On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer

Abstract

We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinitedimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for plane-parallel geometries with isotropic scattering, but the approach and proofs generalize to multi-dimensional problems and more general scattering operators verbatim.

On Robustly Convergent and Efficient Iterative Methods for Anisotropic Radiative Transfer

“…Common discretization methods can be classified into two main approaches based on their semidiscretization in s. The spherical harmonics method [5,19,35] approximates the solution u by a truncated series of spherical harmonics, which allows for spectral convergence for smooth solutions. For non-smooth solutions, which is the generic situation, local approximations in s can be advantageous, which is achieved, e.g., by discrete ordinates methods [26,35,43,44,46], continuous Galerkin methods [7], the discontinuous Galerkin (DG) method [24,32,40], iteratively refined piecewise polynomial approximations [13], or hybrid methods [12,30].…”

“…Effective DSA schemes rely on consistent discretization of the corresponding diffusion approximation, see [40,48] for isotropic scattering, and [41] for two-dimensional problems with anisotropic scattering. The latter employs a modified interior penalty DG discretization for the corresponding diffusion approximation, which has also been used in [47] where it is, however, found that their DSA scheme becomes less effective for highly heterogeneous optical parameters.…”

“…A discrete analysis of DSA schemes for high-order DG discretizations on possibly curved meshes, which may complicate the inversion of the transport part, can be found in [28]. In the variational framework of [40] consistency is automatically achieved by subspace correction instead of finding a consistent discretization of the diffusion approximation. This variational treatment allowed to prove convergence of the corresponding iteration and numerical results showed robust contraction rates, even in multi-dimensional calculations with heterogeneous optical parameters.…”

“…It is the purpose of this paper to generalize the approach of [40] to the anisotropic scattering case, which requires non-trivial extensions as outlined in the next section.…”

“…In this paper we focus on the construction of robustly and provably convergent efficient iterative schemes for the radiative transfer equation with anisotropic scattering. To describe our approach, let us introduce the linear system that we need to solve, which stems from a mixed finite element discretization of (1)-(3) using discontinuous polynomials on the sphere [17,40], i.e.,…”