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.

翻译：本文介绍了一种有界Voronoi剖分的新性质，该性质支持无环网格扫描算法。我们证明了在任何流动方向和维度下，对任意Voronoi剖分的对偶图进行拓扑排序都是可行的，这使得该算法可直接应用于一阶双曲型偏微分方程和玻尔兹曼输运方程的不连续伽辽金离散化，且无需进行通量环修正。我们还提出了一种在对偶网格节点上执行拓扑排序的高效算法，以确保获得有效的扫描顺序。该成果通过提供可扩展并行化求解的鲁棒性框架，拓展了不连续伽辽金方法在多面体网格输运问题中的适用性。为验证其有效性，我们开展了一系列计算实验，展示了用于玻尔兹曼输运方程的不连续伽辽金格式，同时证明了该方法在计算效率和复杂几何适应性方面的优势。