Sparse tensor phase space Galerkin approximation for radiative transport

Abstract

We develop, analyze, and test a sparse tensor product phase space Galerkin discretization framework for the stationary monochromatic radiative transfer problem with scattering. The mathematical model describes the transport of radiation on a phase space of the Cartesian product of a typically three-dimensional physical domain and two-dimensional angular domain. Known solution methods such as the discrete ordinates method and a spherical harmonics method are derived from the presented Galerkin framework. We construct sparse versions of these well-established methods from the framework and prove that these sparse tensor discretizations break the “curse of dimensionality”: essentially (up to logarithmic factors in the total number of degrees of freedom) the solution complexity increases only as in a problem posed in the physical domain alone, while asymptotic convergence orders in terms of the discretization parameters remain essentially equal to those of a full tensor phase space Galerkin discretization. Algorithmically we compute the sparse tensor approximations by the combination technique. In numerical experiments on 2 + 1 and 3 + 2 dimensional phase spaces we demonstrate that the advantages of sparse tensorization can be leveraged in applications.