# An efficient implementation of mass conserving characteristic-based schemes in 2D and 3D

## TL;DR

This paper introduces the ball-approximated characteristics (B-char) method, a computationally efficient algorithm for implementing mass-conserving characteristic-based schemes in 2D and 3D, using ball intersections instead of polytopes.

## Contribution

The paper presents a novel ball-based approximation approach for characteristic schemes that reduces computational complexity while maintaining mass conservation.

## Key findings

- Achieves similar accuracy to polytope-based methods
- Significantly reduces computational cost in 3D implementations
- Ensures local and global mass conservation through optimization

## Abstract

In this paper, we develop the ball-approximated characteristics (B-char) method, which is an algorithm for efficiently implementing characteristic-based schemes in 2D and 3D. Core to the implementation of numerical schemes is the evaluation of integrals, which in the context of characteristic-based schemes with piecewise constant approximations boils down to computing the intersections between two regions. In the literature, these regions are approximated by polytopes (polygons in 2D and polyhedra in 3D) and, due to this, the implementation in 3D is nontrivial. The main novelty in this paper is the approximation of the regions by balls, whose intersections are much cheaper to compute than those of polytopes. Of course, balls cannot fully tessellate a region, and hence some mass may be lost. We perform some adjustments, and also solve an optimisation problem, in order to yield a scheme that is both locally and globally mass conserving. This algorithm can achieve results that are similar to those obtained from an implementation which uses polytopal intersections, with a much cheaper computational cost.