Cycle-Free Polytopal Mesh Sweeping for Boltzmann Transport

TL;DR

This work addresses the challenge of performing efficient, cycle-free mesh sweeps for transport equations on unstructured polytopal meshes. It proves that Voronoi tessellations yield omnidirectionally acyclic directed dual graphs, enabling a topological sort that supports sweep-based DG discretizations for the mono-directional transport and the Boltzmann Transport Equation (BTE). It introduces a Voronoi-Scheduler with $O(N\log N)$ time and $O(dN)$ space and demonstrates that permutation-based sweeping yields a lower triangular system, accelerating forward-substitution solves and enabling scalable parallelism. Numerical experiments on DG discretizations of the transport equation and the BTE, including complex reactor-core geometries, show robust convergence, favorable matrix structure, and improved efficiency without cycle-breaking.

Abstract

We introduce a novel property of bounded Voronoi tessellations that enables cycle-free mesh sweeping algorithms. We prove that a topological sort of the dual graph of any Voronoi tessellation is feasible in any flow direction and dimension, allowing straightforward application to discontinuous Galerkin (DG) discretisations of first-order hyperbolic partial differential equations and the Boltzmann Transport Equation (BTE) without requiring flux-cycle corrections. We also present an efficient algorithm to perform the topological sort on the dual mesh nodes, ensuring a valid sweep ordering. This result expands the applicability of DG methods for transport problems on polytopal meshes by providing a robust framework for scalable, parallelised solutions. To illustrate its effectiveness, we conduct a series of computational experiments showcasing a DG scheme for BTE, demonstrating both computational efficiency and adaptability to complex geometries.

Figures (10)

• Figure 1: An illustration of discrete ordinates discretisation of the unit sphere, highlighting the angular partitioning used for transport directions.
• Figure 2: An illustration of upwind and downwind elements for $\boldsymbol{\omega} \cdot \boldsymbol{\nu}_{1,2} > 0$, with $T_1$ upwind of $T_2$, as shown by the edge direction on the dual graph.
• Figure 3: Primal mesh elements, $\mathcal{T}$ (in black), with the associated directed dual mesh $(\mathcal{T}^, \mathcal{E}^)$, represented by black points and red arrows for a given direction $\boldsymbol{\omega}$.
• Figure 4: An example of a subdomain problem: Algorithm \ref{['alg:Scheduler']} sorts a subset of regions, $R$, from a full Voronoi decomposition, even when Voronoi centres lie outside $R$.
• Figure 5: CPU times for Algorithm \ref{['alg:Scheduler']}, showing performance in generating sweeping permutations for a Voronoi mesh with varying numbers of elements and discrete ordinates.

Theorems & Definitions (12)

• Definition 2.1: Polytopal Mesh
• Definition 2.2: Voronoi Tessellation
• Remark 2.3: Convexity of Voronoi elements
• Lemma 2.4: Error control
• Lemma 3.1: Error control
• Definition 4.1: Dual mesh
• Definition 4.2: Directed dual mesh
• Remark 4.3: Dual of a Voronoi mesh
• Theorem 4.4: Voronoi meshes are omnidirectionally acyclic
• proof