi).The diffusion term is integrated by parts so that its contains only first order derivatives(hence bilinear elements are sufficient to represent the diffusion term). This introducesan additional surface integral over the problem boundary, which will be evaluated usingthe boundary condition information (by omitting this term, the problem boundaries arenaturally reflective).

Mass lumping is invoked for the time derivative and energy exchange terms in order toincrease the diagonal dominance of the resulting matrix and improves the positivity of theresulting solution. Diagonalising the mass matrix converts these terms into finite differ-ence form and makes the scheme locally conservative over the nodal control volume. Theresulting equations are:(

1cdt

+ σa

)En+1

i

∫V

widV +∑j

En+1j

∫V

D∇wi ·∇φ jdV−∫

∂VwiDn·∇En+1dl

=(

1cdt

Eni + σaBn+1

i i

)∫V

widV (3)

where for clarity the group suffix has been dropped and the cross-sections in the lumpedterms have been assumed to be independent of space.

Finally, the form of the weighting function is chosen. For planar geometry, the weightfunction is chosen to be identical to the nodal shape function. In this case, the equations arethe standard Galerkin equations. In axisymmetric geometry the choice of weight functionsis motivated by a desire to maintain spherical symmetry of the radiation intensity. This isachieved by ignoring the variation ofr in the differential volume by using(r/r)φi for theweight function, where ¯r is the volume averaged value ofr in the cell. This latter changemeans that the integration by parts used to derive the “weak” form of the diffusion equationis not strictly valid, aswi is no longer a continuous function across the cell boundaries.

Symmetry preservation is achieved by neglecting the terms associated with this jump dis-continuity at the interfaces between different cells. The trade-off for preserving sphericalsymmetry is that the discrete equations no longer conserve energy exactly; a constant func-tion (throughout the problem domain) cannot be constructed from a linear combination ofthe weight functions. Comparisons with the solution of the Galerkin equations indicate thatthis does not have a detrimental effect on the accuracy of the results from the code. In prac-tical simulations, this error is small in comparison with other mechanisms which spoil theenergy conservation. In particular the hydrodynamics algorithm conserves momentum inpreference to kinetic energy, and total energy is conserved to an accuracy of a few percentin typical simulations.

The diffusion timestepping scheme is based on solving the equations for a single step, fol-lowed by repeating the solution for two half steps, with the material properties etc. updatedat the half step, in an attempt to accurately predict the flow timescales while maximisingthe timestep by using an accurate measure of the error due to the non-linear behaviour.

A related issue for the efficiency of the algorithm are its mesh convergence properties. Theaccuracy of the solution can be improved significantly by evaluating the opacity at a tem-perature based on the average value ofT4 in the cell (which corresponds to the energydensity of the equilibrium radiation field in the cell), rather than using the cell tempera-ture. The improvement in the propagation velocity for a Marshak wave travelling throughoptically thick cells is equivalent to at least a factor two refinement of the computationalmesh.

2.3 SOURCE TERM LINEARISATION

The thermal emission source term is linearised with respect to the material temperature,which reduces the magnitude of the coupling terms in the strongly coupled limit, enablinglarger timesteps to be used in the solution of the diffusion equation.

With a fully implicit discretisation of the time derivative, the material energy equation forgroupg can be written as

CV(Tn+1−Tn)

dt= cσa(En+1

g −Bng)+ Sn+1

g (4)

whereSis apportioned among the photon groups by some means, for example in proportionto the contribution to the emissivity or the radiation spectrum. The superscriptsn andn+1represent the values at the start and end of the current timestep respectively, anddt is thelength of the timestep.

The linearisation of the source term involves the replacement ofBn+1g by

Bng + (Tn+1−Tn) dBg

dT

∣∣∣Tn

. From this, the energy equation can be solved to provide an ex-pression for the temperature change. Inserting this into the exchange term yields

σa(Bn+1g −En+1

g ) = σ′a(Bn+1g −En+1

g )+ σ′adBg

dT

∣∣∣∣Tn

dtSn+1g

CV(5)

whereσ′a = σa/(1+ cdtσaCV

dBgdT

∣∣∣Tn

) and the effective coupling between the material andradiation has been significantly reduced.

Each group is treated independently in this implementation, a fact which could be exploitedto extract parallelism from the algorithm. The material temperature is updated at the endof the radiation diffusion step based on the sum of the changes from each group. Thecoupling between different groups is achieved by means of the time step control. Rosselandaveraging is used to obtain the group mean opacities in both the transport and the energyexchange terms. In this way, a grey calculation is treated as a single group spanning theentire frequency space. The accuracy of this treatment can be improved by using Planckaveraging in the exchange term (this will be discussed at length in section 3.6).

Fleck and Cummings[6] have applied a similar linearisation procedure to the transportequation. In their work the groups are coupled together rather than treated independentlyfor multi-group calculations. Their grey procedure applied independently to each groupis equivalent to the procedure described above. However, the difference for the transportequation is that the linearisation results in the absorption term being balanced by an expres-sion which involves the mean intensity.

Consequently, applying this linearisation to source iteration based schemes for solving thetransport equation requires an efficient acceleration technique to calculate the mean inten-sity (or scalar flux in neutron transport terminology), even when the physical scattering inthe problem is negligible, for cases where the matter and radiation are strongly coupled.

2.4 GENERATION OF NODAL TEMPERATURES AND THE COUPLING TO THEHYDRO

The key difficulty with the scheme is the requirement to generate a set of nodal tempera-tures for use in the energy equation, while the natural centering of the temperature for thehydrodynamics is at the cell centre. This has been overcome to produce an accurate andreliable package. The precise details of the implementation determine the success of thescheme, which is due to the application of monotonic principals to the nodal temperaturedistribution. Shestakov et al.[5] have shown that mapping total energy leads to unaccept-able levels of numerical diffusion. Instead the energy change derived from solving theradiation diffusion equation is mapped back to the cell centres.

The nodal temperature distribution arising from a pure radiation problem will match thecell centred energies except for errors generated by the non-linearity of the equation ofstate. By contrast, for pure hydro problems it will not be possible in general to the producea nodal temperature distribution which is consistent with the cell energies. Consequentlyin order to generate an accurate nodal temperature distribution in radiation-hydrodynamicsimulations, the temperature distribution from the radiation diffusion solution is evolved intime. Changes are applied to account for the work done by the hydrodynamics and non-linearities in the equation of state, but these must be strictly controlled in order to preventthe accuracy of the nodal temperature from being corrupted away from the shocks.

