We introduce a new treatment of the difference formulation[1] for photon radiation transport without scattering in 1d slab geometry that is closely analogous to that of Fleck and Cummings[2] for the traditional formulation. The resulting form is free of implicit source terms and has the familiar effective scattering of the field of transport

Daffin, F.

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UCRL-TR-225368An application of Fleck effectivescattering to the difference formulation forphoton transportFrank DaffinOctober 18, 2006Disclaimer   This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.      This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.  An application of Fleck effective scattering to thedifference formulation for photon transportFrank DaffinUniversity of CaliforniaLawrence Livermore National LaboratoryLivermore, California 94550E-mail: daffin1@llnl.govWe introduce a new treatment of the difference formulation[1] for photon radiationtransport without scattering in 1-d slab geometry that is closely analogous to thatof Fleck and Cummings[2] for the traditional formulation. The resulting form isfree of implicit source terms and has the familiar effective scattering of the field oftransport.Key Words: difference formulation; Monte Carlo methods; photon transport1. INTRODUCTIONSymbolic-Implicit Monte Carlo (SIMC) treatments[3] of the difference formulation ofphoton transport[1] have demonstrated impressive levels of noise reduction. As it has beenapplied to the difference formulation in local thermodynamic equilibrium (LTE), the SIMCmethod enhances the numerical stability of those calculations at the expense of a non-linearsystem solve at each time step. For well-conditioned systems discretized into relativelyfew zones, the computational effort is typically dominated by particle tracking; the expenseof the non-linear solver is inconsequential, and SIMC applied to either the standard orthe differnece formulation enables the use of large time steps while maintaining numericalstability. This can reduce overall computation time.Notwithstanding considerations of scattering and connectivity in higher dimensions, forproblems of sufficiently large spacial extent the matrix to be inverted will be band-diagonal,the width of which is limited by int(cδt/δx)+1, where c is the propagation speed, δt is thetime step size, and δx is the size of the zones used. Additionally, optically thick problemsguarantee that the reach of influence of Monte Carlo particles from symbolic sources willbe limited by attenuation, especially when techniques are utilized for eliminating particlesof insignificant weight. Either way, the band-limiting tends to ease the iterative burdenby reducing the influence of the non-linear portions of the system solve. However, if oneconsiders the costs associated with inverting the large matrices storing Monte Carlo sumsfor a finely-discretized system, then one may find that multi-dimensional, high-fidelitycalculations using SIMC to be impractical.12To improve the numerical stability of the Monte Carlo solution of the standard photontransport equation, Fleck and Cummings[2] introduced a method of that converts a portionof the absorption into an artificial scattering term, weakening the coupling between theradiation transport and material energy equations. While the treatment, now called ImplicitMonte Carlo (IMC), is not perfect[4][5] and can be costly in optically thick problems wherethere are many effective scattering events, it is unconditionally stable and does not requirethe extra storage or computational burden of the solution of the non-linear system.2. FLECK-LIKE TREATMENT OF THE DIFFERENCE FORMULATIONFor optically thick systems in LTE, the departure of the specific photon intensity, I , fromthe planck field, B, is small[1]. The difference