标题： $\mathbb{R}^{4}$中三角剖分的中位数对偶区域的性质及其在高维空间中的扩展 显示英文标题

标题： Properties of median-dual regions on triangulations in $\mathbb{R}^{4}$ with extensions to higher dimensions

摘要： 在计算流体力学领域中的许多时间依赖性问题可以在四维时空设置中求解。 然而，使用现代高阶数值方法求解这些问题在计算上是昂贵的。 为了应对这一问题，目前正在开发高效、基于节点的边方案。 在这些方案中，基于初始三角剖分构建空间-时间域的中位对偶划分。 不幸的是，在 $d \geq 4$的三角剖分上构造中位对偶区域或推导其性质并不直接。 在本工作中，我们提供了任何维度的三角剖分上中位对偶区域的第一个严格定义。 此外，我们提出了计算这些对偶区域几何属性的第一种方法。 我们引入了一种新的方法来计算 $\mathbb{R}^d$中中位对偶区域的超体积。 此外，我们提供了一种新的方法来计算 $\mathbb{R}^{4}$中中位对偶区域面的定向超面积向量。 这些几何属性对于在更高维度中促进基于节点的边方案的构建至关重要。 显示英文摘要

摘要： Many time-dependent problems in the field of computational fluid dynamics can be solved in a four-dimensional space-time setting. However, such problems are computationally expensive to solve using modern high-order numerical methods. In order to address this issue, efficient, node-centered edge-based schemes are currently being developed. In these schemes, a median-dual tessellation of the space-time domain is constructed based on an initial triangulation. Unfortunately, it is not straightforward to construct median-dual regions or deduce their properties on triangulations for $d \geq 4$. In this work, we provide the first rigorous definition of median-dual regions on triangulations in any number of dimensions. In addition, we present the first methods for calculating the geometric properties of these dual regions. We introduce a new method for computing the hypervolume of a median-dual region in $\mathbb{R}^d$. Furthermore, we provide a new approach for computing the directed-hyperarea vectors for facets of a median-dual region in $\mathbb{R}^{4}$. These geometric properties are key for facilitating the construction of node-centered edge-based schemes in higher dimensions.