Given the nodal temperature distribution at the start of the timestep, and information onthe cell centred properties at the start and end of the timestep, an approximate update tothe nodal temperature is derived by assuming the material energy change is distributeduniformly across each cell. Evaluating the material energy change in the nodal controlvolume leads to the following expression for the update in the nodal temperature.

∑c

(∫cρcCVc(Ti−Tc)φidV

)n+1

= ∑c

(∫cρcCVc(Ti−Tc)φidV

)n

(6)

The internal energy dependence has cancelled out, to leave a simple expression for changein the nodal temperature due to the changes in the temperature at the cell centres. Thisapproximation generates a smooth nodal temperature distribution which conserves energyas well as being monotonic with respect to the cell temperatures, but does not fit the precisecell centred temperature distribution locally.

The consistency between the nodal and cell temperatures is improved in the following steps:

• The error in the cell energies∫

cρcCvc(Tc−Tc(Tj))dV is evaluated, whereTc(Tj)= ∑i

∫cTjφ jdV/

∫cdV is the cell centred temperature derived from the nodal temperatures.

• This energy difference is again mapped back to the nodes by recalculating the change inthe energy contained in the nodal volume,

∑c

wic

∫cρcCvc(Tc−Tc(Tj))dV

wherewic are a set of weights normalised so that∑i wic = 1.

• The nodal temperatures are updated and the corresponding error in the cell energies recal-culated. This process is repeated for a specified number of iterations.

Experience shows that repeated application of this correction can lead to divergence of thenodal temperature distribution, at a rate determined by the choice of the weightswic and thedetails of the calculation. To prevent this, at the end of each iteration a monotonic principalis applied to the nodal temperature distribution.

Various different forms for the weightswic have been studied. Using the intersection be-tween the nodal and cell volumes is consistent with the timestep update. Alternatively, itis possible to weight in proportion to the current contribution of the nodal temperature tothe cell temperature associated with the nodes. With the monotonicity filter this has littleeffect on the rate of convergence and the precise nodal temperature distribution that is con-structed, which is more closely associated with the constraints imposed by monotonicity.

The monotonic limiting procedure works as follows. For each node the extremal valuesof the temperatures in the connected cells (for multi-material cells the temperatures of thecomponent materials are treated individually) are generated. Also, the extremal values ofthe temperatures at directly connected nodes are calculated, provided that there is a regionof overlap between the two ranges. Optionally, the indirectly connected nodes can beincluded in this latter step in order to produce a smoother distribution.

Restricting the nodal temperatures to lie within this range guarantees monotonicity of thenodal temperature distribution with respect to both the neighbouring cell temperatures andthe nodal temperatures along the mesh lines. For symmetry axes and merged points thereis enough information to carry out this limiting, but for the nodes along a free boundarythis involves looking outside the problem and consequently they are excluded from thisprocess.

2.5 EXAMPLE ICF CAPSULE SIMULATIONS

The diffusion scheme described above has been used to simulate the implosion of a cryo-genic NIF capsule design. Rather than undertaking a detailed design study for this capsule,this is a proof of principal calculation, with simplified material properties. In order to deter-mine neutron yields for this design, the drive would be tuned in order to control the arrivalof the shocks in the centre of the capsule, to optimise the performance. These calculationswere performed without thermonuclear burn.

Instead, a comparison is made between a symmetric implosion, which demonstrates theability to preserve spherical symmetry for a high convergence target, and a calculation witha 1%P2 asymmetry in the incident flux corresponding to a higher drive on thez-axis. Thecapsule is driven by specifying the temperature time history for the radiation temperaturein the hohlraum, with the temperature imposed in the entire region outside the initial outerradius of the capsule.

In the asymmetric calculations, the shock reflects off ther-axis first due to the enhanceddrive, slowing the implosion in this region. As a result the gas is swept into a pocket onthe z-axis and the ice is allowed to compress onto thez-axis. Both calculations are closeto achieving the conditions required for ignition [7][8], but the asymmetric implosion hasa lower areal density (ρ∆r) on thez-axis, which makes it easier for the alpha particles toescape rather than heating the gas and sustaining the thermonuclear reactions, degrading

the efficiency of any burn. However, this is counteracted by the higherρ∆r on ther-axisfor a proportionately larger mass of ice, and the net effect may be an enhancement of theyield from the capsule for a small enough perturbation.

The density and temperature contours are plotted in figure 1 at 17.0 ns, which is close to thetime of peak compression for the DT ice layer. CORVUS is able to preserve the sphericalsymmetry of the implosion despite the high compressions achieved in the ice layer. At thesame time, it is also important to ensure that physical perturbations of this symmetry suchas that in the drive, can be modelled using the same code and these results show that thealgorithms in CORVUS have both these desirable properties.

3. REVIEW OF DETERMINISTIC TRANSPORT ALGORITHMS

Initial studies are presented on the performance of various deterministic transport algo-rithms applied to thermal radiation transport problems. The method must produce reliableresults for large optical depth cells, while remaining accurate for optically thin problems.In particular, it is important that the scheme has the correct asymptotic limits. The aimof the second part of this paper is to provide an overview of the computational methodsbeing considered, and to evaluate the performance of the different algorithms for thermalradiation transport.

This paper focuses on preliminary investigations on the use of finite element methods forthe angular variable (AWE has sponsored preliminary investigations on the feasibility ofthe extension of the finite element spherical harmonics code EVENT[9] to provide ray ef-fect free solutions for thermal radiation transport). A consistent framework is described fortreating continuous and discontinuous finite element methods in 1D for first and second or-der forms of the transport equation, with a straightforward generalisation to higher dimen-sions. The equations are formed by integrating over each cell individually in the Galerkinformulation and using upwind values in surface terms. The transition between continuousand discontinuous elements simplifies to a modification of the global connectivity arrays.

No attempt has been made to optimise the solution of the equations. The aim of this studyis to compare the merits of the different discretisations in terms of accuracy rather thancomputational efficiency. However, this will be a key issue for multi-dimensional simula-tions.

3.1 SPATIAL DISCRETISATION OF THE TRANSPORT EQUATION

The grey transport equation for the radiation intensityψ, including thermal emission,isotropic scattering and absorption processes can be written as

1c

∂ψ∂t

+ Ω ·∇ψ + σψ = σaB+(σ−σa)14π

∫4π

ψdΩ (7)

whereσ is the total cross-section,σa is the absorption cross section andB is the Planckfunction at the material temperature integrated over all frequencies.

The spatial differencing of the transport equation is considered, along a particular ray di-rectionΩ. The Petrov-Galerkin equations are formed by an equivalent procedure to thatused for the single group diffusion equation. However, in this case the finite element basisfunctionsφi(r) include discontinuous linear elements in addition to the continuous linearelements.