formulation takes advantage of this bytransforming from the photon field, which is large in these systems, to the difference field,D = I − B. It is the smallness of the difference field that we think will work in favor ofthe IMC method.In the SIMC treatments of the difference formulation currently known to the author[6],the reduction in noise is due to the small size of the field and freedom from having tosample the material’s emissivity for emission frequency, a task made difficult by the richstructure in frequency the function exhibits for real materials. We believe similar benefitsmay be realized in an IMC treatment, since the size of the field–determined by the physicsof the problem and not the solution method–should be small. With fewer Monte Carloparticles (samples) of the field required in the calculation to achieve a desired level ofnoise, there should be fewer scattering events to calculate too. We conjecture that twoof IMC’s disadvantages in optically thick problems, namely it’s Monte Carlo noise andfrequent effective scattering, may be mitigated by an analogous treatment of the differenceformulation.The source terms of the difference formulation are treated implicitly in [6] to improvenumerical stability, and it is this implicit treatment that leads to the system of non-linearequations that must be solved for the temperatures of the cells at the end of each time step.Beginning with the difference formulation for photon transport without scattering in 1-dslab geometry,1c∂D∂t+ µ∂D∂x= −σD −(1c∂∂t+ µ∂∂x)B (1)we closely follow the development of the IMC treatment of the standard photon transportformulation[2].The black body field B can be expressed as 12c ur, where ur = aT4 is the radiationenergy density. Closely following Fleck[2], the energy density equation is written as∂ur∂t= βσ(∫ 1−1Idµ+ cur)+ βS (2)where S is an external energy source andβ =∂ur∂um(3)3with um being the material energy density. Integrating both sides of Eq. 2 and taking theappropriate time-centered values, leads to an expression for the radiation energy densityuγr =αβσ∆t1 + αβσc∆t∫ 1−1Iλdµ+unr1 + αβσc∆t+αβ∆tSγ1 + αβσc∆t=2Bγc(4)where the superscripts λ, γ and the factor α define the time-centering of the mean valuesof I , S, B and ur, e.g.:uγr = α(un+1r − unr)+ unr (5)with n being the time-integration cycle index and ∆t = tn+1 − tn. Solving for Bγ givesBγ =12cαβσ∆t1 + αβσc∆t∫ 1−1Iλdµ+Bn1 + αβσc∆t+( 12cαβ∆t1 + αβσc∆t)Sγ (6)Substituting this expression for B into Eq. (1) gives1c∂D∂t+ µ∂D∂x= −σD−(1c∂∂t+ µ∂∂x)( 12cαβσ∆t1 + αβσc∆t∫ 1−1Iλdµ+Bn1 + αβσc∆t+12cαβ∆tSγ1 + αβσc∆t)= −σD −(1c∂∂t+ µ∂∂x)( 12cαβσ∆t1 + αβσc∆t∫ 1−1Iλdµ)−11 + αβσc∆t(1c∂∂t+ µ∂∂x)Bn −12cαβ∆t1 + αβσc∆t(1c∂∂t+ µ∂∂x)Sγ (7)On the right-hand side, the term(1c∂∂t+ µ∂∂x)( 12cαβσ∆t1 + αβσc∆t∫ 1−1Iλdµ)(8)may be written as ( 12cαβσ∆t1 + αβσc∆t)[(1c∂∂t+ µ∂∂x)∫ 1−1Iλdµ](9)Moving the streaming operator within the integral( 12cαβσ∆t1 + αβσc∆t)[∫ 1−1(1c∂∂t+ µ∂∂x)Iλdµ](10)or ( 12cαβσ∆t1 + αβσc∆t)[∫ 1−11c∂Iλ∂t+1µ∂Iλ∂xdµ](11)The integrand looks familiar; it’s just part of the standard transport equation:1c∂I∂t+ µ∂I∂x= −σ (I −B) = −σD (12)4whereD is the difference field in the difference formulation. Thus, the term now becomes( 12cαβσ∆t1 + αβσc∆t)[∫ 1−1−σDλdµ](13)which looks like an effective scattering term for the D-field with scattering cross-sectionσs =(cαβσ∆t1 + αβσc∆t)σ (14)The reader will recognize this as the same effective scattering cross-section that appears inthe transport equation in the work by Fleck and Cummings[2]. The term11 + αβσc∆t(1c∂∂t+ µ∂∂x)Bn (15)is close to the original form for the new source terms in the difference formulation inLTE [6], but here there are no implicit terms Bn+1. The last term is the contribution fromexternal sources,12cαβ∆t1 + αβσc∆t(1c∂∂t+ µ∂∂x)Sγ (16)which is usually known or specified. The transport equation