Deterministic radiative transfer equation solver on unstructured tetrahedral meshes: Efficient assembly and preconditioning

Due to its integro-differential nature, deriving schemes for numerically solving the radiative transfer equation (RTE) is challenging. Most solvers are efficient in specific scenarios: structured grids, simulations with low-scattering materials... In this paper, a full solver, from the discretization of the steady-state monochromatic RTE to the solution of the resulting system, is derived. Using a mixed matrix-ready/matrix-free approach, our solver is able to discretize and solve a 45.7 billion unknown problem on 27 thousand processes in three minutes for a full physics involving scattering, absorption, and reflection. Because of the high dimensionality of the continuous equation, the linear system would have had more than 6×1015 nonzero entries if assembled explicitly. Our approach allows for large memory gains by only storing lower dimension reference matrices. The finite element-based solver is wrapped around open-source software, FreeFEM for discretization, PETSc for linear algebra, and hypre for the algebraic multigrid infrastructure. Overall, deterministic results are presented on arbitrarily-decomposed unstructured grids for radiative transfer problems with scattering, absorbing, and reflecting heterogeneities on up to 27 thousand processes. •Efficient matrix-free discretization of the radiative transfer equation.•Numerically robust preconditioner tested on various unstructured meshes.•Weak and strong scalability proven on up to 27 thousand processes.•Treatment of reflective boundary conditions without orthogonality requirements.