Cycle-Free Polytopal Mesh Sweeping for Boltzmann Transport

TL;DR

This paper presents a new cycle-free mesh sweeping method for Voronoi tessellations, enabling efficient, scalable DG solutions for transport equations on complex geometries without flux-cycle corrections.

Contribution

It introduces a topological sort algorithm for dual Voronoi meshes, expanding DG method applicability to polytopal meshes in transport problems.

Findings

Efficient topological sort algorithm for dual meshes

Cycle-free sweeping applicable in any flow direction and dimension

Demonstrated computational efficiency on complex geometries

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.