An Efficient Implementation of Edge-Based Discretization without Forming Dual Control Volumes

TL;DR

This paper introduces a simplified and faster method to compute edge-based discretization vectors without explicitly forming dual control volumes, facilitating higher-dimensional simulations.

Contribution

The paper presents an efficient algorithm to compute lumped directed-area vectors without forming dual volumes, improving simplicity and computational speed for complex grids.

Findings

Algorithm reduces computation time for directed-area vectors.

Implementation simplifies edge-based discretization process.

Potential to extend to four-dimensional space-time simulations.

Abstract

This paper shows that lumped directed-area vectors at edges and dual control volumes required to implement the edge-based discretization can be computed without explicitly defining the dual control volume around each node for triangular and tetrahedral grids. It is a simpler implementation because there is no need to form a dual control volume by connecting edge-midpoints, face centroids, and element centroids, and also reduces the time for computing lumped directed-area vectors for a given grid, especially for tetrahedral grids. The speed-up achieved by the proposed algorithm may not be large enough to greatly impact the overall simulation time, but the proposed algorithm is expected to serve as a major stepping stone towards extending the edge-based discretization to four dimensions and beyond (e.g., space-time simulations). Efficient algorithms for computing lumped directed-area vectors and dual volumes without forming dual volumes are presented, and their implementations are described and compared with traditional algorithms in terms of complexity as well as actual computing time for a given grid.