An HLLC Scheme to Solve The M
                     1 Model of Radiative Transfer in Two Space Dimensions

Typical external radiotherapy treatments consist in emitting beams of energetic photons targeting the tumor cells. Those photons are transported through the medium and interact with it. Such interactions affect the motion of the photons but they are typically weakly deflected which is not well modeled by standard numerical methods.The present work deals with the transport of photons in water. The motion of those particles is modeled by an entropy-based moment model, i.e. the M 1 model. The main difficulty when constructing numerical approaches for photon beam modelling emerges from the significant difference of magnitude between the diffusion effects in the forward and transverse directions. A numerical method for the M 1 equations is proposed with a special focus on the numerical diffusion effects.

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“…This can be verified by reproducing the computations of [14,1]. The present correction uses similar ideas as the ones proposed in [6,2].…”

Section: A Correction To Accurately Model Transverse Diffusionsupporting

confidence: 65%

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

Pichard

Dubroca

Brull

et al. 2018

Theory, Numerics and Applications of Hyperbolic Problems II

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“…The HLLC flux-a modification of the HLL flux [2,8,24] The HLLC flux based on the approximate Riemann solver is a modification to account for the shortcoming of the HLL flux, offset the influence of intermediate waves. In addition to the wave speed estimates s L and s R in the HLL solver, an estimate s * for the speed of the middle wave is need.…”

Section: 2mentioning

confidence: 99%

A Numerical Study for the Performance of the WENO Schemes Based on Different Numerical Fluxes for the Shallow Water Equations

Lu¹,

Qiu²,

Wang³

2010

JCM

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In this paper we investigate the performance of the weighted essential non-oscillatory (WENO) methods based on different numerical fluxes, with the objective of obtaining better performance for the shallow water equations by choosing suitable numerical fluxes. We consider six numerical fluxes, i.e., Lax-Friedrichs, local Lax-Friedrichs, Engquist-Osher, Harten-Lax-van Leer, HLLC and the first-order centered fluxes, with the WENO finite volume method and TVD Runge-Kutta time discretization for the shallow water equations. The detailed numerical study is performed for both one-dimensional and two-dimensional shallow water equations by addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities.Mathematics subject classification: 65M60, 65M99, 35L65.

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“…By using a closure defined from a minimisation entropy principle, we obtain the M 1 model [33,34,63]. The M 1 model is largely used in various applications such as radiative transfer [7,24,35,71,72,79,80] or electronic transport [33,63]. The M 1 model is known to satisfy fundamental properties such as the positivity of the first angular moment, the flux limitation and conservation of total energy.…”

Section: Introductionmentioning

confidence: 99%

“…The AP frame was also largely extended to the quasi-neutral limit [26][27][28][29][30]43]. In [7], an HLLC scheme is proposed to solve the M 1 model of radiative transfer in two space dimensions. The HLLC approximate Riemann solver considered and relevant numerical approximations of extreme wavespeeds give the asymptotic-preserving property.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

et al. 2017

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Abstract. This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic M1 model in the diffusive regime. The case without electric field and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a numerical scheme which also satisfies the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diffusive limit recovering the correct diffusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical HLL [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35-61.] scheme usually used for the electronic M1 model. It is shown that the new scheme gives comparable results with respect to the HLL scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the HLL scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.Mathematics Subject Classification. 65C20, 65M12.

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An HLLC Scheme to Solve The M 1 Model of Radiative Transfer in Two Space Dimensions

Cited by 78 publications

References 17 publications

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

A Numerical Study for the Performance of the WENO Schemes Based on Different Numerical Fluxes for the Shallow Water Equations

Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases

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An HLLC Scheme to Solve The M 1 Model of Radiative Transfer in Two Space Dimensions

Cited by 78 publications

References 17 publications

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

On the Transverse Diffusion of Beams of Photons in Radiation Therapy

A Numerical Study for the Performance of the WENO Schemes Based on Different Numerical Fluxes for the Shallow Water Equations

Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: Particular cases

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Asymptotic-preserving well-balanced scheme for the electronic M₁ model in the diffusive limit: Particular cases