AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport

We develop a reduction multigrid based on approximate ideal restriction (AIR) for use with asymmetric linear systems. We use fixed-order GMRES polynomials to approximate $A_\textrm{ff}^{-1}$ and we use these polynomials to build grid transfer operators and perform F-point smoothing. We can also apply a fixed sparsity to these polynomials to prevent fill-in. When applied in the streaming limit of the Boltzmann Transport Equation (BTE), with a P$^0$ angular discretisation and a low-memory spatial discretisation on unstructured grids, this "AIRG" multigrid used as a preconditioner to an outer GMRES iteration outperforms the lAIR implementation in hypre, with two to three times less work. AIRG is very close to scalable; we find either fixed work in the solve with slight growth in the setup, or slight growth in the solve with fixed work in the setup when using fixed sparsity. Using fixed sparsity we see less than 20% growth in the work of the solve with either 6 levels of spatial refinement or 3 levels of angular refinement. In problems with scattering AIRG performs as well as lAIR, but using the full matrix with scattering is not scalable. We then present an iterative method designed for use with scattering which uses the additive combination of two fixed-sparsity preconditioners applied to the angular flux; a single AIRG V-cycle on the streaming/removal operator and a DSA method with a CG FEM. We find with space or angle refinement our iterative method is very close to scalable with fixed memory use

Fokker-Planck-Based Acceleration for SN Equations with Highly Forward Peaked Scattering in Slab Geometry

Short mean free paths are characteristic of charged particles. High energy charged particles often have highly forward peaked scattering cross sections. Transport problems involving such charged particles are also highly optically thick. When problems simultaneously have forward peaked scattering and high optical thickness, their solution, using standard iterative methods, becomes very inefficient. In this dissertation, we explore Fokker-Planck-based acceleration for solving such problems

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

Coarse mesh and one-cell block inversion based diffusion synthetic acceleration

DSA (Diffusion Synthetic Acceleration) has been developed to accelerate the SN transport iteration. We have developed solution techniques for the diffusion equations of FLBLD (Fully Lumped Bilinear Discontinuous), SCB (Simple Corner Balance) and UCB (Upstream Corner Balance) modified 4-step DSA in x-y geometry. Our first multi-level method includes a block Gauss-Seidel iteration for the discontinuous diffusion equation, uses the continuous diffusion equation derived from the asymptotic analysis, and avoids void cell calculation. We implemented this multi-level procedure and performed model problem calculations. The results showed that the FLBLD, SCB and UCB modified 4-step DSA schemes with this multi-level technique are unconditionally stable and rapidly convergent. We suggested a simplified multi-level technique for FLBLD, SCB and UCB modified 4-step DSA. This new procedure does not include iterations on the diffusion calculation or the residual calculation. Fourier analysis results showed that this new procedure was as rapidly convergent as conventional modified 4-step DSA. We developed new DSA procedures coupled with I-CI (Cell Block Inversion) transport which can be easily parallelized. We showed that 1-CI based DSA schemes preceded by SI (Source Iteration) are efficient and rapidly convergent for LD (Linear Discontinuous) and LLD (Lumped Linear Discontinuous) in slab geometry and for BLD (Bilinear Discontinuous) and FLBLD in x-y geometry. For 1-CI based DSA without SI in slab geometry, the results showed that this procedure is very efficient and effective for all cases. We also showed that 1-CI based DSA in x-y geometry was not effective for thin mesh spacings, but is effective and rapidly convergent for intermediate and thick mesh spacings. We demonstrated that the diffusion equation discretized on a coarse mesh could be employed to accelerate the transport equation. Our results showed that coarse mesh DSA is unconditionally stable and is as rapidly convergent as fme mesh DSA in slab geometry. For x-y geometry our coarse mesh DSA is very effective for thin and intermediate mesh spacings independent of the scattering ratio, but is not effective for purely scattering problems and high aspect ratio zoning. However, if the scattering ratio is less than about 0.95, this procedure is very effective for all mesh spacing