Finite Volume Solution of the Radiative Transfer Equation Using NVD and TVD Schemes in Unstructured Grids

Abstract

Discretization schemes based on the normalized variable diagram (NVD) and total variation diminishing (TVD) schemes are applied to the solution of the radiative transfer equation (RTE) in unstructured grids. The success of the application of these schemes to Cartesian grids has been previously demonstrated. However, their extension to unstructured grids is not straightforward. A few methods to overcome this difficulty have been proposed, and successfully applied to the solution of a scalar transport equation and to the Navier-Stokes equations. These methods are applied here to the solution of the RTE, along with a new one, recently proposed, for several test cases in which the analytical solution or other reliable numerical solutions are available for comparison. The results demonstrate that although the NVD and TVD schemes are much more accurate than the step and mean flux interpolation schemes, their order of accuracy in the case of unstructured grids is lower than in Cartesian grids. Moreover, in contrast to Cartesian grids, the NVD and TVD schemes are not strictly bounded in unstructured grids, and unphysical solutions may occur. The alternative method proposed to implement the NVD and TVD schemes is generally more accurate than previous ones, but also computationally more demanding.