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

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 infinite-dimensional 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

Angular adaptivity with spherical harmonics for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which uses Pn and filtered Pn expansions, allowing for different expansion orders across space/energy. Our spatial discretisation is specifically designed to use less memory than competing DG schemes and also gives us direct access to the amount of stabilisation applied at each node. For filtered Pn expansions, we then use our adaptive process in combination with this net amount of stabilisation to compute a spatially dependent filter strength that does not depend on a priori spatial information. This applies heavy filtering only where discontinuities are present, allowing the filtered Pn expansion to retain high-order convergence where possible. Regular and goal-based error metrics are shown and both the adapted Pn and adapted filtered Pn methods show significant reductions in DOFs and runtime. The adapted filtered Pn with our spatially dependent filter shows close to fixed iteration counts and up to high-order is even competitive with P0 discretisations in problems with heavy advection.Comment: arXiv admin note: text overlap with arXiv:1901.0492

Scalable angular adaptivity for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which for the first time shows evidence of $\mathcal{O}(n)$ scaling in both runtime and memory usage, where $n$ is the number of adapted angles. This adaptivity uses Haar wavelets, which perform structured $h$-adaptivity built on top of a hierarchical P$_0$ FEM discretisation of a 2D angular domain, allowing different anisotropic angular resolution to be applied across space/energy. Fixed angular refinement, along with regular and goal-based error metrics are shown in three example problems taken from neutronics/radiative transfer applications. We use a spatial discretisation designed to use less memory than competing alternatives in general applications and gives us the flexibility to use a matrix-free multgrid method as our iterative method. This relies on scalable matrix-vector products using Fast Wavelet Transforms and allows the use of traditional sweep algorithms if desired