Manufactured Solutions in the Thick Diffusion Limit

Spatially analytic SN solutions currently exist only under very limited circumstances. For cases in which analytical solutions may not be available, one can turn to manufactured solutions to test the properties of spatial transport discretization schemes. In particular, we show it is possible to use a manufactured solution to conduct such tests in the thick diffusion limit, even though the computed solution is independent of the problem characteristics. We show that a diffusion limit scaling with a manufactured solution source term results in an expression that is valid in the diffusion limit, though it is not of the standard form used in asymptotic diffusion limit analysis. We then derive a necessary, but not sufficient, condition that must be satisfied in order for a spatial discretization of the transport equation to preserve the thick diffusion limit. This condition is stated in terms of the difference between a numerically computed scalar flux solution compared against a known scalar flux. For a sufficiently diffusive problem and optically thick mesh cells, the necessary condition states that if a spatial discretization of the SN equations has the thick diffusion limit, the norm of the difference in the two solutions must converge to zero with decreasing mesh cell spacing. Based on the first observation that the diffusion limit holds for a manufactured solution source term, the known solution can conveniently be taken to be a manufactured solution in a mesh refinement numerical experiment to check whether a spatial discretization satisfies this condition. We present computational examples that verify our analysis and illustrate the expediency of this approach.