Diffusion approximation and computation of the critical size

Abstract

Abstract. This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem.Introduction. The computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts. First, the computation of the critical size is much easier for the diffusion approximation than for the original transport equation. Second, for the critical size one can consider a host media X = nXQ with X0 given and 17 a positive number. The problem of the critical size is then reduced to the computation of the parameter zj. It turns out that when the transport operator is almost conservative, the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate.On the other hand the "physical" boundary condition for the diffusion approximation is of the form

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