Scalable angular adaptivity for Boltzmann transport

Abstract

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which for the first time shows evidence of 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.

Angular adaptivity with spherical harmonics for Boltzmann transport

“…Only a handful of authors have investigated using adaptivity with spherical harmonics in Boltzmann transport applications. We do not discuss other adapted angular discretisations, please see [4,8] for a detailed overview. [19,20] first used different P n expansion orders across space in a 1D test problem, where the expansion order was determined a priori.…”

“…Naturally the aforementioned authors have not investigated the use of P n expansions in voids given the conditioning problems of high order expansions, but unfortunately, even in the smooth problems presented, many of the authors do not show the runtimes of their adaptive methods. Our previous work [8] focused heavily on the scalability of an angular adaptivity scheme based on Haar wavelets. This work follows on from [8] and as such we discuss the scalability and runtime/memory consumption of our adaptive spherical harmonics scheme.…”

“…Performing adaptivity in the angular domain of the BTE is becoming more popular, as high angular resolution is required in many problems to resolve the streaming terms (i.e., when a particle propogates a long distance without interaction). Previously, in the AMCG we have investigated using several different angular discretisations with adaptivity across a range of different Boltzmann transport applications [1,2,3,4,5,6,7,8]. Recently [8] we showed scalable angular adaptivity implemented matrix-free with Haar wavelets that demonstrated O(n) scaling in both runtime and memory usage on some test problems.…”

“…Previously, in the AMCG we have investigated using several different angular discretisations with adaptivity across a range of different Boltzmann transport applications [1,2,3,4,5,6,7,8]. Recently [8] we showed scalable angular adaptivity implemented matrix-free with Haar wavelets that demonstrated O(n) scaling in both runtime and memory usage on some test problems. This Haar wavelet discretisation in angle is equivalent to a P 0 FEM discretisation on the sphere, and like all non-rotationally invariant angular discretisations produces ray-effects in the scalar flux.…”

“…We then turn to the development of a spatially dependent filter strength for our FP n method. Previously we introduced a sub-grid scale FEM discretisation [15,16,17,18,8] that uses less memory than equivalent DG schemes. This scheme explicitly separates our solution into "fine" and "coarse" scales, with the fine scale representing the amount of stabilisation that is applied.…”