We examine the accuracy of a modified finite volume method compared to analytical and Monte Carlo solutions for solving the radiative transfer equation. The model is used for predicting light propagation within a two-dimensional absorbing and highly forward-scattering medium such as biological tissue subjected to a collimated light beam. Numerical simulations for the spatially resolved reflectance and transmittance are presented considering refractive index mismatch with Fresnel reflection at the interface, homogeneous and two-layered media. Time-dependent as well as steady-state cases are considered. In the steady state, it is found that the modified finite volume method is in good agreement with the other two methods. The relative differences between the solutions are found to decrease with spatial mesh refinement applied for the modified finite volume method obtaining <2.4%. In the time domain, the fourth-order Runge-Kutta method is used for the time semi-discretization of the radiative transfer equation. An agreement among the modified finite volume method, Runge-Kutta method, and Monte Carlo solutions are shown, but with relative differences higher than in the steady state.

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SUMMARYA ÿnite di erence solution for a system of non-linear integro-di erential equations modelling the steadystate combined radiative-conductive heat transfer is proposed. A new backward-forward ÿnite di erence scheme is formulated for the Radiative Transfer Equation. The non-linear heat conduction equation is solved using the Kirchho transformation associated with a centred ÿnite di erence scheme. The coupled system of equations is solved using a ÿxed-point method, which relates to the temperature ÿeld. An application on a real insulator composed of silica ÿbres is illustrated. The results show that the method is very e cient.

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This article deals with a numerical solution for combined radiation and conduction heat transfer in a grey absorbing and emitting medium applied to a two-dimensional domain using triangular meshes. The radiative transfer equation was solved using the high order Discontinuous Galerkin method with an upwind numerical flux. The energy equation was discretized using a high order finite element method. Stability and error analysis were performed for the Discontinuous Galerkin method to solve radiative transfer equation. A new algorithm to solve the nonlinear radiative-conductive heat transfer systems was introduced and different types of boundary conditions were considered in numerical simulations. The proposed technique's high performance levels in terms of accuracy and stability are discussed in this paper with numerical examples given.

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