The streaming term is integrated by parts. In combination with fully implicit time differ-encing, this equation can be expressed as,

σ′ψn+1i

∫Vi

widV−∑j

ψn+1j

∫Vi

φ jΩ ·∇widV +∫

∂Vi

ψn+1wiΩ · ndl

=(

1cdt

ψni + σaBn+1

i +(σ−σa)14π

∫4π

ψn+1i dΩ

)∫Vi

widV (8)

whereσ′ = ( 1cdt + σ) and the superscript refers to the time level. Mass lumping has been

invoked for the time derivative, absorption and source terms in order to introduce localconservation (over the nodal shape function volume) for continous methods, together withensuring positivity of the solution for a pure absorber. The Galerkin form of the first orderequation is obtained by settingwi = φi .

For simplicity, the cross-sections have been assumed to be independent of position so thatthey can be taken outside the integrals. The surface term is evaluated in the upstreamdirection for discontinuous trial functions and using the prescribed boundary conditionsat the problem boundary for both continuous and discontinuous methods. Alternatively,the equations for the boundary nodes can be replaced by forcing the solution to satisfy theprescribed boundary conditions exactly.

3.2 SECOND ORDER FORMS OF THE TRANSPORT EQUATION

The second order form of the transport equation considered in this report is the self-adjointangular flux (SAAF) equation[10], which can be obtained in several different ways. It canbe thought of as a Petrov-Galerkin discretisation of the first order equation, with weightfunctionwi = φi +σ′−1Ω ·∇φi in volumeVi andwi = φi on the boundary∂Vi . Alternatively,

it can be obtained by using the fully implicit time discretised form of the first order equationto obtain an expression forψ. This can then be substituted back into the streaming term.The implicit streaming term is replaced by an explicit streaming term, together with animplicit diffusion term.

The Petrov-Galerkin form of the second order equation can be written as

σ′ψn+1i

∫Vi

widV +∑j

ψn+1j

∫Vi

σ′−1Ω ·∇φ jΩ ·∇widV +∫

∂Vi

ψn+1wiΩ · ndl

=(

1cdt

ψni + σaBn+1

i +(σ−σa)14π

∫4π

ψn+1i dΩ

)∫Vi

widV

+∑j

(1

cdtψn

j + σaBn+1j +(σ−σa)

14π

∫4π

ψn+1j dΩ

)∫Vi

σ′−1φ jΩ ·∇widV (9)

The Galerkin form of the second order equation is obtained by settingwi = φi .

The close correspondence between the form of this equation and the radiation diffusionequation is the motivation for studying this second order equation. The radiation diffusionapproximation is recovered by assuming that the radiation field is linearly anisotropic andintegrating this equation over angle. To recover a single diffusion equation, it is necessaryto assume thatσ 1

cdt, which is equivalent to ignoring the time derivative of the radiationflux.

The steady-state diffusion equation can be obtained by simply assuming that the radia-tion field is isotropic. However, for time dependent problems the absence of the radiationflux term leads to a diffusion equation which significantly under-predicts the rate of en-ergy transfer. Consequently, flux-limited diffusion based on using the modified diffusioncoefficient 1/(3σ′) does not produce physically meaningful results.

Variational principals can also be constructed from the first order equation by expressingit as a set of coupled first order equation in terms of the even and odd parity componentsof the radiation intensity[11]. The resulting second order equations have a similar form tothis equation, and can be solved using similar methods. The advantage of solving eitherthe even or odd parity equations is that only half the angular domain needs to be discretisedin order to represent the parity fluxes, and consequently solving these equations requiresonly half the computational work required to solve the SAAF equation. For time dependentproblems the complementary parity component is required.

The difficulty with this approach can be illustrated by considering the behaviour of a timedependent source which produces a collimated beam of photons propagating through a vac-uum. This streaming problem can be modelled in terms of the even and odd parity solutions.However, unless independent solutions of the even and odd parity are generated using con-sistent trial functions (requiring the same computational work load as solving the secondorder equation for the intensity and producing identical results provided a subtle modifi-

cation is made to the boundary conditions for the SAAF equation[12]), the reconstructedintensity distribution will be inconsistent. In particular, it will not vanish for radiation trav-elling in the opposite direction to the beam. Consequently, solutions based on solving forthe even or odd parity flux alone, and deriving the complementary variable from this solu-tion, are better suited to stationary problems or problems which do not involve transparentmaterials.

3.3 DIFFUSIVE BOUNDARY LAYER BEHAVIOUR

The accuracy of the penetration of radiation into diffuse regions (regions which by ne-cessity are modelled with optically thick cells) is sensitive to spatial differencing scheme.Larsen and Morel[13] have shown that schemes based on linear representations of the spa-tial variation within a cell are the most accurate of all the difference schemes that theyhave considered for the first order transport equation in slab geometry. In particular, whilethe solution in the first cell of the diffusive region may be in error, the interior solution isbased on an angular weighting of the incident fluxµ+ 3

2µ2 which closely matches the exactweighting function.

This analysis has been extended to consider thermal radiation transport specifically[14].The diffusion limit has been analysed for linear discontinuous solutions of the system ofmulti-group radiation transport equations coupled to the linearised material energy equa-tion. The non-linearities due to the emission term mean that the accuracy of this coupledset of equations is not as good as the transport equation alone; specifically the temperatureat the surface of the diffuse region may be significantly in error. Despite this, the interiorsolution remains remarkably accurate, demonstrating the suitability of this approach forthermal radiation diffusion problems. Recently, this has been further extended to includeproblems which are optically thin in some parts of the spectrum[15].

The asymptotic analysis of the transport equation has also been extended to two dimen-sional rectangular meshes by Borgers, Larsen and Adams[16], who show that a schemebased on linear expansions in the two coordinate directions needs to be modified in orderto obtain the correct asymptotic behaviour in the diffusion limit. Morel, Dendy, Jr. andWareing[17] have shown that a scheme based on Galerkin weighted bilinear discontinuousfinite elements behaves correctly in this limit; this is the two dimensional analogue of thescheme considered in this report.

For the second order forms of the transport equation solved using continuous finite ele-ments, a hybrid principal has been advocated[18] where the intensity distribution is derivedfrom a linear combination of the solutions of the even and odd parity equations, which sep-arately use a weighting of the incident intensity distribution which can be in error by asmuch as a factor of two[19]. This hybrid scheme can be simulated by solving the SAAFequation using linear elements, coupled to a solution of the first order equation using piece-wise constant elements, by making a minor adjustment to the boundary conditions of the

second order equation[12].

