Space-angle discontinuous Galerkin method for plane-parallel radiative transfer equation

Abstract

The radiative transfer equation (RTE) for a plane-parallel problem involving scattering, absorption and radiation is solved using the discontinuous Galerkin (DG) finite element method (FEM). Both space and angle directions are discretized by the DG method. Thus, while the method has a higher accuracy in angle direction than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods, it removes the continuity constraint implied by the form of basis function for continuous FEMs in space and angle. The discrete formulation of the problem is presented for nonzero phase function and a variety of boundary conditions. The numerical results demonstrate a p+1 convergence rate when the intensity is interpolated by an order p polynomial in both space and angle. The method is validated against exact solutions, and compared with other space-angle and hybrid FEMs for a few benchmark problems. The appropriateness of the DG formulation for problems with discontinuous solution is demonstrated by solving a problem with delta source term, where an in-element averaging of the source term eliminates negative intensity values for high order elements. Finally, a problem with an angular-line source term and a convergence study where the solution order is zero in angle are used to further demonstrate the advantages of the high order space-angle DG formulation; for the convergence study problem, the error was reduced by about 13 binary orders of magnitude, by increasing the order in both space and angle, rather than in space only.