Asymptotic analysis of radiative transfer problems

Abstract.We consider the Pn model to approximate the time dependent transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it has to give the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by Gosse to solve the P1 model without absorption term. It requires the computation of the solution of the steady state Pn equations. This is made by one Monte-Carlo simulation performed outside the time loop. Using formal expansions with respect to a small parameter representing the inverse of the number of mean free path in each cell, the resulting scheme is proved to have the diffusion limit. In order to avoid the CFL constraint on the time step, we give an implicit version of the scheme which preserves the positivity of the zeroth moment.Mathematics Subject Classification. 82C70, 35B40, 74S10.

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“…This regime may be obtained by the introduction of the following scaling in the transport equation [16,17,23]:…”

Section: The P N Equationsmentioning

confidence: 99%

Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit

Cargo

Samba

2010

ESAIM: M2AN

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“…This approximation has been described in [12 and 16], and a first rigorous proof of its validity has been given in [2]. Still, as explained in [2,12,16], this approximation is valid in the interior only. To extend the approximation near the boundary, we are led to study the asymptotic behavior of (1-1).…”

Section: Jr+mentioning

confidence: 99%

“…(2) Even if o was not bounded above, we would have, by using the same method as in Step 3 Even if (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) gives no result on the decay of T¡, we notice that it gives the natural weight for the exponential decay of /, namely I/o.…”

Section: Proposition 3 (Invariant Quantities) If F G Lco(r+;v)+ Satimentioning

confidence: 99%

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The Milne problem for the radiative transfer equations (with frequency dependence)

Golse¹

1987

Trans. Amer. Math. Soc.

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“…In the past, Dorn [15] has used the inverse problem of the time-dependent transport equation for modeling optical tomography. Larsen et al [16] has carried out asymptotic analysis of radiative transfer problems, and Tarvainen et al [17] have worked on finite element modeling of the coupled radiative transfer equation and diffusion approximation considering the time dependency.…”

mentioning

confidence: 99%

Laguerre Runge--Kutta--Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue

Handapangoda

Premaratne

Yeo

et al. 2008

IEEE J. Select. Topics Quantum Electron.

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Abstract-An efficient algorithm for solving the transient radiative transfer equation for laser pulse propagation in biological tissue is presented. A Laguerre expansion is used to represent the time dependency of the incident short pulse. The Runge-KuttaFehlberg method is used to solve the intensity. The discrete ordinates method is used to discretize with respect to azimuthal and zenith angles. This method offers the advantages of representing the intensity with a high accuracy using only a few Laguerre polynomials, and straightforward extension to inhomogeneous media. Also, this formulation can be easily extended for solving the 2-D and 3-D transient radiative transfer equations.

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Asymptotic analysis of radiative transfer problems

Cited by 121 publications

References 27 publications

Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit

Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit

The Milne problem for the radiative transfer equations (with frequency dependence)

Laguerre Runge--Kutta--Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue

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Asymptotic analysis of radiative transfer problems

Cited by 121 publications

References 27 publications

Resolution of the time dependentPnequations by a Godunov type scheme having the diffusion limit

Resolution of the time dependentPnequations by a Godunov type scheme having the diffusion limit

The Milne problem for the radiative transfer equations (with frequency dependence)

Laguerre Runge--Kutta--Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue

Contact Info

Product

Resources

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Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit

Resolution of the time dependentP_nequations by a Godunov type scheme having the diffusion limit