Angular adaptivity in P0 space and reduced tolerance solves for Boltzmann transport

Abstract

Previously we developed an adaptive method in angle, based on solving in Haar wavelet space with a matrix-free multigrid for Boltzmann transport problems. This method scalably mapped to the underlying P$^0$ space during every matrix-free matrix-vector product, however the multigrid method itself was not scalable in the streaming limit. To tackle this we recently built an iterative method based on using an ideal restriction multigrid with frozen GMRES polynomials (AIRG) for Boltzmann transport that showed scalable work with uniform P$^0$ angle in the streaming and scattering limits. This paper details the practical requirements of using this new iterative method with angular adaptivity. Hence we modify our angular adaptivity to occur directly in P$^0$ space, rather than the Haar space. We then develop a modified stabilisation term for our FEM method that results in scalable growth in the number of non-zeros in the streaming operator with P$^0$ adaptivity. We can therefore combine the use of this iterative method with P$^0$ angular adaptivity to solve problems in both the scattering and streaming limits, with close to fixed work and memory use. We also present a CF splitting for multigrid methods based on element agglomeration combined with angular adaptivity, that can produce a semi-coarsening in the streaming limit without access to the matrix entries. The equivalence between our adapted P$^0$ and Haar wavelet spaces also allows us to introduce a robust convergence test for our iterative method when using regular adaptivity. This allows the early termination of the solve in each adapt step, reducing the cost of producing an adapted angular discretisation.