An Adaptive Mesh Refinement Algorithm for the Radiative Transport Equation

The discrete ordinates form of the radiative transport equation (RTE) is spatially discretized and solved using an adaptive mesh refinement (AMR) algorithm. This technique permits local grid refinement to minimize spatial discretization error of the RTE. An error estimator is applied to define regions for local grid refinement; overlapping refined grids are recursively placed in these regions; and the RTE is then solved over the entire domain. The procedure continues until the spatial discretization error has been reduced to a sufficient level. The following aspects of the algorithm are discussed: error estimation, grid generation, communication between refined levels, and solution sequencing. This initial formulation employs the step scheme and is valid for absorbing and isotropically scattering media in two-dimensional enclosures. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single-grid algorithm. For two simple benchmark problems, the AMR algorithm maintains the convergence characteristics of the standard single-grid algorithm, but it does not provide any efficiency gains due to a lack of disparate spatial scales. In a third, more localized problem, however, the AMR algorithm demonstrates significant memory and CPU time reductions.