The diffusion limit of these hybrid principals is not consistent with the diffusion schemedescribed in this report, and the resulting intensity distribution is piecewise discontinuousas it results from a combination of linear continuous and piecewise constant solutions.Another concern is the ill-posedness of the equations for triangular and tetrahedral finiteelements[18]. This is not an issue for the schemes being considered in this report, as thecomputational mesh from the hydrocode is composed of quadrilaterals or hexahedra.

3.4 ANGULAR DISCRETISATIONS

The spatial discretisation of the transport equation has been discussed, which is sufficient toallow solutions to be compared for the transport along a single ray direction. In this sectionthe radiation intensityψ is expanded in terms of space-angle (or phase-space) finite elementbasis functions formed as a product of independent space and angular basis functions. Theform of the equations is unchanged, but the basis functions now depend on angle and theintegrations are over the phase space support volume of the shape functions. Two angulardiscretisations are considered, piecewise constant (or step) elements and linear elements.For 1D slab geometry the rays travel in straight lines. Consequently the streaming termdoes not contain any angular derivative terms, and the surface terms do not need to bemodified for discontinuous angular representations. The effect of mass lumping in anglewill be addressed to determine whether it also has a beneficial effect on the solution of thetransport equation.

3.5 SCATTERING SOURCE ITERATION

For the angular discretisations considered above, the localised support of the spatial andangular finite elements minimises the bandwidth of the matrix associated with the result-ing set of linear equations, provided that the scattering source is treated explicitly. Forpiecewise constant angular elements, the equation for each angular element is indepen-dent. Furthermore, the resulting matrix equation for each angle can be converted into ablock lower triangular form, by ordering the unknowns in such a way that the equations aresolved by sweeping through the mesh in the direction of particle motion, starting from theknown solution on the incoming boundaries.

The scattering source must be iterated to convergence to ensure particle conservation. Thisis important in order to obtain accurate results for problems where the scattering termis significant, whether the scattering is physical or is introduced by the linearisation ofthe thermal emission source. This iteration process converges very slowly for scatteringdominated problems, and consequently a method of accelerating this iteration is required.The discontinuous finite element discretisation of the DSA equations derived by Adams

and Martin[20] is used for the finite element discretisations considered in this paper.

The source iteration is a two step process. The transport equation is solved using theold value of the mean intensityψ0

n+1,l , which produces a modified value of the intensityψn+1,l+ 1

2 . The next step in the process is to calculate an updated mean intensityψ0n+1,l+1.

The unaccelerated procedure setsψ0n+1,l+1 = ψ0

n+1,l+ 12 , and the transport solution is re-

peated for thel +1 iteration.

Integrating the second order transport equation over angle gives,

σ′ψ0n+1,l+ 1

2i

∫Vi

widV +∑j

ψ2

n+1,l+ 12

j:∫

Vi

σ′−1∇φTj ∇widV +

∫∂Vi

ψn+1wiΩ · ndl

=(

1cdt

ψ0ni + σaBn+1

i +(σ−σa)ψ0n+1,li

)∫Vi

widV

+∑j

1cdt

ψ1nj·∫

Vi

σ′−1φ j∇widV (10)

where the subscript indicates the various moments of the intensity. The above expres-sion can also be used to solve for a consistent value ofψn+1,l+1 which conserves particlenumber. The difference between these two expressions gives the following expression forδψn+1,l+1 = (ψn+1,l+1−ψn+1,l+ 1

2),(1

cdt+ σa

)δψ0

n+1,l+1i

∫Vi

widV +∑j

δψ0n+1,l+1j

∫Vi

3σ′−1∇φ j ·∇widV (11)

+∫

∂Vi

δψn+1,l+1wiΩ · ndl = (σ−σa)(ψ0n+1,l+ 1

2i −ψ0

n+1,li )

∫Vi

widV

where the angular correction is assumed to be only linearly anisotropic.

For continuous elements the surface term disappears and this is the discretised diffusionsynthetic acceleration equation. For discontinuous elements, the surface term is evaluatedby splitting the net current into incoming and outgoing partial currents. The correction isassumed to take the formδψ = δψ0−σ′−1Ω ·∇δψ0 appropriate to the diffusion limit. Thesurface integral is transformed into the following expression.∫

∂Vi

δψn+1,l+1wiΩ · ndl =−∫

∂Vi , n·Ω<0

(14

δψ0n+1,l+1 +

16σ′

n·∇δψ0n+1,l+1

)widl

−∫

∂Vi , n·Ω>0

(14

δψ0n+1,l+1− 1

6σ′n·∇δψ0

n+1,l+1)

widl (12)

These partial currents are evaluated using the values in the upwind direction. The meanintensity is updated using the solution of these equations for both the first and second orderforms of the transport equation and the source iteration proceeds as before. The use of thisscheme for the first order equation is argued on the basis that apart from differences in theangular weighting of the flux at the boundary with diffusive regions, the two forms of theequations behave similarly as they approach the diffusion limit, where the application of

DSA is most important. The correction is assumed to be isotropic and consequently thedifference in angular weighting is not an issue.

This discontinuous scheme can be computationally expensive to solve as the correspondingmatrix is asymmetric, and attempts have been made to use the continuous form of theseequations to accelerate discontinuous finite element solutions of the transport equation[21].The need to solve a diffusion equation can be avoided by using multi grid accelerationtechniques for the angular variable. Transport synthetic acceleration (TSA) schemes arebased on the use of lower order angular respresentations to accelerate the convergence ofthe scattering source iteration[22]. These tend to be computationally efficient and easier toparallelise than solving the diffusion equation, although more iterations may be required toconverge the scattering source. These efficiency issues will become important for multi-dimensional simulations.

3.6 GROUP AVERAGED CROSS-SECTIONS FOR TRANSPORT CALCULATIONS

The frequency dependence of the radiative cross-section or opacity is rarely resolved incalculations. In order to achieve as much accuracy as possible for a particular group struc-ture, group mean cross-sections are used, obtained from averaging the opacity over fre-quency in a similar way as multi-group cross-sections are derived for neutron transport.Unlike neutron transport calculations, there is no characteristic spectrum associated withmost problems. Instead the weighting function is obtained by considering the asymptoticbehaviour.

In the diffusion limit Rosseland averaging correctly predicts the transport of radiation,while for an optically thin plasma, Planck averaging ensures that the correct amount ofenergy is emitted by the plasma. The Rosseland average is a reciprocal average of theopacity, which emphasises the minima in the frequency dependent opacity which allow ra-diation to flow, while the Planck average is a linear average which highlights the maximain the opacity, which dominate emission and absorption.

