Edge-based discretizations on triangulations in $\mathbb{R}^d$, with special attention to four-dimensional space

TL;DR

This paper introduces a rigorous framework for median-dual regions in high-dimensional triangulations, enabling efficient node-centered edge-based discretizations for complex space-time problems in computational fluid dynamics.

Contribution

It provides the first rigorous definition of median-dual regions in any dimension, along with new methods for computing their hypervolumes and directed-hyperarea vectors, facilitating advanced discretization schemes.

Findings

Successful numerical experiments in 2D, 3D, and 4D demonstrate scheme effectiveness.

New algorithms improve geometric property computations of median-dual regions.

Analysis confirms computational efficiency and robustness of the proposed methods.

Abstract

Many time-dependent problems in the field of computational fluid dynamics can be solved using space-time methods. However, such methods can encounter issues with computational cost and robustness. In order to address these issues, 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. These methods are node-centered or node-based, as the primary components of the discretization are median-dual regions (polytopes) which surround the mesh nodes. These methods are extremely robust, as the median-dual regions will often maintain a positive volume and other good geometric properties, even in cases when some of the associated simplices have negative volumes, or other poor geometric properties. Unfortunately, it is not straightforward to construct median-dual regions or deduce their properties on triangulations for $d \geq 3$. In this work, we provide the first rigorous definition of median-dual regions on triangulations in any number of dimensions. In addition, 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 faces of a median-dual region in $\mathbb{R}^{4}$. These geometric properties are key for developing node-centered edge-based schemes in any number of dimensions. We conclude our work by analyzing the computational complexity of the edge-based schemes, and performing numerical experiments in two, three, and four dimensions. We successfully demonstrate their effectiveness by obtaining accurate solutions to several canonical problems.