Multilevel methods for transport equations in diffusive regimes

Thomas A. Manteuffel and Klaus Ressel

Computational Mathematics Group, University of Colorado at Denver, Denver, CO 80217-3364


In the diffusion limit, a certain asymptotic limit in which the reciprocal of the total cross section and the absorbtion cross section approach zero, transport theory transitions into diffusion theory. A few mean free pathes away from the boundaries, a solution of the analytic transport equation will converge to a solution of an analytic diffusion equation. For the numerical solution of transport problems it is important to find discretization schemes which have the same property, i.e., the difference scheme for the transport equation must approximate a diffusion operator in this limit.

We consider the single group , steady state, isotropic equation in slab geometry, discretized in angle by discrete ordinates (S_N). A least squares finite element discretization converts these first order equations into a self-adjoint variational formulation. However, by using the moment representation of these semi discrete equations, it can be shown that a least squares discretization does not behave properly in the diffusion limit. For this reason a scaling is applied to the semi discrete equations, so that a least squares discretization will have the correct diffusion limit. A full multigrid solver was developed for the resulting functional. The obtained numerical results are very satisfactory. We believe that the scaling technique proviedes a general framework to find discretization schemes for the transport equation that have the diffusion limit, also for 2 and 3 dimensional geometries.