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 approxi- mation is of the form

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Diffusion approximation and computation of the critical size

Neutron Transport with Anisotropic Scattering

Various concepts of ellipticity of finite-difference approximations to general elliptic partial differential systems are reviewed and introduced, and rules are given for the construction of stable schemes with high approximation orders, even for singular perturbation problems. Fast multi-grid solvers for these discrete schemes are described. These solvers also provide a convenient way of separating the questions of accuracy and stability (using, for example, both central and upstream differencing). The local mode analysis, which accurately predicts the efficiency of multi-grid solvers, is presented. Concrete examples are given in terms of Cauchy-Riemann equations and the steady-state incompressible Navier-Stokes equations. Their multi-grid solution, based on new єdistributive” relaxation schemes, costs about seven work-units.

Multigrid Solutions to Elliptic Flow Problems

Abstract time-dependent transport equations☆

With the use of invariance principles in a systematic fashion, we shall derive not only new analytic formulations of the classical particle processes, those of transport theory, radiative transfer, random walk, multiple scattering, and diffusion theory, but, in addition, new computational algorithms which seem well fitted to the capabilities of digital computers. Whereas the usual methods reduce problems to the solution of systems of linear equations, we shall try to reduce problems to the iteration of nonlinear transformations.Although we have analogous formulations of wave processes, we shall reserve for a second paper in this series a detailed and extensive treatment of this part of mathematical physics.

Joseph Lehner

Papers

On the spectrum of an unsymmetric operator arising in the transport theory of neutrons

An unsymmetric operator arising in the theory of neutron diffusion