ASYMPTOTIC SOLUTIONS OF NUMERICAL TRANSPORT PROBLEMS IN OPTICALLY THICK, DIFFUSIVE REGIMES
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We present an asymptotic analysis of spatial differencing schemes for the discrete-ordinates equations, for diffusive media with spatial cells that are not optically thin. Our theoretical tool is an asymptotic expansion that has previously been used to describe the transition from analytic transport to analytic diffusion theory for such media. To introduce this expansion and its physical rationale, we first describe it for the analytic discrete-ordinates equations. Then, we apply the expansion to the spatially discretized discrete-ordinates equations, with the spatial mesh scaled in either of two physically relevant ways such that the optical thickness of the spatial cells is not small. If the result of either expansion is a legitimate diffusion description for either the cell-average or cell-edge fluxes, then we say that the appropriate flux has the appropriate diffusion limit; otherwise, we say it does not. We consider several transport differencing schemes that are applicable in neutron transport and thermal radiation applications. We also include numerical results which demonstrate the validity of our theory and show that differencing schemes that do have a particular diffusion limit are substantially more accurate, in the regime described by the limit, than those that do not. 1987.
