Angular adaptivity with spherical harmonics for Boltzmann transport

Abstract

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which uses P n and filtered P n 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 P n 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 highorder convergence where possible. Regular and goal-based error metrics are shown and both the adapted P n and adapted filtered P n methods show significant reductions in DOFs and runtime. The adapted filtered P n with our spatially dependent filter shows close to fixed iteration counts and up to high-order is even competitive with P 0 discretisations in problems with heavy advection.

A comparison of element agglomeration algorithms for unstructured geometric multigrid

“…On unstructured grids however, it can be difficult to perform optimal sweeps, particularly in parallel and that motivates our continuing work on investigating alternate smoothers. In this work we use a Jacobi-preconditioned GMRES(3) method as our smoother (as do [16,27,28]), with three iterations on each level as the pre and post smoother. These strong smoothers help make up for our simple operators.…”

Goal-based angular adaptivity for Boltzmann transport in the presence of ray-effects

“…The spatial discretisation we use is a sub-grid scale FEM [5,6,7,8,4,1], which is stable and can be considered low-memory in comparison with standard DG methods. The solution to (1) is written as ψ = φ + θ, where φ and θ are the solutions on the "coarse" and "fine" scales, respectively. We then represent the coarse scale with a continuous finite-element representation, with the fine scale using a discontinuous.…”

“…Computing a goal-based error metric often involves multiplying both a forward and adjoint solution (or forward/adjoint residuals) and if ray-effects are present in either the forward or adjoint solutions, then for the duct problem described the resulting error metric would be zero in those regions. This means that adaptivity will not be Boltzmann Transport Equation (BTE) Ω • ∇ r ψ(r, Ω) + Σ t ψ(r, Ω) − S (ψ(r, Ω)) = S e (r, Ω), (1) where ψ(r, Ω) is the angular flux in direction Ω, at spatial position r. The macroscopic total cross section is Σ t with interaction/source terms given by S (ψ(r, Ω)) and external sources, S e . We write the ψ(r, Ω) dependent source term as the typical angular scattering operator, namely…”

A low-rank method for two-dimensional time-dependent radiation transport calculations

“…An alternative technique employs a basis expansion using the natural basis for the sphere: spherical harmonics. The spherical harmonics or P N method uses a truncated expansion to approximate the angular dependence [4][5][6]. Low-order approximations are also used such as flux-limited diffusion [7,8], simplified P N [9][10][11] along with hybrid methods such as quasi-diffusion [12].…”