now has the form1c∂D∂t+ µ∂D∂x= −σD + 12σs∫ 1−1Dλdµ− 11 + αβσc∆t(1c∂∂t+ µ∂∂x)Bn− 12(cαβ∆t1 + αβσc∆t)(1c∂∂t+ µ∂∂x)Sγ (17)which is free of implicit terms, but has an effective scattering term for the D-field.As ∆t→∞, σs → σ and the transport equation becomes1c∂D∂t+ µ∂D∂x= −σD + 12σ∫ 1−1Dλdµ − 12(1σ)(1c∂∂t+ µ∂∂x)Sγ (18)Note the absence of source terms from the streaming operator on B:(1c∂∂t+ µ∂∂x)Bthe new source terms of the difference formulation. The 1/∆t-dependence of this sourceterm in Eq.17 may lead to an unphysical decoupling between the material energy and thedifference field for large time-step sizes, a behavior observed in the IMC method appliedto the standard formulation[4]. As∆t→ 0, σs → 0 and we recover the original differenceformulation of the radiation transport equation.1c∂D∂t+ µ∂D∂x= −σD −(1c∂∂t+ µ∂∂x)B (19)3. CONCLUDING REMARKSWe have shown how the simple application of methodology used in [2] to improve thenumerical stability of non-linear radiation transport has eliminated symbolic source terms5in the difference formulation. But the old aphorism “there is no free lunch” holds heretoo: one must still deal with the scattering of theD-field and all the well-known problemsit invites (radon-walk techniques [7] may reduce the burden caused by scattering) as wellas the possible problems with the new 1/∆t-dependence of the source terms for largetime steps. It would be interesting to see if an analysis similar to [4] could be applied.Nonetheless, it is our hope that the relatively few Monte Carlo samples required of thedifference formulation will mean fewer scattering events for the D-field in some of themost challenging problems for the standard formulation–namely optically thick systemsin LTE. Further, since the method is free of a system solve, there may be a net savings incomputational effort over an SIMC implementation of the difference formulation when onetries to solve problems composed of many cells.We close by noting that Nick Gentile of this laboratory and Richard Smedley-Stevensonof AWE are busy mining a similar vein.REFERENCES1. Abraham Szo¨ke and Eugene D. Brooks III, The Transport Equation in Optically Thick Media, Journal ofQuantitative Spectroscopy & Radiative Transfer, Vol. 91, 2005, pp. 95-110.2. J. A. Fleck, Jr and J. D. Cummings, An implicit Monte Carlo Scheme for Calculating Time and FrequencyDependent Non Linear Radiation Transport, Journal of Computational Physics, Vol. 8, Issue 3, December1971, pp. 313-342.3. Eugene D. Brooks III, Symbolic Implicit Monte Carlo, Journal of Computational Physics, Vol. 83, No. 2,August 1989, pp. 433-446.4. EdwardW. Larsen and BertrandMercier, Analysis of a Monte Carlo Method for Nonlinear Radiative Transfer,Journal of Computational Physics, Vol. 71, Issue 1, July 1987, pp. 50-64.5. Jeffery D. Densmore and Edward W. Larsen, Asymptotic Equilibrium Diffusion Analysis of Time-DependentMonte Carlo Methods for Grey Radiative Transfer, Journal of Computational Physics, Vol. 199, Issue 1,September 2004, pp. 175-204.6. Eugene D. Brooks III, Michael Scott McKinley, Frank Daffin and Abraham Szo¨ke, Symbolic Implicit MonteCarlo Radiation Transport in the Difference Formulation: a Piecewise Constant Discretization, Journal ofComputational Physics, Vol. 205, Issue 2, May 2005, pp. 737-754.7. J. A. Fleck, Jr. and E. H. Canfield, A RandomWalk Procedure for Improving the Computational Efficiency ofthe Implicit Monte Carlo Method for Nonlinear Radiation Transport, Journal of Computational Physics, Vol.54, Issue 3, June 1984, pp. 508-523.

An application of Fleck effective scattering to the difference formulation for photon transport

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An application of Fleck effective scattering to the difference formulation for photon transport

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