Comparative Study on Accuracy and Solution Cost of the First/Second-Order Radiative Transfer Equations Using the Meshless Method

The direct collocation meshless (DCM) method is applied to solve and evaluate the performance of the second-order radiative transfer equation (SORTE) proposed by Zhao and Liu (Numer. Heat Transfer B, vol. 51, pp. 391-409, 2007). The SORTE transforms the original first-order radiative transfer equation (FORTE) into a form similar to a diffusion equation, so no additional artificial diffusion or upwinding treatment is needed in the numerical discretization for stabilization. In order to investigate the accuracy and cost of the direct collocation meshless method based on the SORTE, two typical radiative transfer problems are considered. These cases are also solved by the DCM approach and the least-squares collocation meshless (LSCM) approach based on the FORTE. Numerical results show that the DCM approach based on the SORTE is more accurate and stable than the DCM approach and the LSCM approach based on the FORTE. The convergence rate of the SORTE-based methods with increase of collocation point number is faster than that of the FORTE-based methods. For obtaining the same target accuracy, the DCM approach based on the SORTE is more efficient than the other two meshless methods based on the FORTE. In addition, the DCM approach based on the SORTE also exhibits higher accuracy in solving radiative transfer problems with complex geometries or discontinuous temperature distributions along the boundary.