The use of a Planck opacity to transport radiation will overly restrict the flow of energywhile the use of a Rosseland opacity to calculate the energy exchange will underestimatethe coupling between the material and radiation. In diffusion calculations, it is possibleto combine these two limits by evaluating the diffusion coefficient using Rosseland meanopacities and the energy exchange terms using Planck mean opacities, which producesaccurate results for coarse group structures, and in particular for grey calculations.

For the first order transport equation the opacity only appears in the absorption term, andconsequently there is no way to incorporate this combination of averages in order to im-prove the accuracy of the results. By transforming to the second order form of the transportequation the streaming term has been converted into two transport terms, which involve thereciprocal of the opacity. Rosseland averaging can be employed in these terms in a way that

allows the transport to be accurately calculated, while Planck averaging can be employed inthe emission and absorption terms in order to ensure that the coupling between the materialand radiation is predicted correctly. This should enable more accurate transport solutionsto be obtained from the same group structure.

3.7 METHODS FOR RAY EFFECTS MITIGATION/ELIMINATION

It is important to identify methods for overcoming the ray effect[23],[24] if discrete ordi-nate like methods are to be used in multi-dimensions. The radiation transport package isintended to be coupled to a hydrodynamics calculation, and the ray effect is likely to sys-tematically perturb the hydrodynamics in contrast to the statistical perturbations introducedby Monte Carlo transport, which are immediately obvious in the calculation.

The use of angular finite elements has been shown to lead to ray effect mitigation in multi-dimensions (at least for the second order form of the transport equation) and this behaviouris well understood[25]. However there is an associated loss of accuracy for problems wherethe ray effect is less important due to the additional numerical diffusion introduced by thecoupling in angle. An alternative strategy is to use an adaptive angular mesh for discontin-uous treatments of the angular variable. However the resolution required to eliminate theray effect may still not be achievable in practical simulations.

By using rotationally invariant angular basis functions, the spherical harmonic method pro-duces ray effect free solutions. An alternative strategy for eliminating the ray effect is toproduce an equivalent set of equations by modifying the form of the discrete ordinate equa-tions through the addition of a fictitious source[26]. While a variety of schemes have beenproposed which are capable of this transformation, in practice problems are experiencedin trying to accelerate the source iteration with this fictitious source, and consequently theefficiency of the discrete ordinate method is severely compromised.

The strategy investigated in this report is the projection of the piecewise constant angularsolution onto a low order spherical harmonic expansion, thus preserving the lower ordermoments of the angular variation, while smoothing out the variation of the solution overangle. This is based on ideas from the radiative transfer methods used in astrophysics,specifically the SHDOM method of Evans[27]. This also has the advantage of reducing thestorage requirements for the transport solution, since this is stored in terms of the expansioncoefficients. The solution of the transport equation itself is unchanged, and there is only asmall overhead associated with the projection between the finite element and spherical har-monic representations. This method is a palliative which offers the potential for obtainingray effect free solutions for time dependent problems, but its true effectiveness can only bedetermined by solving the transport equation in multi-dimensions.

3.8 COMMENTS ON THE POTENTIAL EFFICIENCY OF THE DIFFERENT DIS-CRETISATIONS

The combination of the spatial and angular discretisations determines the most efficientmethod for solving the resulting equations. The first order equation does not produce asymmetric set of equations and consequently it is more efficient to solve this equationusing discontinuous treatments of the spatial and angular variables and iterate on the scat-tering source. The resulting equations can be solved by sweeping through the mesh inthe direction of particle travel, with a suitably modified approach for curvilinear coordi-nate systems. The use of discontinuous spatial elements is also required in order to avoidoscillations in the solution due to the discontinuous change in gradient associated with amaterial interface, which may not be well resolved in real applications.

The second order equation is attractive from the point of view that it contains an implicitdiffusion-like term which can be solved accurately using linear finite elements and an ex-plicit streaming term, which can be solved accurately using a wider variety of techniquesthan can practically be applied to the implicit form of this term in the first order equation.For instance, continuous linear angular elements could be employed in the streaming term,while mass lumping in angle can be used to simplify the diffusion-like term.

The diffusion term, akin to the false diffusion term included in the streamline diffusionmethod for solving advection-diffusion equations[30], damps the oscillations from contin-uous spatial solutions associated with material boundaries, so that they are not present inthe solution of the second order equation. A discontinuous solution may produce betterresults, but at the cost of destroying the symmetry of the system of equations. This sym-metry is important for continuous angular representations which require the resulting set ofmatrix equations to be solved directly (specifically for thePn method, and the direct solu-tion for piecewise linear angular elements). The positive-definiteness and symmetry of thismatrix allows a wider variety of iterative techniques to be employed, which are capable ofsolving this matrix equation efficiently.

It may also be possible to use a continuous angular representation in the first order equationand move the terms which result from the coupling to neighbouring angles onto the right-hand side of the equations and iterate these to convergence. The use of continuous linearangular elements for either the first or second order equation offers the benefit of ray effectmitigation, but with a corresponding increase in the complexity of the equations.

3.9 COMPUTATIONAL IMPLEMENTATION

The computational implementation is based on the work of Miller, Lewis and Rossow[28],which provides a simple way of evaluating the required shape function integrals for the bi-linear finite element basis functions used in phase space. A unique numbering of the phase

space nodes for each element is determined, which allows piecewise constant elements tobe simulated, together with the arbitrary location of discontinuities in space and/or angle.The resulting equations are solved iteratively using the biconjugate gradient method[29]for general asymmetric sparse matrices.

4. RESULTS

A sequence of results is presented comparing the performance of the different schemes fora series of simple thermal radiation transport test problems. Initial comparisons are madefor streaming problems, compared against analytic solutions. The smoothness of the initialradiation front has a significant impact of the accuracy of the various schemes. Specifically,square and Gaussian pulses are used in order to compare the accuracy achieved in relationto the smoothness of the initial conditions. This is extended to problems with attenuationand then to participating media, in particular the coupling to the material temperature isinvestigated.

This sequence of calculations acts as a good way to validate the correctness of the algo-rithms as coding errors are immediately apparent on inspection of the results. The initialpart of the streaming problem tests the discretisation of this operator, in particular the ac-curacy of the propagation velocity together with the amount of numerical diffusion of theinitial profile, up to the stage at which the solution begins to interact with the boundary. In-adequacies in the transmissive boundary treatment are shown by the reflection of spurioussignals from the boundary.

