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Journal ArticleDOI

Asymptotic equilibrium diffusion analysis of time-dependent Monte Carlo methods for grey radiative transfer

01 Sep 2004-Journal of Computational Physics (Academic Press Professional, Inc.)-Vol. 199, Iss: 1, pp 175-204
TL;DR: In this paper, the authors apply the same analysis to the Fleck-Cummings, Carter-Forest, and N'kaoua Monte Carlo approximations for grey (frequency-independent) radiative transfer.
About: This article is published in Journal of Computational Physics.The article was published on 2004-09-01. It has received 49 citations till now. The article focuses on the topics: Photon transport in biological tissue & Dynamic Monte Carlo method.
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Journal ArticleDOI
TL;DR: The filtered P"7 solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem in 2D Cartesian geometry and is more robust and less oscillatory than standard P"N solutions and more accurate than discrete ordinates solutions of comparable order.

115 citations

Journal ArticleDOI
TL;DR: In this article, a combination of spectral, velocity-dependent, Implicit Monte Carlo and discrete-diffusion Monte Carlo methods was proposed for neutrino transport calculations in core-collapse supernovae.
Abstract: Monte Carlo approaches to radiation transport have several attractive properties such as simplicity of implementation, high accuracy, and good parallel scaling. Moreover, Monte Carlo methods can handle complicated geometries and are relatively easy to extend to multiple spatial dimensions, which makes them potentially interesting in modeling complex multi-dimensional astrophysical phenomena such as core-collapse supernovae. The aim of this paper is to explore Monte Carlo methods for modeling neutrino transport in core-collapse supernovae. We generalize the Implicit Monte Carlo photon transport scheme of Fleck & Cummings and gray discrete-diffusion scheme of Densmore et al. to energy-, time-, and velocity-dependent neutrino transport. Using our 1D spherically-symmetric implementation, we show that, similar to the photon transport case, the implicit scheme enables significantly larger timesteps compared with explicit time discretization, without sacrificing accuracy, while the discrete-diffusion method leads to significant speed-ups at high optical depth. Our results suggest that a combination of spectral, velocity-dependent, Implicit Monte Carlo and discrete-diffusion Monte Carlo methods represents a robust approach for use in neutrino transport calculations in core-collapse supernovae. Our velocity-dependent scheme can easily be adapted to photon transport.

112 citations

Journal ArticleDOI
TL;DR: A semi-implicit, linear discontinuous Galerkin method for the spherical harmonics (P"N) equations for thermal radiative transfer in planar geometry that is novel in that the material coupling terms are treated implicitly and the streaming operator is treated explicitly using a second-order accurate Runge-Kutta method.

74 citations

Journal ArticleDOI
TL;DR: A robust implicit Monte Carlo (IMC) algorithm based on more accurately updating the linearized equilibrium radiation energy density that can avoid the nonphysical overheating that occurs in standard IMC when the time step is large is developed.

51 citations

Journal ArticleDOI
25 Apr 2016
TL;DR: In this paper, the Implicit Monte Carlo (IMC) equations were re-derived and outfit with a Monte Carlo interpretation, and the IMC equations were compared with other approximate forms of the radiative transfer equations and presented a new demonstration of their equivalence to another well-used linearization.
Abstract: In 1971, Fleck and Cummings derived a system of equations to enable robust Monte Carlo simulations of time-dependent, thermal radiative transfer problems. Denoted the “Implicit Monte Carlo” (IMC) equations, their solution remains the de facto standard of high-fidelity radiative transfer simulations. Over the course of 44 years, their numerical properties have become better understood, and accuracy enhancements, novel acceleration methods, and variance reduction techniques have been suggested. In this review, we rederive the IMC equations—explicitly highlighting assumptions as they are made—and outfit the equations with a Monte Carlo interpretation. We put the IMC equations in context with other approximate forms of the radiative transfer equations and present a new demonstration of their equivalence to another well-used linearization solved with deterministic transport methods for frequency-independent problems. We discuss physical and numerical limitations of the IMC equations for asymptotically ...

50 citations


Cites background from "Asymptotic equilibrium diffusion an..."

  • ...... Asymptotic equilibrium diffusion limit Even in the absence of spatial grid effects, the IMC equations do not preserve the “asymptotic equilibrium diffusion limit” due to the linearization error (Densmore and Larsen, 2004)....

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  • ...…are unconditionally stable and monotone, that the time-dependent source and effective scattering terms can be implemented efficiently, and that there are cases for which they preserve the equilibrium diffusion limit (Cheatham, 2010; Densmore and Larsen, 2004; Martin and Brown, 2002, 2001c;…...

    [...]

  • ...This approach does contain temporal truncation errors due to usingT 4 n+1 as the emission term and time-explicit data for the opacities and emission spectrum, but, because it rigorously treats the nonlinearity, it preserves the thick diffusion limit (Densmore and Larsen, 2004)....

    [...]

  • ...It can be shown that theCarter-Forest equations are temporally exact for linear problems, that they are unconditionally stable and monotone, that the time-dependent source and effective scattering terms can be implemented efficiently, and that there are cases for which they preserve the equilibrium diffusion limit (Cheatham, 2010; Densmore and Larsen, 2004; Martin and Brown, 2002, 2001c; Wollaber and Larsen, 2011)....

    [...]

  • ...This approach does contain temporal truncation errors due to usingT 4n+1 as the emission term and time-explicit data for the opacities and emission spectrum, but, because it rigorously treats the nonlinearity, it preserves the thick diffusion limit (Densmore and Larsen, 2004)....

    [...]

References
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Book
29 Nov 2005

750 citations

Journal ArticleDOI
TL;DR: In this article, a method for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations is presented, which consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter e, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration.
Abstract: A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter e, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in e, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to e−1. Similarly ψ0 decays at a rate proportional to e−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half‐space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from ...

402 citations

Journal ArticleDOI
TL;DR: In this article, a flexible and accurate method for solving nonlinear, frequency-dependent radiative transfer problems by a Monte Carlo technique is developed, based upon the concept of effective scattering, wherein a fraction of the radiative energy absorbed is instantaneously and isotropically reradiated in a manner analogous to a scattering process.

360 citations

Journal ArticleDOI
TL;DR: In this article, a theoretical method is described for assessing the accuracy of transport differencing schemes in highly scattering media with optically thick spatial meshes and numerical results are presented that demonstrate the validity and accuracy of the analysis.

326 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider the case when the mean free path is small compared to typical lengths in the domain (e.g., the size of a reactor) and derive a uniform asymptotic expansion of the solution of the problem.
Abstract: We consider initial−boundary value and boundary value problems for transport equations in inhomogeneous media. We consider the case when the mean free path is small compared to typical lengths in the domain (e.g., the size of a reactor). Employing the boundary layer technique of matched asymptotic expansions, we derive a uniform asymptotic expansion of the solution of the problem. In so doing we find that in the interior of the domain, i.e., away from boundaries and away from the initial line, the leading term of the expansion satisfies a diffusion equation which is the basis of most computational work in reactor design. We also derive boundary conditions appropriate to the diffusion equation. Comparisons with existing results such as the asymptotic and P1 diffusion theories, the PN approximation, and the extrapolated end point condition for these approximations, are made. Finally the uniform validity of our expansions is proved, thus yielding the desired error estimates.

132 citations