Space-angle discontinuous Galerkin method for radiative transfer between concentric cylinders

Abstract

The integro-differential radiative transfer equation (RTE) for concentric cylinders problem involving scattering, absorption and emission is solved using the discontinuous Galerkin (DG) finite element method (FEM). The space-angle DG method directly solves the cylindrically-symmetric RTE as a three-dimensional problem, where a 1D spatially domain in radial distance r is twice extruded in the cosine of polar angle (μ) and the difference in azimuthal angle (φ˜) directions. Thus, the method has a higher accuracy than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods. This is reflected by numerically verified convergence rate of p+1 for smooth problems and space-angle polynomial interpolation order of p. The axisymmetric RTE formulation is more complicated than the plane-parallel formulation, for having two independent angle directions (μ and φ˜) and an extra derivative term with respect to φ˜ in the differential equation. This results in a complex characteristic structure in r−φ˜ plane with strong discontinuity lines in radiation intensity I. A method of characteristics is formulated and implemented to verify the DG formulation and demonstrate its accuracy when such strong discontinuities persist in the solution, specifically when there are no scattering and absorption terms. The relaxation of inter-element continuity constraint of continuous FEMs by this DG method implies its superiority in numerically capturing such discontinuities. Finally, a benchmark problem pertained to heat radiation in a gray gas and another one with nonzero phase function demonstrate the effectiveness of the method in modeling black-body and scattering angular integration terms.