Adding attenuation helps to demonstrate the positivity of the scheme for cells which aremany mean free paths wide. Multi-material configurations can be studied to examine theimprovement gained by the use of discontinuous solution methods. Finally, starting withthe zero solution and specifying a constant isotropic incoming flux at one of the boundariestests this aspect of the boundary conditions and the ability to propagate a steep front throughthe mesh. Again, attenuation can be included and finally coupling to the material, so thatthe final test is of the thermal radiation wave solution.

4.1 COMPARISONS FOR A PURELY STREAMING PROBLEM

The time dependent transport equation for radiation streaming in a vacuum in a particulardirection

1c

∂ψ∂t

+ Ω ·∇ψ = 0 (13)

is analogous to the advection equation solved in two step Eulerian and Arbitrary LagrangianEulerian (ALE) hydrocodes, where the Lagrangian form of the Euler equations is solved

followed by an advection phase where the fluid variables are re-mapped onto either anEulerian or an arbitrary grid.

The solution of this transport problem can be considered as a projection of the solutionadvancing through space in the direction of propagation, back onto the spatial mesh, re-peated every timestep. In contrast to the hydrodynamics, the propagation velocity doesnot vary over space. The results from the various discretisations are compared in figure2 for Gaussian and square wave profiles, using 100 uniform spatial zones and 1000 equaltimesteps. Continuous and discontinuous trial functions can be constructed from the trialspace of linear functions. The transport equation is solved for a single ray direction, chosenso thatΩ ·∇→ d

dx.

For the streaming problem the results are similar for both the first and second order equa-tions, the key difference being the amount of oscillation behind the wave for continu-ous solutions, which persists for the square wave; a difficulty alluded to by Mihalas andWeaver[11] for radiative transfer calculations. The presence of discontinuities in the initialsolution is akin to the discontinuities which arise in the solution of the transport equationat a material interface, which if it is not resolved by the mesh is likely to produce similaroscillations in the solution.

There is a fundamental difference between the continuous and discontinuous solutions ofthis transport equation. Continuous solutions involve both upstream and downstream infor-mation, and information ahead of the wave can have a significant influence on the resultingflow. By contrast discontinuous solution schemes only make use of information from theupstream direction, and the solution method emulates the physical phenomenon by beingignorant about conditions ahead of the wave. Discontinuous solutions are to be preferredfor this reason.

The amount of erosion of the peak function value and the significance of the oscillationsbehind the front are related to the size of the timestep. Small timesteps (much less than thestability timestep for explicit methods) are required in order to produce accurate solutions.The timestepping algorithm used to resolve the non-linearities of the radiation diffusionscheme does not perform particularly well for this problem, allowing too large a timestep,leading to too much erosion of the peak of the Gaussian profile.

These results show that in the streaming limit, radiation propagates at roughly the correctaverage velocity, although the arrival of the photons is smeared out over time. This isan extreme case. In typical ICF applications the propagation of radiation is either slowedby interactions with the material (and the direct contribution from steaming photons isnegligable) or the radiation field reaches a steady-state equilibrium (for example in thecentre of a hohlraum). However, this behaviour should be noted in applications, in case ithas an impact on the interpretation of the results.

One possibility for further improvement of the streaming results is to adopt a total variationdiminishing (TVD) scheme[31], developed for solutions of the Euler equations, in order

to determine the values of the “fluxes” (not to be confused with the radiation flux) arisingfrom the surface integral. The piecewise constant finite element (or step scheme) guaran-tees positivity provided the signal travels less than a mesh width in a timestep. However,this scheme is only first order accurate in space, which leads to excessive diffusion of thesolution.

Flux corrected transport (FCT) methods ensure that the positivity of the solution is retainedwhile counteracting the numerical dissipation associated with first order methods by usingan anti-diffusion “flux”, calculated from a higher order solution of the equations (usinglinear discontinuous elements for example), in an attempt to sharpen up the features of theadvected profile. A limiter is then applied to the second order surface ”fluxes” in order toensure that oscillations from this higher order solution do not contaminate the final solution.For more details on this procedure see for example Zalesak[32].

4.2 PURE ABSORBER

The inclusion of absorption in the transport equation leads to an exponential decay in theintensity as a function of distance which must be represented in the solution of the transportequation. The spatial differencing needs to be able to model this variation accurately with-out generating negative intensities. Some treatments of the spatial variation, in particulardiamond differencing, only produce positive solutions provided that the mesh width is com-parable with the mean free path and consequently they are not suitable for thermal radiationdiffusion problems with optically thick cells. For the linear discontinuous discretisation ofthe first order equation, mass lumping guarantees the positivity of the solution in 1D slabgeometry for any computational mesh[14].

The characteristics of the discretised transport equation for a pure absorber can be studiedin steady-state, as the characteristics of the time dependent behaviour were brought out inthe study of the streaming limit. The steady-state transport equation for a pure absorber is

Ω ·∇ψ + σψ = 0 (14)

Figure 3 compares the results of the various finite element schemes for a pure absorberdivided into a semi-transparent region (σ = σa = 0.01 for 0< x< 10) and an opaque region(σ = σa = 1.0 for 10< x< 20). Unit incident intensity is specified atx = 0 for the singleray direction used in the streaming problem. The transport equation is solved by dividingthe problem into ten uniform spatial zones in order to study the behaviour of the solutionin the presence of a discontinuity in material properties, when the dimensions of the spatialmesh are greater than the mean free path in the opaque material. In these simulations, thesolution is generated by solving the time dependent equation, and allowing the solution toconverge to the steady state, for a constant incoming flux.

As expected, severe oscillations are present in the linear continuous solution, confirming

the conclusions from the analysis of the streaming problem. The continuous solution of thesecond order equation is much better behaved, except that the value of the solution in thetransparent region does not match the value from the boundary condition. This is becausethe boundary condition has been only weakly imposed, by including it in the surface inte-gral term; the continuous solution can be forced to satisfy the boundary condition directly ifnecessary. The overall accuracy of the discontinuous schemes is very good, especially forthe second order equation, where the solution closely matches that from a mesh convergedcalculation.

The replacement of the exponential variation by a linear function can lead to significanterrors in the solution, even for schemes where the positivity of the solution is guaranteed.Higher order schemes have been developed which produce more accurate solutions, whichare of particular importance for shielding problems in neutron transport. Of particularinterest is the set of schemes which incorporate an exponential variation in the differencingapproach, but this can prove computationally expensive.

