Asymptotic diffusion limit of the symbolic Monte-Carlo method for the transport equation

Clouët, J.-F.; Samba, G.

doi:10.1016/j.jcp.2003.10.008

Cited by 17 publications

(35 citation statements)

References 9 publications

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“…We have extended the piecewise linear treatment of the material state variable for the Symbolic Implicit Monte Carlo (SIMC) transport method, originally developed by Clouet and Samba [6], to the case of the fully non-linear time dependent equations of photon transport, under conditions of local thermodynamic equilibrium (LTE), in the difference formulation. The use of the difference formulation removes two hurdles preventing practical application of the method for the standard formulation of photon transport: the difficulty of sampling a transport source term that depends upon the material opacity at the location sampled, and the Monte Carlo noise that becomes an impasse when attempting to use the method in the diffusion limit.…”

Section: Discussionmentioning

confidence: 99%

“…(1) and (2), using a SIMC method that employs a piecewise linear finite element treatment of the material state variable. Our method is similar to that of Clouet and Samba [6]. In a nutshell, our algorithm is organized as follows:…”

Section: Solution Methodsmentioning

confidence: 99%

“…The spatial basis functions, permitting discontinuity at zone boundaries, are the same as those that were employed by Clouet and Samba [6] in their linearized treatment of the transport equation. We have explored the addition of temporal basis functions in an attempt to extend the second order accurate treatment to the time variable, but find that the result is only conditionally stable.…”

Section: The Basis Functionsmentioning

confidence: 99%

“…This defect causes a faster-than-physical propagation velocity for a Marshak (thermal) wave, and excessive energy transport even under steady state conditions. Clouet and Samba [6] have shown that a piecewise linear (in space) discretization of the material state variable produces the correct diffusion limit. They did this in the context of a linearized form of the transport problem for a time independent solution; see [7] for an example of this linearization in a diffusion context.…”

Section: Introductionmentioning

confidence: 99%

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Piecewise linear discretization of Symbolic Implicit Monte Carlo radiation transport in the difference formulation

Brooks

Szke

Peterson

2006

Journal of Computational Physics

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We describe a Monte Carlo solution for time dependent photon transport, in the difference formulation with the material in local thermodynamic equilibrium (LTE), that is piecewise linear in its treatment of the material state variable. Our method employs a Galerkin solution for the material energy equation while using Symbolic Implicit Monte Carlo (SIMC) to solve the transport equation. In constructing the scheme, one has the freedom to choose between expanding the material temperature, or the equivalent black body radiation energy density at the material temperature, in terms of finite element basis functions. The former provides a linear treatment of the material energy while the latter provides a linear treatment of the radiative coupling between zones. Subject to the conditional use of a lumped material energy in the vicinity of strong gradients, possible with a linear treatment of the material energy, our approach provides a robust solution for time dependent transport of thermally emitted radiation that can address a wide range of problems. It produces accurate results in the diffusion limit.

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Section: Discussionmentioning

confidence: 99%

Section: Solution Methodsmentioning

confidence: 99%

Section: The Basis Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Piecewise linear discretization of Symbolic Implicit Monte Carlo radiation transport in the difference formulation

Brooks

Szke

Peterson

2006

Journal of Computational Physics

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“…The original form of SIMC which relies on piecewise constant basis functions does not approach the diffusion limit and suffers from teleportation error problems [9]. A piecewise linear SIMC was shown to approach the diffusion limit and is much less affected by teleportation error [8,20]. Teleportation error also occurs in IMC for small time steps [10].…”

Section: Introductionmentioning

confidence: 99%

An extension of implicit Monte Carlo diffusion: Multigroup and the difference formulation

Cleveland

Gentile

Palmer

2010

Journal of Computational Physics

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Implicit Monte Carlo (IMC) and Implicit Monte Carlo Diffusion (IMD) are approaches to the numerical solution of the equations of radiative transfer. IMD was previously derived and numerically tested on grey, or frequency-integrated problems [1]. In this research, we extend Implicit Monte Carlo Diffusion (IMD) to account for frequency dependence, and we implement the difference formulation [2] as a source manipulation variance reduction technique. We derive the relevant probability distributions and present the frequency dependent IMD algorithm, with and without the difference formulation. The IMD code with and without the difference formulation was tested using both grey and frequency dependent benchmark problems. The Su and Olson semi-analytic Marshak wave benchmark was used to demonstrate the validity of the code for grey problems [3]. The Su and Olson semi-analytic picket fence benchmark was used for the frequency dependent problems [4]. The frequency dependent IMD algorithm reproduces the results of both Su and Olson benchmark problems. Frequency group ref nement studies indicate that the computational cost of ref ning the group structure is likely less than that of group ref nement in deterministic solutions of the radiation diffusion methods. Our results show that applying the difference formulation to the IMD algorithm can result in an overall increase in the f gure of merit for frequency dependent problems. However, the creation of negatively weighted particles from the difference formulation can cause signif cant numerical instabilities in regions of the problem with sharp spatial gradients in the solution. An adaptive implementation of the difference formulation may be necessary to focus its use in regions that are at or near thermal equilibrium.

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Accurate and Efficient Radiation Transport in Optically Thick Media – by Means of the Symbolic Implicit Monte Carlo Method in the Difference Formulation

Szöke

Brooks

McKinley

et al.

Lecture Notes in Computational Science and Engineering

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The equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit. One source of this difficulty is that in thick, absorbing media thermal emission is almost completely balanced by strong absorption. In a previous publication [SB03], the photon transport equation was written in terms of the deviation of the specific intensity from the local equilibrium field. We called the new form of the equations the difference formulation. The difference formulation is rigorously equivalent to the original transport equation. It is particularly advantageous in thick media, where the radiation field approaches local equilibrium and the deviations from the Planck distribution are small. The difference formulation for photon transport also clarifies the diffusion limit. In this paper, the transport equation is solved by the Symbolic Implicit Monte Carlo (SIMC) method and a comparison is made between the standard formulation and the difference formulation. The SIMC method is easily adapted to the derivative source terms of the difference formulation, and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media.

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Asymptotic diffusion limit of the symbolic Monte-Carlo method for the transport equation

Cited by 17 publications

References 9 publications

Piecewise linear discretization of Symbolic Implicit Monte Carlo radiation transport in the difference formulation

Piecewise linear discretization of Symbolic Implicit Monte Carlo radiation transport in the difference formulation

An extension of implicit Monte Carlo diffusion: Multigroup and the difference formulation

Accurate and Efficient Radiation Transport in Optically Thick Media – by Means of the Symbolic Implicit Monte Carlo Method in the Difference Formulation

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