In contrast, linear elements are likely to be adequate for modelling thermal radiation prob-lems. Where there are regions with large optical depths, the cross-sections decrease stronglyas the materials absorb energy from the radiation field, due to their temperature dependenceand the expansion of the material. In addition, the strong coupling in optically thick regionstends to bring the radiation into thermal equilibrium with the material, and the heat frontpropagates due to absorption and re-emission from the material, rather than directly fromthe source. The directionality present in the radiation field disappears and the radiationtransport is replaced by a random walk, which can be described by the radiation diffusionequation. This transition from streaming to diffusion is studied in detail in the next section.

4.3 THERMAL RADIATION WAVE SOLUTIONS

The transition to diffusive flow and the Marshak wave limit is investigated for a semi-infinite slab of cold material with a constant isotropic flux of radiation applied at one end.Graphical solutions to this problem have been generated by Szilard and Pomraning[33]using a linear discontinuous spatial discretisation combined with a discrete ordinates treat-ment of the angular variable using a Gauss-Legendre quadrature set. Comparisons arealso made with diffusion simulations, both unmodified and with a complex flux limitingprescription. These results show that while flux limiting can improve the accuracy of theresults there is no substitute for an accurate transport solution when these effects are im-portant.

The solutions for the various spatial discretisations considered in this paper are comparedin figure 4 for 10 uniform piecewise constant angular elements. The radiation field from thesolutions of the second order equation is slightly advanced at the radiation front comparedwith the solution of the first order equation, and none of the schemes is completely suc-cessful in preventing radiation from propagating faster than the speed of light. Despite this

behaviour, the various schemes are of comparable accuracy and the material temperatureprofiles are in close agreement; these results compare well with the results from reference[33]. An interesting feature of these solutions is the oscillations in the radiation temperatureprofile due to the discretisation of the angular directions and the associated discretisationof the wave speeds.

This behaviour is studied as a function of the angular discretisation. In figure 5 the resultsfrom different angular discretisations, for linear discontinuous spatial elements in the firstorder equation, are compared. The use of linear angular elements helps to smooth out theoscillations. Nevertheless these are still prominent in the early time streaming behaviour.By representing the natural discontinuity atµ = 0 in calculations, the solution of the equa-tions in the forwardµ> 0 and backwardµ< 0 directions are completely decoupled, ratherthan enforcing continuity atµ = 0, and this produces a corresponding increase in the ac-curacy of the solution[34]. Coupling acrossµ = 0 affects the equilibration in the sourceregion, and this can be seen in the temperature profiles at the intermediate time; at earlytime, the solution is dominated by the forward directions and at late time the radiation inthe source region is close to isotropic.

Figure 6 shows the effect of mass lumping in space and angle, rather than just space (forlinear continuous angular elements with a discontinuity atµ = 0). The disadvantage ofthe un-lumped solution is the presence of faint images of the solutions from neighbouringangles introduced by the non-diagonal mass matrix. Mass lumping removes these spuri-ous “after images”, but the solution is more sensitive to the singularity arising from theboundary condition, with a much greater overshoot at the wave front; the inward flux in-stantaneously becomes non-zero, rather than rising smoothly in time.

Figure 7 shows the effect of converting the representation of the angular variation in thetransport equation (using 10 piecewise constant angular finite elements) to a truncated Leg-endre polynomial expansion. AP5 representation allows an an adequate representation ofthe angular variation, while damping the oscillations in the wave profile due to the angulardiscretisation. The only drawback with this procedure is that the truncation of the seriesexpansion leads to the generation of negative intensities, but these are allowed to developin order to preserve the moments of the angular distribution.

CONCLUSIONS

The details of a successful discretisation of the radiation diffusion equation in two-dimen-sions using bilinear finite elements has been presented. The generation of a set of nodalmaterial temperatures has been described. This is the key to the success of the scheme.Sample results for an ICF capsule implosion calculation, for a symmetric and an asymmet-ric radiation field are presented, which demonstrate the capability of the coupled radiationdiffusion-hydrodynamics code.

The streaming analysis shows that all the second order accurate spatial discretisations pro-duce some oscillations in the shape of the profile. This behaviour is exaserbated by con-sidering a square pulse of radiation, which does not occur in typical thermal radiationproblems. Instead the radiation source generally increases slowly in magnitude up to amaximum as the hohlraum heats up and then decreases slowly as the system cools down.The small timesteps required in order to prevent the excess numerical diffusion of the Gaus-sian pulse are a concern. In particular, the inability of the timestepping scheme used in thediffusion solver to detect this error warrants further investigation.

The results for the multi-material pure absorber rule out the use of continuous spatial el-ements in the first order equation, while the second order equation produces the most ac-curate solutions for this problem. The accuracy of thermal wave solution for the variousspatial discretisation of the transport equation considered in this report is encouraging. TheDSA procedure is sufficient to accelerate the solution of these equations as equilibrium isapproached and the equilibrium solutions are consistent with the solution of the radiationdiffusion scheme implemented in CORVUS.

The key issue with solutions of the second order equation is the accuracy for problemswhere the incident flux on a diffuse region is significantly anisotropic. By contrast, thesolutions of the first order equation are expected to be significantly more accurate for thesetypes of problem. If this is important, then hybrid forms of the second order equation canbe used. Unfortunately this will modify the behaviour in the diffusion limit.

A comparison of the various angular discretisations indicates that the oscillations in theprofile of the radiation front introduced by discretising the speed of propagation, persisteven with a continuous treatment of the angular variable. Mass lumping in angle has adetrimental effect on the solution and should not be employed for second order angularelements, as it is not required to ensure conservation over the spatial control volume.

One of the most interesting results is the elimination of the oscillations in the profile of theradiation front, for piecewise constant angular elements, by replacing the angular variationof the intensity by a truncated Legendre polynomial expansion. The effectiveness of thisprocedure for correcting ray effects in multi dimensions should be investigated.

The next step is to extend this work to two dimensional planar and axisymmetric geome-tries.

REFERENCES

[1] I Ashdown. Radiosity: A Programmer’s Perspective. John Wiley & Sons, Inc., 1994.

[2] A J Barlow. Mesh adaptivity and material interface algorithms in a two dimensionallagrangian hydrocode. InProceedings of the Hydrocode Workshop, Banff, Canada,11–15 April 1996.

[3] A J Barlow. ALE in CORVUS. InProc. of conference on New Models and Numer-ical codes for Shock-wave Processes in Condensed Media, St. Catherine’s College,Oxford, England, 15–19 September 1997.

[4] A J Barlow. Friction in CORVUS a 2D ALE code. InProc. of the 22nd InternationalSymposium on Shock Waves, Imperial College, London, 1999.

[5] A I Shestakov, J A Harte, and D S Kershaw. Solution of the diffusion equation byfinite elements in lagrangian hydrodynamic codes.JCP, 76:385–413, 1988.

[6] J A Fleck, Jr. and J D Cummings. An implicit Monte Carlo scheme for calculatingtime and frequency dependent nonlinear radiation transport.JCP, 8:313–342, 1971.

[7] J D Lindl. Phys. Plasmas, 2:3933, 1995.

[8] T R Dittrich et al. Reduced scale national ignition facility capsule design.Phys.Plasmas, 5:3708, 1998.

[9] C R E De Oliveira and A J H Goddard. EVENT - a multidimensional finite element-spherical harmonics radiation transport code. InOECD Proceedings, 3-D determinis-tic radiation transport computer programs. Features, applications and perspectives.OECD Chateau de la Muette, Paris, France., 2–3 December 1996.

[10] J E Morel and J M McGhee.Nucl. Sci. Eng., 123(3), 1999.

[11] D Mihalas and R Weaver. Time-dependent radiative transfer with automatic fluxlimiting. JQSRT, 28(3):213–222, 1982.

[12] C J Gesh. Finite Element Methods for the Second Order Forms of the TransportEquation. PhD thesis, Texas A&M, 1999.

[13] E W Larsen and J E Morel. Asymptotic solutions of numerical transport problems inoptically thick, diffusive regimes II.JCP, 83:212–236, 1989.

[14] J E Morel, T A Wareing, and K Smith. A linear-discontinuous spatial differencingscheme for SN radiative transfer calculations.JCP, 128:445–462, 1996.

[15] M L Adams and P F Nowak. Asymptotic analysis of a computational method fortime- and frequency-dependent radiative transfer.JCP, 146:366–403, 1998.

[16] C Borgers, E W Larsen, and M L Adams. The asymptotic diffusion limit of a lineardiscontinuous discretization of a two-dimensional linear transport equation.JCP,98:285–300, 1992.

[17] J E Morel, J E Dendy, Jr., and T A Wareing. Diffusion-accelerated solution of thetwo-dimensional SN equations with bilinear-discontinuous differencing.Nucl. Sci.Eng., 115:304–319, 1993.

[18] C J Gesh and M L Adams. Even- and odd-parity finite element solutions to thick dif-fusive problems in cartesian geometry. InProc. Int. Topl. Mtg on Advances in Math-ematics and Computation, Reactor Physics and Environmental Analysis, Madrid,Spain, 27–30 September 1999.

[19] M L Adams. Even-parity finite element transport methods in the diffusion limit.Prog.in Nucl. Energy, 25:159–198, 1991.

[20] M L Adams and W R Martin. Diffusion synthetic acceleration of discontinuous finiteelement transport iterations.Nucl. Sci. Eng., 111:145–167, 1982.

[21] T A Wareing, J M McGhee, J E Morel, and S D Pautz. Discontinuous finite elementSN methods on 3-d unstructured grids. InProc. Int. Topl. Mtg on Advances in Math-ematics and Computation, Reactor Physics and Environmental Analysis, Madrid,Spain, 27–30 September 1999.

[22] M R Zika and E W Larsen. Fourier mode analysis of slab-geometry transport itera-tions in spatially periodic media. InProc. Int. Topl. Mtg on Advances in Mathematicsand Computation, Reactor Physics and Environmental Analysis, Madrid, Spain, 27–30 September 1999.

[23] K D Lathrop. Ray effects in discrete ordinate equations.Nucl. Sci. Eng., 32:357–369,1968.

[24] K D Lathrop. Remedies for ray effects.Nucl. Sci. Eng., 45:255–268, 1971.

[25] L L Briggs, W F Miller, Jr., and E E Lewis. Ray-effect mitigation in discrete ordinate-like angular finite element approximations in neutron transport.Nucl. Sci. Eng.,57:205–217, 1975.

[26] W F Miller, Jr. and W H Reed. Ray effect mitigation methods for two-dimensionalneutron transport theory.Nucl. Sci. Eng., 92:391–411, 1977.

[27] K F Evans. The spherical harmonics discrete ordinate method for three-dimensionalatmospheric radiative transfer.J. Atmos. Sci., 55:429–446, 1998.

[28] W F Miller, Jr., E E Lewis, and E C Rossow. The application of phase-space finiteelements to the one-dimensional neutron transport equation.Nucl. Sci. Eng., 51:148–156, 1973.

[29] W H Press, S A Teukolsky, W T Vetterling, and B P Flannery.Numerical Recipesin FORTRAN: The Art of Scientific Computing. Cambridge University Press, secondedition, 1992.

[30] C Johnson.Numerical solution of partial differential equations by the finite elementmethod. Cambridge University Press, 1995.

[31] P K Sweby. High resolution schemes using flux limiters for hyperbolic conservationlaws. SIAM J. Numer. Anal., 21:995–1011, 1984.

[32] S T Zalesak. Fully multidimensional flux-corrected transport algorithms for fluids.JCP, 31:335–362, 1979.

[33] R H Szilard and G C Pomraning. Numerical transport and diffusion methods in ra-diative transfer.Nucl. Sci. Eng., 112:256–269, 1992.

[34] W R Martin and J J Duderstadt. Finite element solutions of the neutron transportequation with applications to strong heterogeneities.Nucl. Sci. Eng., 62:371–390,1977.

Figure 1: Comparison of symmetric and asymmetric NIF capusle implosion calculations

Figure 2: Comparison of various finite element discretisations of the streaming operator:1st order step (thick-line), discontinuous (red), continuous (green), and 2nd order discon-tinuous (yellow), continuous (blue)

Figure 3: Comparison of various finite element discretisations for a pure absorber: 1st orderdiscontinuous (red), continuous (green), and 2nd order discontinuous (yellow), continuous(blue) compared with the exact solution.

Figure 4: Comparison of various spatial discretisations for the thermal radiation wave.Radiation temperature (thick lines) and material temperature plotted at times 0.1, 1 and 10:1st order discontinuous (red), 2nd order discontinuous (yellow) and continuous (blue).

Figure 5: Comparison of various angular discretisations for the thermal radiation wave:piecewise constant (red), linear discontinuous (yellow), linear continuous exceptingµ = 0(green) and linear continuous includingµ = 0 (blue).

Figure 6: The effect of mass lumping in angle: un-lumped (left) and lumped (right) inten-sities plotted along the angular mesh lines, at early time (0.1).

Figure 7: The results of the spherical harmonic projection for the thermal radiation wave:P1 (red),P3 (yellow), P5 (green) andP9 (blue) expansions.